MATHEMATICAL PAPEES.
ILonDon: 0. J. CLAY AND SONS,
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AVE MARIA LANE.
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THE COLLECTED
MATHEMATICAL PAPERS
OF
AETHUE CAYLEY, Sc.D., F.E.S.,
LATE SADLEEIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE.
VOL. XI.
CAMBRIDGE :
AT THE UNIVERSITY PRESS.
1896
[All Rights reserved.]
CAMBRIDGE :
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.
ADVEETISEMENT.
rilHE present volume contains 93 papers numbered 706 to 798, published,
with the exception of one series, for the most part in the years 1878
to 1883. This series is constituted by the articles which Professor Cayley
wrote for the Encyclopaedia Britannica between the years 1878 and 1888 ;
it seemed desirable to place these together in the same volume, in spite
of the departure from the chronological arrangement which governs the
sequence of the papers in the volumes generally. The Syndics of the
University Press desire to acknowledge their obligation to Messrs Adam
and Charles Black, Publishers of the ninth Edition of the Encyclopedia
Britannica, for their courteous consent in allowing these articles to be
included in the Collected Mathematical Papers. Exact references to the
volumes, from which the articles are extracted, will be found in the Table
of Contents.
The frontispiece to the present volume is a reproduction by Mr
A. G. Dew-Smith, of Trinity College, of a photograph of Professor Cayley
which he made in the year 1885. The Syndics of the Press desire to
acknowledge their obligation to Mr Dew-Smith.
ADVERTISEMENT.
The Table for the eleven volumes is
Vol. I. Numbers 1 to 100,
„ II. „ 101 „ 158,
„ III. „ 159 „ 222,
„ IV. „ 223 „ 299,
„ V. „ 300 „ 383,
„ VI. „ 384 „ 416,
„ VII. ;, 417 „ 485,
„ VIII. „ 486 „ 555,
„ IX. „ 556 „ 629,
„ X. „ 630 „ 705,
„ XL „ 706 „ 798.
A. K FORSYTH.
21 November, 1896.
Vll
CONTENTS.
[An Asterisk means that the paper is not printed in full.]
PAGE
706. On the distribution of electricity on two spherical surfaces . 1
Phil. Mag., Ser. 5, t. v. (1878), pp. 54—60
707. On the colouring of maps . . . . . . . . 7
Geogr. Soc. Proc., t. i. (1879), pp. 259—261
708. Note sur la theorie des courbes de I'espace .... 9
Assoc. Fran9., Compt. Rend., t. ix. (1880), pp. 135—139
709. On the number of constants in the equation of the surface
0 . ...... . . . 14
Tidsskrift for Mathematik, Ser. 4, t. iv. (1880), pp. 145—148
710. On a differential equation ..... ... 17
Collectanea Mathematica, in memoriam Dominici Chelini, (Milan,
Hoepli, 1881), pp. 17—26
711. On a diagram connected with the transformation of elliptic
functions .......... 26
British Association Report, 1881, p. 534
712. A partial differential equation connected with the simplest case
of Abel's theorem ...... 27
British Association Report, 1881, pp. 534, 535
713. Addition to Mr. Rowes "Memoir on Abel's theorem" . 29
Phil. Trans., t. CLXXII. (1881), pp. 751—758
CONTENTS.
PAGE
• 37
714. Various notes ...-••
Messenger of Mathematics, t. vn. (1878), pp. 69: 115: 124: 125
715. Note on a system of algebraical equations
Messenger of Mathematics, t. vn. (1878), pp. 17, 18
716. An illustration of the theory of the ^-functions . 41
Messenger of Mathematics, t. vn. (1878), pp. 27—32
717. On the triple theta-functions . . ••• ^
Messenger of Mathematics, t. vn. (1878), pp. 48—50
718. Addition to Mr. Geneses paper " On the theory of envelopes" . 50
Messenger of Mathematics, t. vn. (1878), pp. 62, 63
719. Suggestion of a mechanical integrator for the calculation of
(Xdx+Ydy) along an arbitrary path . . 52
Messenger of Mathematics, t. vn. (1878), pp. 92—95; British Asso
ciation Report, 1877, pp. 18 — 20
720. Note on Arbogast's method of derivations . 55
Messenger of Mathematics, t. vn. (1878), p. 158
721. Formula involving the seventh roots of unity . .56
Messenger of Mathematics, t. vn. (1878), pp. 177—182
722. A problem in partitions ..... . . 61
Messenger of Mathematics, t. vn. (1878), pp. 187, 188
723. Various notes .....-• .63
Messenger of Mathematics, t. vin. (1879), pp. 45, 46: 126: 127
724. On the deformation of the model of a hyperboloid . . . 66
Messenger of Mathematics, t. vin. (1879), pp. 51, 52
(i 'it* r/yy
725. New formula for the integration of 7^ + Tv~® ' ' ' ^
\l •**• v •*
Messenger of Mathematics, t. vin. (1879), pp. 60 — 62
726. A formula by Gauss for the calculation of log 2 and certain
other logarithms . . . . . • • 70
Messenger of Mathematics, t. vm. (1879), pp. 125, 126
CONTENTS. ix
PAGE
727. Equation of the wave-surface in elliptic coordinates . . . 71
Messenger of Mathematics, t. vm. (1879), pp. 190, 191
728. A theorem in elliptic functions . . . . . .' . 73
Proc. Lond. Math. Soc., t. x. (1879), pp. 43—48
729. On a theorem relating to conformable figures . . . . 78
Proc. Lond. Math. Soc., t. x. (1879), pp. 143—146
730. [Addition to Mr. Spottiswoode's paper " On the twenty-one
coordinates of a conic in space "] ..... 82
Proc. Lond. Math. Soc., t. x. (1879), pp. 194—196
731. On the binomial equation xp — 1=0; trisection and quarti-
section ........... 84
Proc. Lond. Math. Soc., t. xi. (1880), pp. 4—17
732. A theorem in spherical trigonometry . . . . . 97
Proc. Lond. Math. Soc., t. XL (1880), pp. 48—50
733. On a formula of elimination ... . . . 100
Proc. Lond. Math. Soc., t. xi. (1880), pp. 139—141
734. On the kinematics of a plane . . . . . . . 103
Quart. Math. Journ., t. xvi. (1879), pp. 1—8
735. Note on the theory of apsidal surfaces . . . . . ill
Quart. Math. Journ., t. xvi. (1879), pp. 109—112
736. Application of the Newton- Fourier method to an imaginary
root of an equation . . . . . . . . 114
Quart. Math. Journ., t. xvi. (1879), pp. 179—185
737. On a covariant formula . . . . . . . . 122
Quart. Math. Journ., t. xvi. (1879), pp. 224—226
738. Note on a hypergeometric series . . . . . . 125
Quart. Math. Journ., t. xvi. (1879), pp. 268—270
739. Note on the octahedron function . . . . . . 128
Quart. Math. Journ., t. xvi. (1879), pp. 280, 281
C. XI. b
CONTENTS.
PAGE
740. On certain algebraical identities . . . ... 130
Quart. Math. Journ., t. xvi. (1879), pp. 281, 282
741. On a theorem of Abel's relating to a quintic equation . . 132
Camb. Phil. Soc. Proc., t. in. (1880), pp. 155—159
742. On the transformation of coordinates . . . . . 136
Camb. Phil. Soc. Proc., t. in. (1880), pp. 178—184
743. On the Newton- Fourier problem . . . • • • 143
Camb. Phil. Soc. Proc., t. in. (1880), pp. 231, 232
744. Table of AmOn •*- II (w) up to m = n = 20 . . . . _ 144
Camb. Phil. Trans., t. xin. (1883), pp. 1—4
745. On the Schwarzian derivative, and the polyhedral functions . 148
Camb. Phil. Trans., t. xin. (1883), pp. 5—68
*746. Higher Plane Curves .-. 217
Salmon's Higher Plane Curves, (3rd ed., 1879), Preface
747. Note on the degenerate forms of curves . . . . . 218
Salmon's Higher Plane Curves, (3rd ed., 1879), pp. 383—385
748. On the bitangents of a quartic . . . . . . . 221
Salmon's Higher Plane Curves, (3rd ed., 1879), pp. 387—389
"''749. Solid Geometry 224
Salmon's Treatise on the analytic geometry of three dimensions,
(3rd ed., 1874), Preface
750. On the theory of reciprocal surfaces . . . . . 225
Salmon's Treatise on the analytic geometry of three dimensions,
(3rd ed., 1874), pp. 539—550
751. Note on Riemann's paper " Versucli einer allgemeinen Auffass-
ung der Integration und Differentiation," Werke, pp. 331 —
344 . . . 235
Mathematische Annalen, t. xvi. (1880), pp. 81, 82
752. On the finite groups of linear transformations of a variable ;
with a correction ......... 237
Mathematische Annalen, t. xvi. (1880), pp. 260—263; 439, 440
753. On a theorem relating to the multiple theta-functions . . 242
Mathematische Annalen, t. xvn. (1880), pp. 115 — 122
CONTENTS. xi
PAGE
754. On the connection of certain formulce in elliptic functions . 250
Messenger of Mathematics, t. ix. (1880), pp. 23 — 25
755. On the matrix (a, b ), and in connection therewith the function
I
U 4
ax + b
c^+d • • . 252
Messenger of Mathematics, t. ix. (1880), pp. 104 — 109
756. A geometrical construction relating to imaginary quantities . 258
Messenger of Mathematics, t. x. (1881), pp. 1 — 3
757. On a Smith's Prize question, relating to potentials . . . 261
Messenger of Mathematics, t. XL (1882), pp. 15 — 18
758. Solution of a Senate-House problem . . . . . 265
Messenger of Mathematics, t. XL (1882), pp. 23 — 25
759. Illustration of a theorem in the theory of equations . . 268
Messenger of Mathematics, t. xi. (1882), pp. Ill — 113
r ax
760. Reduction of i , _ .T-J to elliptic integrals . . . . 270
Messenger of Mathematics, t. xi. (1882), pp. 142, 143
761. On the theorem of the finite number of the covariants of a
binary quantic . . . . . . . . . 272
Quart. Math. Journ., t. xvn. (1881), pp. 137—147
762. On Schubert's method for the contacts of a line with a surface 281
Quart. Math. Journ., t. xvn. (1881), pp. 244—258
763. On the theorems of the 2, 4, 8, and 16 squares . . . 294
Quart. Math. Journ., t. xvn. (1881), pp. 258 — 276
764. The binomial equation xp — 1=0; quinquisection . . . 314
Proc. Lond. Math. Soc., t. xn. (1881), pp. 15, 16
765. On the flexure and equilibrium of a skew surface . . . 317
Proc. Lond. Math. Soc., t. xn. (1881), pp. 103—108
766. On the geodesic curvature of a curve on a surface. . . 323
Proc. Lond. Math. Soc., t. xn. (1881), pp. 110—117
b 2
CONTEXTS.
PAGE
767. On the Gaussian theory of surfaces .
Proc. Lond. Math. Soc., t xii. (1881), pp. 187—192
768. JVofe OH Landen's theorem 337
Proc, Lond. Math. Soc,, t xm. (1882), pp. 47, 48
769. Oi o formula relating to elliptic integrals of the third kind . 340
Proc, Lond. Math. Soc., t xm. (1882), pp. 175, 176
770. On the 34 concomitants of the ternary cubic . 342
American Journal of Mathematics, t, IT. (1881), pp. 1 — 15
771. Spi'dnien of a literal table for binary qualities, otherwise a
partition table . . . 357
American Journal of Mathematics, t. iv. (1881), pp. 248 — 255
772. On the analytical forms called trees 365
American Journal of Mathematics, t. iv. (1881), pp. 266 — 268
773. On the S-square imaainaries .
American Journal of Mathematics, t. IT. (1881), pp. 293—296
774. Tables for the binary sextic . £•-
Americau Journal of Mathematics, t. iv. (188 IX pp. 379 — o>4-
775. Tables of corariante of the binary sejctic. 377
Written in 1894 : now first published.
776. On the Jacobian sejctic equation ... 3M)
Quart. Math. Journ., t mil. (1882), pp. 52—65
777. A solvable case of the quintic equation , . . 402
Quart, Math. Journ., t xvm. (1882), pp. 154—157
778. [Addition to Mr. Hudson's paper "On equal- roots of equations"] 405
Quart. Math. Joxirn., t. mil. (1882), pp. 226— lit
[Xote on Mr. Jefrcry's paper " On certain quartic curves,
ichich haw a cusp at infinity 9 whereat the line at injinity
is a tangent"] . ... .. , . . 408
Proc, Lond. Math. Soc., t. XIY. (1883X P* &
CONTEXTS. xiii
PiOK
780. [Addition to J/r. Hammond's paper " Xote on an exceptional
case in which the fundamental postulate of Professor
Sylvesters theoiy of tamisage fails " ' | . . 409
Proc. Loud. Math. Soc., t. xiv. (1883), pp. 88—91
781. On the automorphic transformation of the binary cubic
fimction ..'.".'.*. . . . . 411
Proa Lond. Math. Soc., t xiv. (1883), pp. 103 — 108
782. On Monge's " Memoire sur la theorie des deblais et des
remblais" . . .» . . . . . 4^7
Proc. Lond. Math. Soc., t. xiv. (1883), pp. 139 — 142
783. On Jfr. Wilkinson's rectangular transformation . . 421
Proc. Lond. Math. Soc., t xiv. (1883), pp, 222—229
784. Presidential Address to the British Association, Southport,
September 1888 ........ 429
British Association Report, 1883, pp. 3 — 37
785. Cun-e ......... 460
Encyclopaedia Britannica, 9th ed., t, vi. (1878), pp. 716 — 728
786. Equation . . . . . . . . 490
Encyclopaedia Brit&nnica, 9th ed, t viu. (1878), pp. 497 — 509
787. Function 522
Encyclopjedia Britannica, 9th ed., t. ix. (1879), pp, 818—824
788. Galois . . . . 543
Encyclopaedia Britannica, 9th ed., t. x. (1879), p. 48
789. Grauss 544
Encyclopaedia Britannica, 9th ed., t. x. (1879), p. 116
790. Geometry (analytical) 546
Encyclopaedia Britannica, 9th ed., t. x. (1879), pp. 408—420
791. Landen 5S.S
Encyclopaedia Britannica, 9th ed., t xiv. (1882), p. 271
792. Locus 585
Encyclopaedia Britannica, 9th ed., t. xiv. (1882), pp. 764, 765
CONTENTS.
PAGE
793. Monge . . . ; . . . • • 586
Encyclopaedia Britannica, 9th ed., t. xvi. (1883), pp. 738, 739
794. Numbers (partition of) . . . . • • • • 589
Encyclopaedia Britannica, 9th ed., t. xvn. (1884), p. 614
795. Numbers (theory of) ....... • 592
Encyclopaedia Britannica, 9th ed., t. xvn. (1884), pp. 614—624
796. Series ......... -. 617
Encyclopaedia Britannica, 9th ed., t. xxi. (1886), pp. 677—682
797. Surface ....... " .- . . . : 628
Encyclopaedia Britannica, 9th ed., t. xxn. (1887), pp. 668 — 672
798. Wallis (John] .......... 640
Encyclopaedia Britannica, 9th ed., t. xxiv. (1888), pp. 331, 332
Portrait ......... To face Title.
XV
CLASSIFICATION.
ANALYSIS.
Calculation of log 2 ; 726.
Series, 796.
Prime roots of unity, 721.
8-square imaginaries, 773.
Squares, theorems of 2, 4, 8, 16; 763.
Difference-table for AmO" -4- H (m) ; 744.
Equations, theory of, 736, 741, 743, 759, 776, 777, 778, 786.
Numbers, theory of, 731, 764, 795.
Partitions, 722, 771, 794.
Trees, 772.
Matrices, 755.
Elimination, 733.
Transformation of cubic function, 781.
Covariantive forms and tables, 737, 761, 770, 774, 775, 780.
Fractional differentiation, 751.
Mechanical integrator, 719.
Differential Equations, 725.
Schwarzian derivative, 745.
Hypergeometric series, 710, 738.
Finite groups, 752.
Polyhedral functions, 739, 745.
Elliptic functions, 728, 740, 754, 760, 768, 769.
Transformation of elliptic functions, 711.
Abel's theorem, 712, 713.
Theta Functions, 716, 717, 753.
Function, 787.
Conformal representation, 729.
XVI CLASSIFICATION.
GEOMETRY.
Analytical geometry in general, 790, 792, 797.
Plane Curves, 746*, 785.
Degenerate forms of curves, 747.
Quartic Curves, 748, 779.
Deblais et remblais, 782.
Tortuous curves, 708, 785.
Geodesic curvature, 766.
Theory of surfaces, general, 709, 749*, 767.
Transformation of coordinates, 742, 783.
Reciprocal surfaces, 750.
Wave-surface, 727.
Apsidal surfaces, 735.
Deformation and flexure of surfaces, 724, 765.
Hypergeometry, 730.
Schubert's numerative geometry, 762.
VARIOUS SUBJECTS.
Spherical Trigonometry, 732.
Kinematics of a plane, 734.
Maps, colouring of, 707.
Electricity, distribution of, on spherical surfaces, 706.
Potential, 757.
Presidential Address to the British Association, 784.
Biographical articles ; Galois, 788.
Gauss, 789.
Landen, 791.
Monge, 793.
Wallis (John), 798.
MISCELLANEOUS
714, 715, 718, 720, 723, 756, 758.
706]
706.
ON THE DISTRIBUTION OF ELECTRICITY ON TWO SPHERICAL
SURFACES.
[From the Philosophical Magazine, vol. v. (1878), pp. 54 — 60.]
IN the two memoirs " Sur la distribution de I'e'lectricite a la surface des corps
conducteurs," Mem. de I'Inst. 1811, Poisson considers the question of the distribution
of electricity upon two spheres : viz. if the radii be a, b, and the distance of the
centres be c (where c > a + b, the spheres being exterior to each other), and the
potentials within the two spheres respectively have the constant values h and g, then —
for Poisson's /(-) writing <£(#), and for his F [j] writing <£(;») — the question depends
on the solution of the functional equations
c — x
(x) = g,
c — x ' \c — xi
where of course the x of either equation may be replaced by a different variable.
It is proper to consider the meaning of these equations : for a point on the axis,
at the distance x from the centre of the first sphere, or say from the point A, the
a2 (o?\
potential of the electricity on this spherical surface is a$x or — <p f — ) , according as
the point is interior or exterior; and, similarly, if x now denote the distance from
the centre of the second sphere (or, say, from the point B), then the potential of
TO /7i2\
the electricity on this spherical surface is b<3?x or -<£( — ), according as the point is
x \x J
interior or exterior; </>(#) is thus the same function of (ss, a, b) that <£(#) is of
C. XI. 1
2 ON THE DISTRIBUTION OF ELECTRICITY [706
(x, b, a). Hence, first, for a point interior to the sphere A, if a: denote the distance
from A, and therefore c — x the distance of the same point from B, the potential of
the point in question is
62 , / 62 \
= ad>x + - 3> -- ;
c — x \c-xj
and, secondly, for a point interior to the sphere B, if x denote the distance from B
and therefore c—x the distance of the same point from A, the potential of the
point is
c—x \c-x
The two equations thus express that the potentials of a point interior to A and of
a point interior to B are =h and g respectively.
It is to be added that the potential of an exterior point, distances from the points
A and B = x and c — x respectively, is
b-
,
*
I-J'
c — x \c
and that, by the known properties of Legendre's coefficients, when the potential upon
an axial point is given, it is possible to pass at once to the expression for the potential
of a point not on the axis, and also to the expression for the electrical density at a
point on the two spherical surfaces respectively. The determination of the functions
(J>(x) and <&(x) gives thus the complete solution of the question.
I obtain Poisson's solution by a different process as follows: — Consider the two
functions
a2 (c — x) ax + b
T^jsZ: » = — T7[ » suppose,
and
b- (c — x) OLX + 8
-„ — ~— -, = — -%, suppose;
c2 — a2 — ex <yx+ o
and let the ?ith functions be
— 3^ and — ^
cnx + dn <ynx + bn
respectively.
Observing that the values of the coefficients are
(a, b ) = ( -a2, a-c ), and (a, 8 ) = ( -b2, b*c ),
c, d — c , c2 — b- 7, 8 — c ,
so that we have
and consequently that the two equations
(\+l)2_(a
X ad -be'
706] ON TWO SPHERICAL SURFACES.
are in fact one and the same equation
(X + l )2 (c2-a2-62)2
for the determination of X, then (by a theorem which [686, 687] I have recently
obtained) we have the following equations for the coefficients
( »„, bn ), ( On, 0n )
of the nth functions ; viz. these are
and similarly
„ „
b) + (X" - X) (- d* + b)},
ca?-a)};
- a)}.
Observe that these equations give, as they ought to do,
a0# + bo = #, c0# + d0=l, a1cc+ bj
and similarly
= yx + 8.
a?
Substituting in the first two equations - - in place of x, and in the second two
~~
0 ~~ (X)
equations - - in place of x, we obtain the following results which will be useful : —
C "" — 3C
2 + bn (c — #) = a2 (ynx + Sn),
cna2 + dn (c - a?) = (a
n (c - a?) =
the las.t two of which are obtained from the first two by a mere interchange of
letters ; it will therefore be sufficient to prove the first and second equations.
For the first equation we have
ana2 + bn (c - a?) =
A, —
{(\n+1 - 1) [aa2 + b (c - a?)] + (X" - X) [- da2 + b (c - a)]},
1—2
4 ON THE DISTRIBUTION OF ELECTRICITY [706
where the term in { } is
= (X'l+1 - 1) [- a4 4- a2c (c - x)] + (\n - X) [a2 (62 - c2) 4- a2c (c - x)] ;
viz. this is
= a2 {(\n+1 - 1) (c2 - a- - ex) + (Xn - X) (62 - ca?)} ;
or it is
= a2 {(Xn+1 - 1) (yx +8) + (\n - X) (yx - a)},
whence the relation in question.
The proof of the second equation is a little more complicated. We have
1 /a 4- d\n-1
cna2 4- d» (c - x) = f - — ^ J {(Xn+1 - 1) [ca2 4- d (c - a?)] + (Xn - X) [ca2 - a (c - a;)]},
where the term in { } is
= (Xn+I - 1) [- ca2 + (c2 - 62) (c - x)] + (Xn - X) [- ca2 + a2 (c - x)].
Comparing this with
" /3) + (X»« - X) (-
where the term in { } is
= (X«+2 - 1) [62 (c - x)] + (\n+1 - X) [- c (c2 - a2 - 62) + (c2 - a2) (c - a?)],
it is to be observed that the quotient of the two terms in { } is in fact a constant :
this is most easily verified as follows. Dividing the first of them by the second, we
have a quotient which when x = c is
- 1) (- ca2) + (\n - X) (-ca2) a2(X"+1 - 1 +X»-X) a2(X+l)
(Xn+1 - X) {- c (c2 - a2 - 62)} (\n+l - X) (c2 - a2 - 62) ' ~ (c2 - a2 - 62) X '
and when # = 0 is
(X»+1-l)c(c2-a2-62)
2-62) = (XM+1-l)(c2-a2-62) _ c- - a2 - 62
- X) 62c ' ~(Xn+2-l+Xw+1-X)62' = 62(X + 1) :
these two values are equal by virtue of the equation which defines X ; and hence the
quotient of the two linear functions having equal values for x = c and x = 0, has
£2 _ fj2 _ J2
always the same value ; say it is = ,2. . Hence, observing that a + d = a + 8,
= c2-a2-62, the quotient, cna~ + dn (c - cc) divided by an+1# + /3w+1, is
X + l c2 - a2 - 62 _].
-a2-62' 62(X + D ' ~6"2
or we have the required equation
n (c - x) =
706] ON TWO SPHERICAL SURFACES. 5
Considering now the functional equations, suppose for the moment that g is = 0 ;
the two equations may be satisfied by assuming
klr±
— r
c0x + d0 das +
We in fact, from the foregoing relations, at once obtain
a2 a3 , [ w &)2 ) o?¥L
— x c—x
62
c—x c — x
To satisfy the first equation we must have M = aL ; viz. this being so, the equation
becomes
- -- = — =-
c — x \c-xj c0x + d0
or, since C0# + d0=l, the equation will be satisfied if only aL = l, whence also M=\.
9 7 2 f
And the second equation will be satisfied if only - — = bM ; viz. substituting for L, M
their value, we find &> = ab.
Supposing, in like manner, that h = 0, g retaining its proper value, we find a like
solution for the two equations ; and by simply adding the solutions thus obtained, we
have a solution of the original two equations
c — x \c — X;
o? . f a? \ ,, . ,
— <z> I + 6<P (x) = q :
' — x r \c-xj
viz. the solution is
JL(\_ h | 1 ab } _ f ab (ab)2
' * ^ rt ] si M i s3 rt««i>-l " t/lrt>-vilV* «h HM _l_ n
We have a general solution containing an arbitrary constant P by adding to the
foregoing values for </># a term
Pb(a-b)
Va2 (c - 0) - a? (c2 - 62 - c#) '
and for <!># a term
= _____ Pa (b - a) ___
V6'2 (c — x) — x (c2 — a2 — ex)
6 THE DISTRIBUTION OF ELECTRICITY ON TWO SPHERICAL SURFACES. [706
as may be easily verified if we observe that the function
a2 (c — x) — x (c2 — b" — ex),
writing therein for x, becomes
c-oc
{b2(c-x)-x(c*-a2-cx)}:
(c-x)*
and similarly that
/j2 //» __ sr>\ «_ /r» ( V*- __ /^2 ^_ /*/y»^
t/ \ L* **/ / tv i v vv v«*/ /j
J2
writing therein for x, becomes
c - x
= rr~, ^ la* (c ~ *) ~ x (G' ~ b" ~ cx)}-
More generally, the terms to be added are for tj>x a term as above, where P denotes
Q% iff _ ,£\
a function of x which remains unaltered when x is changed into — — , and for
c- — b2 — ex
<£>x a term as above with P' instead of P, where P' denotes what P becomes when
x is changed into - . But these additional terms vanish for the electrical problem,
C ~~ $/
and the correct values of (f>x, <&x are the particular values given above.
It is to be remarked that the function
a2 (c — x)
c- - b2 - ex
is =
c — x
viz. considering x as the distance of a point X from A, then taking the image of X
in regard to the sphere B, and again the image of this image in regard to the
sphere A, the function in question is the distance of this second image from A. And
similarly the function
62 (c - x) b-
is = — ;
ex
c —
viz. considering here x as the distance of the point X from B, then taking the image
of X in regard to the sphere A, and again the image of this image in regard to
the sphere B, the function in question is the distance of this second image from B.
It thus appears that Poisson's solution depends upon the successive images of X in
regard to the spheres B and A alternately, and also on the successive images of X
in regard to the spheres A and B alternately. This method of images is in fact
employed in Sir W. Thomson's paper " On the Mutual Attraction or Repulsion between
two Electrified Spherical Conductors," Phil. Mag., April and August, 1853.
707]
707.
ON THE COLOURING OF MAPS.
[From the Proceedings of the Royal Geographical Society, vol. I., no. 4 (1879),
pp. 259—261.]
THE theorem that four colours are sufficient for any map, is mentioned somewhere
by the late Professor De Morgan, who refers to it as a theorem known to map-makers.
To state the theorem in a precise form, let the term "area" be understood to mean
a simply or multiply connected* area: and let two areas, if they touch along a line,
be said to be " attached " to each other ; but if they touch only at a point or points,
let them be said to be "appointed" to each other. For instance, if a circular area
be divided by radii into sectors, then each sector is attached to the two contiguous
sectors, but it is appointed to the several other sectors. The theorem then is, that
if an area be partitioned in any manner into areas, these can be, with four colours
only, coloured in such wise that in every case two attached areas have distinct
colours ; appointed areas may have the same colour. Detached areas may in a map
represent parts of the same country, but this relation is not in anywise attended
to: the colours of such detached areas will be the same, or different, as the theorem
may require.
It is easy to see that four colours are wanted; for instance, we have a circle
divided into three sectors, the whole circle forming an enclave in another area; then
we require three colours for the three sectors, and a fourth colour for the surrounding
area: if the circle were divided into four sectors, then for these two colours would
* An area is "connected" when every two points of the area can be joined by a continuous line lying
wholly within the area; the area within a non-intersecting closed curve, or say an area having a single
boundary, is "simply connected"; but if besides the exterior boundary there is one or more than one
interior boundary (that is, if there is within the exterior boundary one or more than one enclave not
belonging to the area), then the area is "multiply connected." The theorem extends to multiply connected
areas, but there is no real loss of generality in taking, and we may for convenience take the areas of the
theorem to be each of them a simply connected area.
s
ON THE COLOURING OF MAPS. [707
be sufficient, and taking a third colour for the surrounding area, three colours only
would be wanted; and so in general according as the number of sectors is even or
odd, three colours or four colours are wanted. And in any tolerably simple case it can
be seen that four colours are sufficient. But I have not succeeded in obtaining a
general proof: and it is worth while to explain wherein the difficulty consists.
Supposing a system of n areas coloured according to the theorem with four colours
only, if we add an (w+l)th area, it by no means follows that we can without
altering the original colouring colour this with one of the four colours. For instance,
if the original colouring be such that the four colours all present themselves in the
exterior boundary of the n areas, and if the new area be an area enclosing the n
areas, then there is not any one of the four colours available for the new area.
The theorem, if it is true at all, is true under more stringent conditions. For
instance, if in any case the figure includes four or more areas meeting in a point
(such as the sectors of a circle), then if (introducing a new area) we place at the
point a small circular area, cut out from and attaching itself to each of the original
sectorial areas, it must according to the theorem be possible with four colours only
to colour the new figure; and this implies that it must be possible to colour the
original figure so that only three colours (or it may be two) are used for the
sectorial areas. And in precisely the same way (the theorem is in fact really the
same) it must be possible to colour the original figure in such wise that only
three colours (or it may be two) present themselves in the exterior boundary of the
figure.
But now suppose that the theorem under these more stringent conditions is true
for n areas: say that it is possible with four colours only, to colour the n areas
in such wise that not more than three colours present themselves in the external
boundary : then it might be easy to prove that the n + l areas could be coloured
with four colours only: but this would be insufficient for the purpose of a general
proof; it would be necessary to show further that the n + l areas could be with the
four colours only coloured in accordance with the foregoing boundary condition; for
without this we cannot from the case of the n + l areas pass to the next case of
n + 2 areas. And so in general, whatever more stringent conditions we import into
the theorem as regards the n areas, it is necessary to show not only that the n + l
areas can be coloured with four colours only, but that they can be coloured in
accordance with the more stringent conditions. As already mentioned, I have failed
to obtain a proof.
708]
708.
NOTE SUR LA THEORIE DES COURSES DE L'ESPACE.
[From the Compte Rendu de I' Association Frangaise pour I'Avancement des Sciences (1880),
pp. 135—139.]
EN considerant dans 1'espace ime courbe d'espece donne'e, determinee au moyen
d'un nombre suffisant de points, la courbe n'est pas determinee uniquement; mais on
a par les points un certain nombre de telles courbes. Par exemple, la courbe unicursale
d'ordre^ 2p depend, comme on voit sans peine, de 8p constantes et sera ainsi
determinee par 4p points (le cas p = l est urie exception): on ne connait pas, je
pense, le nombre des courbes par les 4>p points; mais pour le cas particulier p = 2
(c'est-a-dire pour une courbe quartique de seconde espece, ou autrement dit, une
courbe excubo-quartique) ce nombre est =4: theoreme de'montre' par moi depuis
longtemps par des considerations geome'triques. (Voir Salmon, Geometry of three
dimensions, 3e ed. 1874, p. 319.) Ce n'est que dernierement que j'ai conside're la
question analytique, de trouver les equations d'une courbe excubo-quartique qui passe
par 8 points donne's ; et meme j'ai pris pour les 8 points une disposition qui n'est
pas tout a fait generale : 1'investigation elle-meme, et la forme du resultat, m'ont
paru assez interessantes pour que je les soumette a 1'Associatiou.
En considerant sur une courbe excubo-quartique 4 points donne's, le plan passant
par 3 quelconques de ces points rencontre la courbe dans un seul point ; et 1'on
obtient ainsi encore 4 points sur la courbe: voila mon systeme de 8 points donnes,
savoir en partant de 4 points quelconques, je prends un point quelconque dans chacun
des plans qui passent par 3 de ces points, et j'obtiens ainsi les autres 4 points. Et
par un tel systeme de 8 points, je cherche a faire passer une courbe de 1'espece dont
il s'agit.
En prenant cc=Q, y = Q, z = §, w = 0, pour les equations des plans du tetraedre
forme' par les 4 premiers points, les coordonndes de ces points seront (1, 0, 0, 0),
(0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1): et pour les coordonnees des 4 autres points,
je prends (0, &, zlt Wl), (x,, 0, *8, w2), («„ y3, (), w,\ (x., yti z4, 0).
c. xi.
10 NOTE SUR LA THEORIE DES COURBES DE L'ESPACE. [708
Les equations de la courbe sont x : y : z : w = P : Q : R : S, ou P, Q, R, S
sont des fonctions (*)(#, I)4 d'un parametre variable 0 ; il s'agit de faire passer une
telle courbe par les 8 points.
Je prends a, ft, 7, B, a, b, c, d pour les valeurs du parametre 0 qui correspondent
aux 8 points respectivement.
Pour que la courbe passe par les premiers 4 points, il faut et il suffit que les
Equations soient de la forme
. 0-a r>0-b nO-c ^0-d_
: w = A 7: : B „ — ,= : O .,
0=~a • "0--0 ' "0-y ' "0-8>
les conditions pour les autres 4 points seront alors
• 2/i : zi '• wi =
: 2/3
=
•
Ea~
b
ra
— c
a
-d
a —
ft '
a
-7 '
-B'
ib
— a
•
cb
— c
X
-d
-Ab
— a
Lb
-7
-B'
1°
— a
• 7? C ~
b
Dc
-d
c
— a.
c —
ft
c
-B'
d
— a
•Bd~
b
cd
— c
' d
— a.
' d-
ft
d
~7
Evidemment il y a deux Equations qui donnent la valeur de B : C, et qui servent
ainsi pour eliminer cette quantite. De cette maniere on obtient six equations que
j'ecris comme voici :
a — b.d — c a — .d — @
_
a — c . d — b ' a —7? .d —
_wty3 _a — d.c — b a — /3 . c — B
_ ZjWz a — c .b — d a — 8 .b —
Z.M! a — d.b — c' a — 7.6 — 8 '
z2x4 b — c .d — a b — a.d — 7
z x2 b — a.d — c' b — y.d— a
x2w3 b — a . c — d b — B . c — a.
x3w2 b— d.c — a'b — a.c — 8
^2/4 _c — a.d — b c — {3 .d — a.
^2/3 c — b.d — a' c — a.d — /3 '
II Z
savoir X, p., v, CT, K, p denotent ici les quantites donnees X = 4 , etc. Le nombre
des equations independantes est 5, car Ton a identiquement Xfwi&Kp = 1. Je remar-
que que 1'on peut faire sur le parametre 0 une transformation lineaire quelconque
(h0 + i) : (j0 + k), et introduire ainsi 3 constantes arbitrages ; on peut done prendre a
708] NOTE SUR LA THEORIE DES COURBES DE I/ESPACE. 11
volonte 3 valenrs du parametre 6, c'est-a-dire les valeurs de 3 quelconques des quantite's
a, /3, 7, 8, a, b, c, d ; et cela dtant les 5 equations donneront les valeurs des autres
5 quantites. Si au moyen des equations on elimine a, /3, 7, 8, on obtient entre
a, b, c, d une equation qui sera, comme on va voir, de 1'ordre 4 par rapport a
chacune de ces quantite's : en prenant comme donnees a, b, c il y aura done 4 valeurs
de d; et pour 1'une quelconque de ces valeurs, celles de a, /3, 7, 8 seront de'termine'es
uniquement : il y aura ainsi 4 courbes qui passent chacune par les 8 points ; ce qui
est le the'oreme dont il s'agit.
J'introduis, pour abreger, la notation
a — d, b — d, c — d, b — c, c— a, a — b,
= f, g; h, a, b, c:
on a done identiquement
a, b, c = g-h, h-f, f-g,
a + b + c = 0,
fa -f gb + he = 0.
Les Equations prennent ainsi la forme
he a — y.d — ft
A* — ~~ T~ 7^ -j " 9 6tC. .
gb a — p . d — 7
ou, en introduisant pour plus de commodite, les symboles
L, M, N, P, Q, R,
pour designer respectivement
gb% he fa he gb fa
"~ C~ ^> ~ ~~ -- ~ ~~ —~ -- -
he
les equations seront
C~ > ~ ~c~ ' -- U v> ~ ~c~ > —~, -- L-
he fa ^ gb fa he gb
T —
-
ii*-_a — /3 . c — 8
~a- 8.C-/3'
AT— a~ ^ • b ~ V
-a-y.b-S'
p = ^r7Trf^a'
n _b — 8 . c — a
r, _ c — /3 .d — a
c — a. rf — /?'
avec la relation identique LMNPQR — 1 ; il s'agit entre ces 5 equations d'eliminer
a, £, 7, S.
2—2
12 NOTE SUR LA THEORIE DES COURBES DE I/ESPACE. [708
J'e'cris ot = a — <f), les facteurs b — a, c — a, d — a de P, Q, R deviennent ainsi
respectivement — c + <f>, g + 4>, — f+</>; cela dtant, les valeurs de P, Q, R servent a
exprimer /3, 7, 8 en fonction de <£ : substituant ces valeurs de /3, 7, B dans celles
de L, M, N, on obtient sans peine
h f(c-<ft) + cP(-f+<fr)
gP b(-
M - — b (~
valeurs qui donnent, comme cela doit etre, LMNPQR = 1 : il faut entre ces equations
e'liminer ^>.
En retablissant \, p, v, CT, K, p au lieu de L, M, N, P, Q, R, ces equations
deviennent
a +
a _
~g A =
(evidemrnent i;r)%= 1), ou j'ecris £, ?;, ^ pour denoter les expressions -X-sr, etc., et ou
O
les valeurs des coefficients X, Y, etc., sont
X = fc (fa + whc), 7 = - f2a - ^hc2,
X2 = be (he + *gb), F2 = he2 - «gb-\
Les deux premieres equations donnent
ou, ce qui est la meme chose,
et Ton n'a qu'a substituer la valeur de ces coefficients.
On a
X,Y2 - X,Y, = fb (gb + pfa) (he2 - «gb2) - be (he + *gb) (- gb2 +
= fghb2c2 - fg2bV + f 2habc2p - f 2gab3/cp + ghb3c2 + g2b4c/e - f 2habc-/j - f 2gab2c*/j
= ghb2c2 (f + b) + g2b4 (- f + c) p - f2gab2 (b + c) Kp
= ghb2c2h + g2b4 (- g) p + f2gab2a*p
= gb2 (h2c2 - g
708] NOTE SUR LA THEORIE DBS COURBES DE I/ESPACE. 13
et de meme
Done
2 - X,Y= he2 (f2a2 - h2cV
X F, - Xl Y= f2a (g2b2 - fVp
- yit/j he2 (f2a2 - h2c2-sr
a
f2a (g2b2 - f2a2/3 + h2c2CT/3) = 0,
on enfin en multipliant par — ai>, et dans un terrae — g2b2h2c2/Ayptzr/c, au lieu de
ecrivant — , 1'equation devient
A
(fa)4 vp + (gb)* + (hc^ 1 - (gb)2 (he)2 (1 + i
- (he)2 (fa)2 ^ («• + ^) - (fa)2 (gb)2 (i; + p) = 0,
on, comme on peut 1'ecrire,
' 2, (gb)2, (hc)2)2 = 0.
C'est la deuxieme d'un systems de trois equations e'quivalentes ; savoir, en multipliant
par -- et en reduisant par \PVGT &p = \, on obtient la premiere forme: et, en multipliant
par \K et reduisant de me"me, on obtient la troisieme forme : le systeme est
2j (gb)2, (hc)2)2 = 0,
En ecrivant he = — fa — gb, on obtient une equation de la forme (*) (fa, gb)4 = 0,
savoir une Equation quartique pour avoir fa : gb, c'est-a-dire, le rapport anharmonique
(a — d) (b — c) : (b — d) (c — a) : en considerant a, b, c comme donnees, il y a done 4
valeurs de d : et Ton a deja vu que les valeurs a, /3; 7, S sont donnees rationnelle-
ment en fonctions de a, b, c, d: le theoreme est done demontre.
Cambridge, juillet, 1880.
14 [709
709.
ON THE NUMBER OF CONSTANTS IN THE EQUATION
OF A SURFACE PS-QR = 0.
[From the Tidsskrift for Mathematik, Ser. -t, t. iv. (1880), pp. 14-5—148.]
THE very important results contained in Mr H. Valentiuers paper "Xogle
Ssetninger om fuldsta?ndige Skjaeringskurver mellem to Flader" may be considered
from a somewhat different point of view, and established in a more simple manner,
as follows*.
Assuming throughout n > p + q, p > q, and moreover that P, Q, R, S denote
functions of the coordinates (x, y, z, w) of the orders p, q, n — q,n—p respectively :
then the equation of a surface of the order n containing the curve of intersection of
two surfaces of the orders p and q respectively, is
P,Q
R, S
so that the number of constants in the equation of a surface of the order n satisfying
the condition in question is in fact the number of constants contained in an equation
of the last-mentioned form. Writing for shortness
the number of constants contained in a function of the order p is = ap + 1 ; or if
we take one of the coefficients (for instance that of xp) to be unity, then the number
* Idet vi med stor Glsede optage Prof. Cayley's simple Forklaririg af den Reduktion af Konstanttallet i
Ligningen PS-QR=Q, som Hr. Valentiuer havde paavist (Tidsskr. f. Math. 1879, S. 22), skulle vi dog
bemaerke, at Grunden til, at dennes Bevis er bleven saa vanskeligt, er den, at ban tillige bar villet bevise,
at der ikke finder nogen ydtrligere Reduktion Sted.
709] ON THE DUMBER OF CONSTANTS IN THE EQUATION OF A SURFACE. 15
of the remaining constants is = ap ; viz. ap is the number of constants in the equation
of a surface of tb order p. As regards the surface in question
P, Q
R, S
= 0,
we may it is ciar take P, Q, R each with a coefficient unity as above, but in the
remaining functin S, the coefficient must remain arbitrary : the apparent number of
constants is thus= Op + ag+ a»_p +a,t_9+ 1 ; but there is a deduction from this number.
The equatioi may in fact be written in the form
P + *Q, Q
8+J3Q
= 0,
where a represens an arbitrary function of the order p — q, and /3 an arbitrary function
of the degree n-p-q: we thus introduce (ap_q + 1) + (an-p-9 4-1), = ap-q + (in-p-.q + 2,
constants, and b means of these we can impose the like number of arbitrary relations
upon the constats originally contained in the functions P, Q, R, S respectively (say
we can reduce o zero this number ap_9 + (in-v-q + 2 of the original constants) : hence
the real number »f constants is
ap + ttq + dn_p + dn_q + 1 — (ap_q + a-,^p^q + 2),
1
= <o suppose ;
viz. this is the rquired number in the case n > p + q, p>q-
If however =p + q, or p = q, or if these relations are both satisfied, then there is a
P Q I
further deduction of 1, 1, or 2 : in fact, calling the last-mentioned determinant ' jL ,
then the four caes are
S'
+ IP' + kS' + klR
S'+ IR
p + q,
p>y>
p/, q
=
P', Q'
R, S'.
p>v>
p, qf \ =
p' + kR, q
R,
*-*
F, q
F, Q' + kP'
R, S' + kR
*K
P' cy
R S'
J~V j U
=
F + kR, q
R',
where k, I dence arbitrary constants: these, like the constants of a and /3, may be
used to impose arbitrary relations upon the original constants of P, Q, R, S; and
hence the numbr of constants is = &>, to - 1, &>-!, eo - 2 in the four cases respectively;
where as above
af
16 ON THE NUMBER OF CONSTANTS IN THE EQUATION OF A SURFACE. [709
If n = 4, there is in each of the four cases one system of values of p, q ; viz. the
cases are
P> (1 =
21 No. = as + a, + a, + a. - a, - a, - 1 = 9 + 3 + 9 + 19 - 3 - 3 - 1, = 33,
31 „ a3 + a, + a, + a, - az - a0 - 2 = 19 + 3 -!- 3 + 19 - 9 - 0 - 2, = 33,
11 „ a1 + a1 + os + a3-a0-a2-2= 3 + 3 + 19 + 19 - 0 - 9 - 2, = 33,
22 „ a2+a, + a2 + a2-a0-a0-3= 9 + 9 + 9 + 9 - 0 - 0 - 3, = 33,
and the number of constants is in each case = 33. This is easily verified : in the first
case we have a quartic surface containing a conic, the plane of the conic is therefore
a quadruple tangent plane; and the existence of such a plane is 1 condition. In the
second case the surface contains a plane cubic ; the plane of this cubic is a triple
tangent plane, having the points of contact in a line ; and this is 1 condition. In
the third case the surface contains a line, which is 1 condition : hence in each of
these cases the number of constants is 34 — 1, =33. In the fourth case, where the
surface contains a quadriquadric curve, we repeat in some measure the general reasoning :
the quadriquadric curve contains 16 constants, and we have thus 16 as the number
of constants really contained in the equations P = 0, Q = 0 of the quadriquadric curve:
the equation PS — QR = 0, contains in addition 9+10, =19 constants, but writing it
in the form P (S + kQ) — Q(R + kP) = 0, we have a diminution =1, or the number
apparently is 16 + 19 — 1, =34. But the quadriquadric curve is one of a singly infinite
series P + IR = 0, Q + IS = 0 of such curves, and we have on this account a diminution
= 1; the number of constants is thus 34—1, =33 as above: the reasoning is, in fact, the
same as for the case of a plane passing through a line ; the line contains 4 constants,
hence the plane, qua arbitrary plane through the line, would contain 1+4, =5 constants ;
but the line being one of a doubly infinite system of lines on the plane the number is
really 5 — 2, = 3, as it should be.
Cambridge, 2nd Sept., 1880.
710]
17
710.
ON A DIFFEBENTIAL EQUATION.
[From Collectanea Mathematical in memoriam Dominici Chelini, (Milan, Hoepli 1881)
pp. 17—26.]
IN the Memoir on hypergeometric series, Crelle, t. xv. (1836), Kummer in effect
considers a differential equation
(a'z* + 2b'z + c) dz* = (ay? + 2bx + c) da?
z*(z-iy a?(x-l)* .'
viz. he seeks for solutions of an equation of this form which also satisfy a certain
differential equation of the third order. The coefficients a, b, c are either all arbitrary
they are two or one of them, arbitrary; but this last case (or say the case
where the function of x is the completely determinate function a? + 2bx + c) is scarcely
considered: a', b', c' are regarded as determinate in terms of a, b, c; and z is to
found as a function of x independent of a, b, c: so that when these coefficients
re arbitrary, the equation breaks up into three equations, and when two of the
coefficients are arbitrary, it breaks up into two equations, satisfied in each case by
he same value of ,; and the value of z is thus determined without any integration-
these cases will be considered in the sequel, but they are of course included in the
a' b> C are reSarded as havinS a»y g^en values
Writing for shortness X = ax* + 2bx + c, in general the integral
f Ndx
) D^/X '
where D is the product of any number n of distinct linear factors x-p and N is
a rational and integral function of * of the order n at most, and therefore also the
integral
D
C. XL
[NXdx
~J D^X'
3
18 ON A DIFFERENTIAL EQUATION. [710
a
where N is now of the order n — 2 at most, is expressible as the logarithm of
quasi-algebraical function, that is, a function containing powers the exponents of which
are incommensurable (for instance, x^'z is a quasi-algebraical function) : in fact, the integral
is of the form
/•/.. A B \ dx
(M+- - + +
J\ x—p x — q
where each term is separately integrable,
[dx _ J^ lo f b
JVJ~ \/a°gl
dx I {(ap + b)x + (bp + c) + *JP-
(x-p)*fit~ ~VP °gl *-P
where P is written to denote ap- + 2bp + c: the integral is thus = log H, where
is a product of factors
ax + b + ^a.^X, ^ ^^ -'etc->
x— p
raised to powers —^ , ^-= , etc. : hence, if we have a differential equation
va vP
N'dz Ndx N'^Zdz _ N\/Xdx
Ol "r^i ~~~ ~r\ 5
where £ (= aV + 26'^ + c'), and ^V, D' are functions of z such as X, N, D are of
# ; then, taking log C for the constant of integration, the general integral is
log n' = log c + log n :
viz. we have the quasi-algebraical integral n' — (7O = 0.
The constants a, b, c, p, q, ... etc. may be such that the exponents are rational,
and the integral is then algebraical: in particular, for the differential equation
+ I4>z + I dz VarM-14r+ Idx
z(z-\ x(x-\)
the general integral is in the first instance obtained in the form
(z + I + </Z) (z - 1 )2 _ (x + 1 + VZ) (a; - 1 )2
"
which, observing that (2#+ 2)2 - X = 3 (aj - I)2, may also be written
(* + 1) (^-34^ + 1) + ^^ = c (a?+l)(^-34a?+l) +
V^ 0 - I)2 Va? (a- - 1)"
710] ON A DIFFERENTIAL EQUATION. 19
I had previously obtained the solution
and I wish to show that this is, in fact, the particular integral belonging to the value
C = 1 of the constant of integration : for this purpose I proceed to rationalise the general
integral as regards z.
Writing for a moment
P = (z+l)(z"- 34* + 1),
Q = (z- + Uz + 1) \/z' + 14z + 1,
R
where
O + 1) (& - 34a? + 1) + (a?
the integral is P + Q + R = 0; or rationalising, it is
(p, _ QJ _ 2#2 (p, + Q2) + R± = Q .
we have
P- = (l, -66, 1023, 2180, 1023, -66, l\z, I)6,
Q-J = (1, 42, 591, 2828, 591, 42, l\z, 1)«,
and thence
P^-Q^ = (0, -108, 432, -648, 432, -108, 0\z, I)6,
P2+Q2 = 2(1, -12, 807, 2504, 807, - 12, l\z, If.
Writing the equation in the form
it thus becomes
(1, -12, 807, 2504, 807, -12, l\z, 1)« -z(z- I)4 \M* + v-^(- = 0,
where Jf has its above-mentioned value; and if we now assume (7=1, then
(a? + 1) (a8 - 34# + 1) + (a;2
108 = (as + 1) (a;2 - 34a; + 1) - (^ + 14a; + l)Vfe"+ 14,-g +"l
M ' V#O-1)2
and thence
M , (108? =f^_10Sy _(^ + 1)^-34^+ I)2
2 : H *
= ^ iy.(l, -12, 807, 2504, 807, -12, l"$x, I)6:
3—2
20 ON A DIFFERENTIAL EQUATION. [710
and the rationalised equation is
(1, -12, 807, 2504, 807, -12, \\z, I)6
#(# — I)
This is a sextic equation in z, of the form
n _12, 807, 2504, 807, -12, ijx, 1)6 = 0.
4
where
X, /*, i/ = -12-fl, 807 + 4Q, 2504 -6O,
if H denote the function of x which enters into the equation ; and writing z + - = 0, this
2
becomes
But the equation in z is satisfied by the value z = xt and therefore the equation in 9 by
the value 6 = x + - = a suppose, we have therefore
10
a3 - 3a + X (a2 - 2) + /*« + v = 0,
and thence subtracting, and throwing out the factor 6 — a,
0-2 + 0a + a2-3
viz. writing for X, p, a their values, this is
x
or, what is the same thing,
where
n= l -(1, -12, 807, 2504, 807, -12, Ija, I)6.
t27 (A? ^~ J. J
Hence in the quadric equation, the coefficients, each multiplied by (x- I)4, are
12+i— (1, -12, 807, 2504, 807, -12, l\x, I)6,
00
and
12 1
/ 12 1\
(x - I)4 1 a? - 1 2« + 806 -- + — )
' \ x a?/
.^ + lVl, _12, 807, 2504, 807, -12, l$a?, I)6,
which are respectively rational and integral quartic functions of x ; and, writing for 0 its
value, the equation finally is
1, 188, 646, 188, \\x, I)4 . (1, -644, 3334, -644, \\x, I)4
e**v. -=o
710] ON A DIFFERENTIAL EQUATION. 21
Writing
4/- 1 - £ D * + £ rr 1 ~ *'£ r» 1 + *£ / ---
f-v^ 4.-f+f. jB=r^r C=ir4' jD=r-^|' (*=^-i as usual),
this is
^-J.OO-^O
or, what is the same thing,
that is,
for we have
i (A* _(!> 28, 70, 28,
(£»-!)«
4 , rwv (1. -28, 70, -28,
And substituting these values, the coefficients will be rational functions of £4, that is, of
x, and it is easy to verify that they have in fact their foregoing values.
It thus appears that for C=l, besides the values x and -, we have for z onlv the
x J
values
viz. that the only solution is
The example shows that although the differential equation
V + 26^ + c'dg Voa;--3 + Zbx + c dx
can be integrated generally in a quasi-algebraical or algebraical form as above, yet
we cannot from the general solution deduce, at once or easily, the various particular
integrals comprised therein: nor can we find for what values of the constants a, b, c
and a', b', c' the differential equation admits of a simple solution, or say of a solution
where z is expressed as an explicit (irrational) function of x.
In the cases considered by Kummer there is a second (or it may be also a
third) differential equation of the like form, the equations being each of them satisfied
by the same value of z : hence eliminating the differentials dan, dz, the relation between
x and z is of the form
P'_P
Q'~Q'
22 ON A DIFFERENTIAL EQUATION. [7LO
where P, Q are quadric functions of x\ P', Q' quadric functions of z. But P and
Q may contain a common factor, and the integral is then expressible in the form
P'
x= rr, , the quotient of two quadric functions of z ; or P' arid Q' may have a common
HJ
p
factor, and the integral is then expressible in the form z = ^ , the quotient of two
quadric functions of x; or there may be a common factor of P, Q, and also a common
factor of P' and Q', and the integral is then of the form * = j^, the quotient of two
linear functions of x.
In the general case the differential equation is
\(aP' + bQ')dz*
~~~
where a, b are arbitrary constants, X is a constant the value of which can in each
P
particular case be at once determined ; so when the integral is Z—-Q> tne differential
equation is
X (az + b) dz2 _ (aP + bQ) dxz
z*(z-I)* a?(x-].)-
where a, b are arbitrary constants, but A, is now a linear function of z the value
of which can in each particular case be at once determined. When the integral
is z=^, the differential equation is
X (az* + 26z + c) dz* _ (aL- + 2bLM + cM-} dx-
z* (z-\f a-2(#-l)2
containing the three arbitrary constants a, b, c; X is a constant the value of which can
be at once determined.
There are in all 6 integrals of the form z = ^-,, for which the differential equation
P I
contains three arbitrary constants : 18 integrals of the form z = ^ f and of course the
P'\ P P'
same number of integrals of the form # = 7y)> and 9 integrals of the form -Q =/y> f°r a^
of which the differential equation contains two arbitrary constants. It is to be remarked
that Kummer, considering the values of z as a function of x, obtains the 72 rational and
irrational values mentioned in his equations (31), (35), (36), (37), (38), and (39) : but the
72 values are made up as follows, viz. the 18 values of z as a rational function of x, the
36 irrational values obtained from the 18 expressions of x as a rational function of z, and
the 18 irrational values of z obtained from the 9 integrals in which neither of the
variables is a rational function of the other: 18 + 36 + 18 = 72.
710]
ON A DIFFERENTIAL EQUATION.
23
The several integrals together with the expressions of the functions
a'z2 + Ib'z + c' and ax1 + 2bx + c
which enter into the differential equation are as follows :
a'z2 + 2b'z + c' - ax2 + 2bx + c =
1.
3.
3 =
X
l-x
1
X
1
az2 + 2bz + c
j>
5)
»
))
))
aa2 + 26a? + c
«(^--l)2-26(«-l) + c
a + 2bx + ex2
a-2b(x-l) + c(x-iy
l-x
X
x-l
x-l
a(x—l)2+ 2bx (x — 1 ) + ex2
X
(x+ ly
az2 + 6s
>>
5)
»
n
D
a(x+ l)2 + b(x-lf
a (2x - I)2 + 6
a (x - 2)2 + bx*
a (x+ 1)2 + 46ai
« (2x — I)2 + 4:bx (x - 1)
a (Y *>\3 4.A /->. I \
U- J
(2.x- I)2
(x-'2\
( x )
(x+ir
4:X
(2x-l£
402 (X— 1)
(x-Vf
4(*-l)
/*-iy
6s + c
»
»
M
))
>i
b(x-iy + c(x + if
b + c(2x-l)2
U+ 1)
( l Y
\2x-l)
( x Y
U-2;
4:X
4:bx + c(x + I)2
46a; (a; — 1 ) + c (2aj - 1 )2
-4b(x-l)+c(x-2f
(x+iy
4x(x-l)
~(2x-iy
4 (aj-1)
(*'-2)2
24
z =
ON A DIFFERENTIAL EQUATION.
a'z* + 2b'z + c' — ay? 4- 2bx + c —
4.
4a3
as2 - (a + c) z + c
a (x— I)2 + 4ca:
-4*(*-l)
»
4aa; (oj — 1 ) + c
4(«-l)
»
— 4a (a; - 1 ) + ca;2
ar2
4x
?>
4aw + c (a; - 1 )2
(a;-!)2
-1
5)
a + 4raj (a; — 1)
4as(*-l)
x2
!?
aar2 — 4c (a; — 1)
4(£C- 1)
5.
6. [• same as 2, 3, 4 interchanging cc and s.
7.
9.
10.
2:2
a (z — I)2 + 46z
as2 + 46 (s — 1)
4« («-!) =
t^_1)^-4a?(a;-l)
(2-l)2_ 4(a-l)
-46(*-l)
'-!) —
r2
j - ,
4 (as- 1)
4 «-
4az(z
- 4a (a - 1) +
c =
4ax + 6 (a;- I)2
• 4a (a; — 1) - 6ar
« + 46a?(a;- 1)
4aa; (a?— 1) + b
4ax (x — 1 ) + b
4a (x — 1) + bx2
a(x- 1 )2 + 46a3
aa;2 — 46 (x — 1)
10] ON A DIFFERENTIAL EQUATION. 25
The six functions of the set (1), that is,
., 1 1 x x —1
®> *^> ~ > 1 - > - T~ » ~~ 1
x I— as x — 1 #
form a group; and by operating with the substitutions of this group, and of the like
group
I 1 2 z-1
~2> z> i-2> i^i> -2~>
upon any value of z in the sets (2), (3), (4), for instance upon z = (— ~V, we form all
\*E — 1 /
the 18 functions of these sets.
In any one of these sets (2), (3), and (4), comparing two forms (the same or
different), for instance in the set (2), writing y for z and then in one form z for x,
/a? + iy , /2 + i\" ,
»=U-f) and '-\f=V' whence
or
)2 ,
-' whence =
we obtain either the equations of the set (1) or those of the sets (8), (9) and (10); and
whether we use the set (2), (3) or (4), the only new equations obtained are thus the 9
equations of the sets (8), (9) and (10). These several equations present themselves
however in different forms: for instance, instead of the equation
(z — 1)2_ 4>ae
~4z~ ~^la'
we may obtain
If, to get rid of this variety of form, we multiply out the denominators, the 9
equations are
0= x-z-- Zx-z- 2x2-+ xn--l2x2+ 22- 2x- 22+ 1,
0 = x*z- - Ifaz + IQx + IGz - 16,
0 = 16#V - IQx-z - IQxz- + IGxz - 1,
0 = a?z- - 2x22 + xn-+ 16x2 - 162
0 = 16x-2 - 16x2 - z- +2.0-1,
0= lQx-2 -I6x*-16xz+ z' + lQx
0= x-z" — 2x2- +16^+ 2--16x
0= 16x2-- x2-16xz + 2x + 1,
0= 16xz* + x2- 16x2 -162- +162
These 9 equations are derivable all from any one of them by the changes of the set (1)
upon x and z.
Cambridge, 3rd June, 1879.
C. XL A
[711
711.
ON A DIAGRAM CONNECTED WITH THE TRANSFORMATION OF
ELLIPTIC FUNCTIONS.
[From the Report of the British Association for the Advancement of Science, (1881), p. 534.]
THE diagram relates to a known theorem, and is constructed as follows. Consider
the infinite half-plane y= + ; draw in it, centre the origin and radius unity, a
semicircle; and draw the infinite half-lines # = -£, and «=J; then we have a
region included between the lines, but exterior to the semicircle. The region in
question may be regarded as a curvilinear triangle, with the angles 60°, 60°, and 0°.
The region may be moved parallel to itself in the direction of the axis of x, through
the distance 1; say this is a "displacement"; or we may take the "image" of the
region in regard to the semicircle. Performing any number of times, and in any
order, these two operations of making the displacement and of taking the image, we
obtain a new region, which is always a curvilinear triangle (bounded by circular
arcs) and having the angles 60°, 60°, 0°; and the theorem is that the whole series
of the new regions thus obtained completely covers, without interstices or over
lapping, the infinite half-plane. The number of regions is infinite, and the size of
the successive regions diminishes very rapidly. The diagram was a coloured one,
exhibiting the regions obtained by a few of the successive operations.
ao> + /3
The analytical theorem is that the whole series of transformations, <o into —£-$ ,
where a, /3, y, 8 are integers .such that a&-j3y<=I, can be obtained by combination
of the transformations to into o> + 1 and o> into .
712] 27
A PARTIAL DIFFERENTIAL EQUATION CONNECTED WITH THE
SIMPLEST CASE OF ABEL'S THEOREM.
[From the Report of the British Association for the Advancement of Science, (1881).
pp. 534, 535.]
CONSIDER a given cubic curve cut by a line in the points (a?,, yjt (Xn, ys),
0»3> 2/3); taking the first and second points at pleasure, these determine uniquely the
third point. Analytically, the equation of the curve determines yl as a function of
xi, and 7/2 as a function of x.2: writing in the equation
a?3 = Xo?, + (1 - X) #o, y3 = \yt + (1 - X) yst
we have X by a simple equation, and thence x.,- viz. x, is found as a function of
aii, a?2, and of the nine constants of the equation. Hence forming the derived equations
(m regard to aclt ara) of the first, second, and third orders, we have (1 + 2 +3 + 4=) 10
equations from which to eliminate the 9 constants; xs, considered as a function of
x, and a?s, thus satisfies a partial differential equation of the third order, independent
of the particular cubic curve.
To obtain this equation it is only necessary to observe that we have, by Abel's
theorem,
dXi dx2 ,dxs_
y -T y T ^F~ — v,
•«1 ^2 -^3
where X, is a given function of x, and ylt that is, of #, ; X2 and X3 are the like
functions of #2 and xs respectively. Hence, considering x, as a function of ^ and x.r
we have
^=_£ dx^__X^
dx, X,' dx,~ Z2'
4—2
2g A PARTIAL DIFFERENTIAL EQUATION. [712
and consequently
dx3 ^_^2.
dx^ dx., Xi '
where X,, X, are functions of x,, x, respectively: hence taking the logarithm and
differentiating successively with regard to x, and x,, we have
which is the required partial differential equation of the third order.
This differential equation has a simple geometrical signification. Consider three
consecutive positions of the line meeting the cubic curve in the points 1, 2, 3;
1' 2', 3'; 1", 2", 3" respectively: qua equation of the third order, the equation
should in' effect determine 3" by means of the other points. And, in fact, the three
positions of the line constitute a cubic curve; the nine points are thus the inter
sections of two cubic curves, or, say, they are an "ennead" of points; any eight of
the points thus determine uniquely the ninth point.
713] 29
713.
ADDITION TO MR HOWE'S MEMOIR ON ABEL'S THEOREM.
[From the Philosophical Transactions of the Royal Society of London, vol. 172, Part in.
(1881), pp. 751—758. Received May 27,— Read June 10, 1880.]
IN Abel's general theorem y is an irrational function of x determined by an
equation ^ (y) — 0, or say ^ (x, y} = 0, of the order n as regards y : and it was shown
by him that the sum of any number of the integrals considered may be reduced to
a sum of 7 integrals; where 7 is a determinate number depending only on the form
of the equation x (*. y) = 0, and given in his equation (62), [(Euvres Completes, (1881),
t. I. p. 168] : viz. if, solving the equation so as to obtain from it developments of y
in descending series of powers of x, we have*
a
%!/*! series each of the form y = <*?•* + ...,
* The several powers of x have coefficients: the form really is y = Alx^ + ..., which is regarded as
J^
representing the MI different values of y obtained by giving to the radical x^ each of its MI values, and
the corresponding values to the radicals which enter into the coefficients of the series: and (so understanding
it) the meaning is that there are MJ such series each representing MI values of y. It is assumed that the
_1
series contains only the radical x1*1 , that is, the indices after the leading index ^ are TOI ~ 1 ™l ~ 2 • a
4 a Mi Mi Mi
series such as y = A1x* + B1x» + ... , depending on the two radicals x^, x~ represents 15 different values, and
would be written y-A-^x^ + ..., or the values of Wj and MI would be 20 and 15 respectively: in a case like
7/1
this where -- is not in its least terms, the number of values of the leading coefficient Al is equal, not to
Mi, but to a submultiple of nl. But the case is excluded by Abel's assumption that ^, ^...., are fractions
Ml /*o
each of them in its least terms.
30 ADDITION TO MR ROWE's [713
(so that w = n,^1 + na/ig+...+wt/A*), then 7 is a determinate function of nlt wi,, //,;
Mr Rowe has expressed Abel's 7 in the following form, viz. assuming
then this expression is
7 =
*>»•
or, what is the same thing, for n writing its value
where in the first sum r, s have each of them the values 1, 2, ...,&, subject to the
condition 6- > r ; in each of the other sums n, m, and /z, are considered as having the
suffix r, which has the values 1, 2, ..., k.
It is a leading result in Riemann's theory of the Abelian integrals that 7 is the
deficiency (Geschlecht) of the curve represented by the equation %(%, y) = 0: and it
must consequently be demonstrable a posteriori that the foregoing expression for 7 is
in fact = deficiency of curve % (x, y} = 0. I propose to verify this by means of the
formulae given in my paper "On the Higher Singularities of a Plane Curve," Quart.
Math. Jour., vol. vii., (1866), pp. 212—223, [374].
It is necessary to distinguish between the values of which are >, =, and < 1 ;
and to fix the ideas I assume k = 7, and
?HI 7H2 m3 , i
— , — , — , eacn > I,
— = 1 ; say m4 = /u,4 = X, and ?i4 = 6 ;
but it will be easily seen that the reasoning is quite general. I use 2' to denote
a sum in regard to the first set of suffixes 1, 2, 3, and 2" to denote a sum in
regard to the second set of suffixes 5, 6, 7. The foregoing value of n is thus
n = 2V + \6 + 2' V.
Introducing a third coordinate z for homogeneity, the equation %(#, y) = 0 of
the curve will be
where it is to be observed that ( )"^' is written to denote the product of n^
W, MI
different series each of the form yz^~l - A^ - ... ; these divide themselves into n,
713] MEMOIR ON ABEL'S THEOREM. 31
groups, each a product of /^ series; and in each such product the ^ coefficients At
!_
are in general the ^ values of a function containing a radical a^ and are thus
different from each other: it is in what follows in effect assumed not only that this
is so, but that all the n^^ coefficients A} are different from each other* : the like
/ A\A»
remarks apply to the other factors. It applies in particular to the term [y — #M
A
viz. it is assumed that the coefficients A in the \6 series y = Ax* + ... are all of
them different from each other. These assumptions as to the leading coefficients
really imply Abel's assumption that — , ...,— are all of them fractions in their least
LL, ILl.
\
terms, and in particular that - is a fraction in its least terms, viz. that \ = 1 : I
A,
retain hoAvever for convenience the general value X, putting it ultimately =1.
In the product of the several infinite series, the terms containing negative powers
all disappear of themselves ; and the product is a rational and integral function
F(a, y, z) of the coordinates, which on putting therein z=l becomes = x(®, y).
The equation of the curve thus is F(ac, y, z) = 0 ; and the order is
viz. if K is the order of the curve x (A>> 2/) = °> tnen K — %'nm + \0 + ^"np.
The curve has singularities (singular points) at infinity, that is, on the line z = 0
viz.-
First, a singularity at (z = 0, x = 0), where the tangent is x — 0, and which,
writing for convenience y = 1, is denoted by the function
/ *»i y^i-M,
where observe that the expressed factor indicates nl branches [jr— «"»-*»l , or
_wi^
say iHCmi-ft) partial branches z - a?"'-*, that is, n, (m^ - ^} partial branches
»»!
z = A1xmi~^ + ..., with in all 11^(11^-^} distinct values of A^: and the like as regards
the unexpressed factors with the suffixes 2 and 3.
Secondly, a singularity at 0=0, 2/ = 0), where the tangent is y = 0, and which,
writing for convenience x—I, is denoted by the function
* This assumption is virtually made by Abel, (I. c.) p. 162, in the expression "alors on aura en general,
excepte quelques cas particuliers que je me dispense de considerer : h(y' -y") = hy', &c.": viz. the meaning is
that the degree of y' being greater than or equal to that of y", then the degree of y'-y" is equal to that
of y" : of course when the degrees are equal, this implies that the coefficients of the two leading terms must
be unequal.
32 ADDITION TO MR HOWE'S [713
where observe that the expressed factor indicates «„ branches \z - y** '"5J , or
_»*5
say n5(/4B-m6) partial branches z - y**~m* , that is, *,(/!, -«,) partial branches
IN
2 = ^5^-^ + ..., with in all n5(/*B-wi5) distinct values of ^45: and the like as
regards the unexpressed factors with the suffixes 6 and 7.
Thirdly, singularities at the 0 points (z = 0, y-Ax = 0), A having here 0 distinct
values, at any one of which the tangent is y-Ax = 0, and which are denoted by
the function
but in the case ultimately considered X is = 1 ; and these are then the 0 ordinary
points at infinity, (z = 0, y — Ax = 0).
According to the theory explained in my paper above referred to, these several
singularities are together equivalent to a certain number B' + K of nodes and cusps;
viz. we have
«' = S(a-l),
hence
Assuming that there are no other singularities, the deficiency
s>r
This should be equal to the before-mentioned value of 7 ; viz. we ought to have
(K - 1) (K - 2) - M + 2 (a - 1) = ^nrmrnsfjis + 2n-mfji - 2wm - 2w/4 - S?i + 2,
or, as it will be convenient to write it,
M = K-- %K + 2 (a - 1) -
which is the equation which ought to be satisfied by the values of M and
calculated, according to the method of my paper, for the foregoing singularities of
the curve.
We have as before
s>r
The term 2nrmrn8ps, written at length, is
+ 0\ ( n^^
+ n-,m5 ( n6n6 + n7/u7)
713] MEMOIR ON ABEL'S THEOREM. 33
which is
= 2/wrwrng/As + 6\ (S'nw + 2'V) + 2'?i?>i . 2'V + "Z"
s>r s
We have moreover
2wm = 2'wra + 6\ + 2"nm,
2V = 2V + 0X + 2' V>
2?i = 2'n + 0 + 2"w.
We next calculate 2 (a — 1).
For the singularity
_
each branch U — xm^-^j gives a = TOJ — ^, and the value of 2 (a — 1) for this
singularity is
% (TO! - /A: - 1) + w2 (m2 - yu2 - 1) + ??3 (ms - ^ - 1),
which is
= S'wi?i — 2'w/i — S'/?.
For the singularity
/ Mg \H5-ms
each branch I j — 7/^-»n5J gives a = ^5 - ??i5, and the value of S (a — 1) for this
singularity is
ns (p5 - ms - 1) •+ w6 (>6 - ??zg - 1 ) + w7 (yu7 - m7 -I),
which is
= 2' V - 2"nm - 2'V.
For each of the ^ singularities
we have a = X and the value of 2 (a— 1) is =0(X — 1) : this is = 0 for the value
X = l, which is ultimately attributed to X.
The complete value of 2 (a — 1) is thus
= S'wm - 2"nm - 2V + 2'V ~ %'n - 2"?i + 0\-0.
Substituting all these values, we have
M = (2'nm + 2'V)2 + 26>X (2W + 2'V) + (^)2
- 3 (2'nm + 2'V) ~ 30X
+ 2'nm — 2"rm — 2'w/t + 2"7iyu. — 2'w — 2"w + ^X — ^
— 2^'nrmrnsfjLs — W\ (Z'nrn + 2'V) ~ 22'nm . 2"w/A — 22"?irwr??syu,s
.« > r « > r
- 2'tfm/j, - (6>X)2 - 2"w«m/A
+ 2'wrn + 0X + 2"wm
+ 2V + 0* + 2'V
+ 2'n + 6> + 2"w,
c. xi. 5
34 ADDITION TO MR ROWE's [713
or, reducing,
M= (2nm - ^!nm — 'Z'tfm/j, — 2^'nrmrnsfis
and it is to be shown that the two lines of this expression are in fact the values
of M belonging to the singularities
...f and *— Jf«"
/
respectively. We assume \ = 1, and there is thus no singularity (y —
I recall that, considering the several partial branches which meet at a singular
point, M denotes the sum of the number of the intersections of each partial branch
by every other partial branch : so that for each pair of partial branches the inter
sections are to be counted twice. Supposing that the tangent is x — 0, and that for
any two branches we have z1 = A^x^, zz = A^pc^ (where p1} p2 are each equal to or
greater than 1), then if p2 = pi, and z1 — z2 = (Al — A^x^ where A± — A,, not =0 (an
assumption which has been already made as regards the cases about to be considered),
then the number of intersections is taken to be =p± ; and if p1 and p2 are unequal,
then taking p2 to be the greater of them, the leading term of ^ — z.2 is = A^x^, and
the number of intersections is taken to be =pl ; viz. in the case of unequal ex
ponents, it is equal to the smaller exponent.
/ _J2i_Y»iG"i-W
Consider now the singularity (z - xm^^j ...; and first the intersections of
a partial branch z — xm^~^ by each of the remaining nl(m1— fa) — I partial branches
777
of the same set : the number of intersections with any one of these is = - — ;
A?7
and consequently the number with all of them is = - - [MI (m^ — /ij) — 1]. But we
mi — /MI
obtain this same number from each of the n1 (m^ — /^) partial branches, and thus the
whole number is
"?7?
n, (m, - /&!> - - Dh (wa - /ij) - 1], = n, m, [n, (Wj - ^) - 1].
ml — fa
Taking account of the other sets, each with itself, the whole number of such
intersections is
^1^1 [MJ (ml - /AI) — 1] + w2m2 [??2 (™2 — ^2) - 1] + n3m3 [n3 (m3 — fi3) — 1],
which is
713] MEMOIR ON ABEL'S THEOREM. 35
Observe now that — > — , that is, D < c2 1 an(j that, these being each < 1 we
/^i A^2 iYl>\ T¥Li)
thence have 1 > 1 - Ma , that is, -3 — ^ > — — ^2 : and we thus have
Considering now the intersections of partial branches of the two sets
ml \ni (M|^t|} / 7/3
TO — 1 J I
— 3C l J 3/HCl I .S' — ^/ 2
?^}
respectively, a partial branch s - a;^i-^. gives with each partial branch of the other
set a number = — -_- — ; and in this way taking each partial branch of each set,
the number is
??! (ml — /AJ) . n* (m2 — fj,2) .— — , = n^rt^n^ (m.2 — /i2) :
and thus for all the sets the number is
which is
where in the first sum the 2' refers to each pair of values of the suffixes. But the
intersections are to be taken twice ; the number thus is
Adding the foregoing number
2'n2rn2 — S/n*0t/u — %
the whole number for the singularity in question is
= (2'nm)2 - S'rcra - S
8>r
(<*» \%(M;-m5)
^_y^-m5J <>>; taking each set with itself, the
number of intersections is
[ns (fi8 - r»6) - 1] + n,/^ [n7 (& -m?}- I],
which is
KV/ o » K1// o ^<//
= ft n-/j.- — z wmfji — 2 TI/A.
5—2
36 ADDITION TO MR ROWE's MEMOIR ON ABEL'S THEOREM. [713
We have here — 5>— 6; each of these being less than 1, we have 1 -<!-— ^ ',
fJ'S A*6 ^5 ^6
that is, Mg~Wg<^g^: — , or — ^— > — ^^ ; and so
Hence considering the two sets
.% (M6 -We)
yLlg
a partial branch of the first set gives with a partial branch of the second set _m
intersections: and the number thus obtained is
n5 (/*, - ra5) . n6 (/*, - m6) . — — , = w5w6/*6 (ps - m5).
MB '%
For all the sets the number is
or taking this twice, the number is
where in the first sum the 2" refers to each pair of suffixes. Adding the foregoing
value
S"n>2 - 2' Vm/i - 2'V,
the whole number for the singularity in question is
s>r
and the proof is thus completed.
Referring to the foot-note (ante, p. 31), I remark that the theorem 7 = deficiency,
is absolute, and applies to a curve with any singularities whatever : in a curve which
has singularities not taken account of in Abel's theory, the "quelques cas particuliers
que je me dispense de considerer," the singularities not taken account of give rise
to a diminution in the deficiency of the curve, and also to an equal diminution of
the value of 7 as determined by Abel's formula; and the actual deficiency will be
= Abel's 7 — such diminution, that is, it will be = true value of 7.
714] 37
714.
VARIOUS NOTES.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 69, 115, 124, 125.]
An Identity.
THE following remarkable identity is given under a slightly different form by
Gauss, Werke, t. in., p. 424,
3
then
On two related quadric functions.
Assume
fa = a2 (c - a?) - ar (c2 - 62 - car),
•^ra; = J2 (c — x) — X (c2 — O? — CX) '.
U~
In the first of these for oc write - ; then
c — x
a?(c-x)\ a*(c-x? ¥ _a26_
- ^ - * 2 9 1
38 VARIOUS NOTES. [714
A Trigonometrical Identity.
cos (b - c) cos (b + c + d) + cos a cos (a + d)
i
= cos (c — a) cos (c + a + d) + cos b cos (6 + d)
= cos (a - b) cos (a + b + d) + cos c cos (c + d)
= cos a cos (a + d) + cos b cos (b + d) + cos c cos (c + d) — cos d.
Extract from a Letter.
" I wish to construct a correspondence such as
(x + iy)x + O + iy) = X + iY,
or, say, for greater convenience
4 (a? + iy)s - 3 (x + iy) = X + iY;
viz. if
x + iy = cos u,
then
Suppose 3«0 is a value of 3u corresponding to a given value of X + i Y, then the
three values of x + iy are of course cos^0, cos(w0±-^-); but I am afraid that the cal-
V o /
culation of «0, even with cosh and sinh tables, would be very laborious. Writing
X + iY = R (cos ® + i sin ®),
the intervals for © might be 5°, 10° or even 15°, those of R, say O'l from 0 to 2,
and then 0'5 up to 4 or 5 ; and 2 places of decimals would be quite sufficient ; but
even this would probably involve a great mass of calculation.
It has occurred to me that perhaps a geometrical solution might be found for
the equation X + iY= cos 3u."
October 31, 1877.
715] 39
715.
NOTE ON A SYSTEM OF ALGEBEAICAL EQUATIONS.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 17, 18.]
ASSUME
x + y +z =P,
yz + zx + xy = Q ,
xyz — R,
A=x (nyz + Q)-w2 (mx + P\
B =y (nzx + Q) — W2 (my + P\
C =z (nxy + Q) — up (mz + P),
© = - mnR + PQ.
Then
(mz + P)B-(my + P)C
= (myz + Py) (nzx + Q) — (myz + Pz) (nxy + Q)
= myz (nzx + Q — nxy — Q) + Pnxyz + PQy — Pnxyz — PQz
= mnxyz (z — y) — PQ (z — y)
= (z-y} {mnxyz - PQ} =(y-z)@;
whence, identically,
(mz +P)B - (my + P)C = (y-z)&,
(mx + P)C - (mz +P)A=(z-ae)&,
(my + P)A- (mx + P)B =(x-y)S.
Hence any two of the equations .4 = 0, B = 0, (7 = 0 imply the third equation.
40 NOTE ON A SYSTEM OF ALGEBRAICAL EQUATIONS. [715
We have
A=x {(n + l)yz + zx + ocy}-wz {(m + 1) x + (y + 2)}
= (x- - w2) (y + z) - x [(m + 1) wz - (n + 1) yz\
and similarly for B and G. The three equations therefore are
x
_ _
x- — wz~ (m + 1) w* — (n + 1) yz '
x + y
z^-'uf- (m + 1) w*-(n
and any two of these equations imply the third equation.
716] 41
716.
AN ILLUSTKATION OF THE THEORY OF THE ^-FUNCTIONS.
[From the Messenger of Mathematics, vol. vii. (1878), pp. 27 — 32.]
IF X be a given quartic function of x, and if u, or for convenience a constant
multiple au, be the value of the integral I -™^ taken from a given inferior limit to
J v \-A- )
the superior limit as; then, conversely, x is expressible as a function of u, viz. it is
expressible in terms of ^-functions of u, where ^u, or say %(u, g) (g a parameter
upon which the function depends), is given by definition as the sum of a series of
exponentials of u; and it is possible from the assumed equation «*« f-J^r, and the
J vv&)
definition of ^n, to obtain by general theory the actual formulas for the determination
of x as such a function of u.
I propose here to obtain these formulae, in the case where X is a product of
real factors, in a less scientific manner, by connecting the function <bu (as given by
such definition) with Jacobi's function @, and by reducing the integral t-—L- by a
• ^ ^' '
linear substitution to the form of an elliptic integral; the object being merely to
obtain for the case in question the actual formulae for the expression of x in terms
of ^-functions of u.
The definition of *bu or, when the parameter is expressed, S- (u, $) is
where s has all positive or negative integer values, zero included, from - oc to + oo
(that is, from - S to + 8, S = oo ) ; the parameter g, or (if imaginary) its real part,
must be positive.
C. XL 6
OF THE
UNIVERSITY
42 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. [716
Evidently ^w is an even function: *r(-u*) = ^u. Moreover, it is at once seen that
we have
S- (u + TT) = <$u, *(u + t'ft) = - e%~2iu STM,
whence also
S- (u + m-rr + raft),
where m and w are any positive or negative integers, is the product of <bu by an
exponential factor, or say simply that it is a multiple of *bu.
Writing u = -$ift, we have ^ (- i»8) - ^ (i*8), that is>
and therefore also
The above properties are general, but if ft be real, then k, K, K', q being as in
Jacobi (consequently k being real, positive, and less than 1, and K and K' real and
positive), and assuming ft = -^- , or, what is the same thing,
i%Ku\
the function S- is given in terms of Jacobi's © by the equation !h»~8f - V; or,
what is the same thing, ©w = ^ ^-^ } .
\Z/L/
We hence at once obtain expressions of the elliptic functions sn u, en u, dn u in
terms of ^, viz. these are
cn
//^'\
- V U)
Consider now the integral
dx
f dx f
-77^^ -^r~ — ji » =
Jav{(— )« — CL.X — b.x — c.x — a} Ja
suPPose.
where a, b, c, d are taken to be real, and in the order of increasing magnitude, viz.
it is assumed that b -a, c -a, d - a, c - b, d - b, d - c are all positive; x considered
as the variable under the integral sign is always real; when it is between a and b
or between c and d, X is positive, and we assume that */(X) denotes the positive
value of the radical; but if x is between b and c, X is negative, and we assume
716] AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. 43
that the sign of \/(X) is taken so that -77-^ is equal to a positive multiple of i,
TV* /
and this being so the integral is taken from the inferior limit a to the superior
limit x, which is real.
Take x a linear function of y, such that for
x = a, b, c, d,
y = 0, 1, ri , x , respectively,
so that, x increasing continuously from a to d, y will increase continuously from 0 to x .
We have
b — a . d — c
~d — b . c — a'
b — d x — a
b — a x — d'
_d — a x — b
b — ax — d'
,2 _d — ax — c _
c — a x — d'
and, thence,
— y • — * = - A - • r — rr« >
c - a V Vc - a/ (x - d?
where A /( ------ ) is taken to be positive, and the sign of \f(X) is fixed as above. Then
V Vc - a/
for y between 0 and 1 or > , 2 , y . 1 — y . 1 — k2y will be positive, and *J(y . 1 — y . I — k2y)
will also be positive ; but y being between 1 and y^ , y.l—y.I— k*y will be negative,
K
and the sign of the radical is such that -r, — = - = - r^ is a positive multiple of i.
- y - 1 -
We have moreover
7 d — a , , ,. dx
CM/ = 7— — (d — b)-, -- = •
y 6 - a v '(fB^-dy'
and therefore
7
- 6 . c - a)
where *j(d — b.c— a) is positive ; or, say,
dy ., , . [ dx
= >J(d — b.c — ft) I
6—2
44 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. [716
Hence, writing y = zz = sn2 u, we have
\ r ^
2w = V(« — b.c — a)
i a
and it is to be further noticed that to
x = a, b, c, d,
correspond
sn u= 0, 1, T , oo ,
or we may say
u = 0, K, K+iK',
Writing for shortness
2
= «>
— b.'c — a)
we have
dx
and moreover
or if for a moment we write
r° dx
~WY\ = A> &c"
J« v(^)
then these equations are
aK = B -A,
Hence B+ C-2A = D-A, that is, A -B- C + D = 0, or B-A=D-C, that is,
where observe as before that x= a to x = b, or as = c to x= d, X is positive, and the
radical >J(X) is taken to be positive.
We have also
fb
oK = B- A =
716] AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. 45
where, as before, from b to c, X is negative, and the sign of the radical is such that
_ is a positive multiple of i; the last formula may be more conveniently written
,
dx
where, from b to c, — X is positive, and \J(—X} is also taken to be positive.
Collecting the results, we have
dx 2 7o b - a.d-c
• b . c — ct) ' d — b . c — a'
and also
d — b . c — a'
and then conversely
a (d — b) + d (b — a) sn2 u _
& 7~5 T~\ . " / 7 \ rt 1
or, what is the same thing,
b — d . x — a
sn2 u = , — , ,
b — a . x — d
d — a . x — b
b — a . x — d '
d — a . x — c
c — a . x — d '
en-5 u =
dn2 w =
where, in place of the elliptic functions we are to substitute their ^--values; it will
be recollected that $, the parameter of the ^--functions, has the value
dx fb dx
\ K J j b V(~~ X)
and, as before,
Hence, finally, a, ^, k', K, § denoting given functions of a, &, c, d, if as above
we have conversely
b — d. x — a
b — d . x — d
c-a.x-d
which are the formulae in question.
_ ,
— K
46 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. [716
The problem is to obtain them (and that in the more general case where a, b, c, d
have any given imaginary values) directly from the assumed equation
dx
= au,
and from the foregoing definition of the function S-.
It may be recalled that the function *bu is a doubly infinite product
u
TTiTT + (n + £) '
m and n positive or negative integers from — oo to +00; I purposely omit all further
explanations as to limits; or, what is the same thing,
iK'\ '
and consequently that, disregarding constant and exponential factors, the foregoing
expressions of
b — d.ac — a d — a.x — b d — a.x — c
b — a.x — d' b—a.x — d' c—a.x — d'
X Y Z
are the squares of the expressions ^., ™., ™-, where X, Y, Z, W are respectively of
the form
( i
Mnn + 7-- , nn
(m, n)} ' ( (m, n}\ '
— — \ I ) -i.AJ.j.\-i. i t „ . i j
m, n)j { (m, n)\
where (m, n) = 2mK + 2niK', and the stroke over the m or the n denotes that the
2m or the 2n (as the case may be) is to be changed into 2m + 1 or 2n -f 1. But
this is a transformation which has apparently no application to the ^--functions of
more than one variable.
717] 47
717.
ON THE TEIPLE THETA-FUNCTIONS.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 48 — 50.]
As a specimen of mathematical notation, viz. of the notation which appears to
me the easiest to read and also to print, I give the definition and demonstration of
the fundamental properties of the triple theta-functions.
Definition.
*(U, V, F) = Sexp. @,
where
® = (A, B, C, F, G, H) (I, m, n)* + 2(U, V, W)(l, m, n),
2 denoting the sum in regard to all positive and negative integer values from
— ao to +00 (zero included) of I, m, n respectively.
^- ( U, V, W) is considered as a function of the arguments ( U, V, W), and it
depends also on the parameters (A, B, C, F, G, H).
First Property. ^(U, V, F) = 0, for
U = %{x7ri + (A, H, G)(«, ft, 7)},
V = ^{yTri + (H, B,F)(*,0, 7)},
W=${z7ri + (G, F, G)(a, 0, 7)},
x, y, z, a, /3, 7 being any positive or negative integer numbers, such that ax + fty + yz
= odd number.
Demonstration. It is only necessary to show that to each term of S- there corre
sponds a second term, such that the indices of the two exponentials differ by an odd
multiple of iri.
48 ON THE TRIPLE THETA-FUNCTIONS. [71 7
Taking I, m, n as the integers which belong to the one term, those belonging to
the other term are
(where observe that one at least of the numbers a, /3, 7 being odd, this system of
values is not in any case identical with I, m, n). The two exponents then are
@, =(A, B, C, F, G, H)(l, m, n)*+2(U, V, W)(l, m, n),
and
©', = (4, B, C, F, G, H)(l + a, m + & n + y}*-2(U, V, W)(l + a, ml/3, w + 7);
viz. the value of ©' is
= (A, B, C, F, G, H}(1, m, iif + (A, B, C, F, G, #)(«, /3, 7)2
+ 2(4, B, C, F, G, H)(l, m, n)(a, 0, 7)
- 2 (U, V, W)(l + a, m+/3, n + 7),
and we then have
©'-0 = 2(4, B, C, F, G, H)(l, m, ?i) (a, /?, 7)
+ (4, B, C, F, G, H}(CL, /3, 7)2
-2(U, V, W)(2l+a, 2m + @, 2?i+7).
Substituting herein for U, V, W their values, the last term is
= - {(21 + a) x + (2m + (3)y + (2n + 7) z]
-2(4, B, C, F, G, H)(l, m, n)(«, & 7)
- (4, B, C, F, G, H}(a, 0, 7)2>
and thence
©' _ © = _ {(ft + a) x + (2m + /3) y + (2w + 7) ^} TTI,
which proves the theorem.
As to the notation, remark that, after (4, B, C, F, G, H) has been once written
out in full, we may instead of
(4, B, C, F, G, H}(1, m, n^, &c., write (4, ...)(«, m, r<)2, &c.,
and that we may use the like abbreviations
(4, ...)(l, m, n), to denote (4, H, G)(l, m, n) respectively,
(H, ...)(l, m, n), „ (H, B, F) (I, m, n)
(G, ...)(l, m, n), „ (G, F, C) (I, m, n)
These are not only abbreviations, but they make the formulas actually clearer, as
bringing them into a smaller compass; and I accordingly use them in the demonstration
which follows.
ON THE TRIPLE THETA-FUNCTIONS. 49
Second Property. If Ult F1; W^ denote
xiri+(A, H, £)(«, 13, 7),
(H, B, F)(a, /3, 7),
(Q, F, C) (a, /3, 7),
respectively, where x, y, z, a, 0, 7 are any positive or negative integers (zero values
admissible), then
.*f(U, V, W),
or say
= exp. {-(A, ...)(«, 13, 7)2}-exp. {- 2(
, F, F).
Demonstration. Writing *(#;, Flf TTO = 2 . exp. ®1; then in the expression of 0t
we may in place of I, m, n write I -a, m-0, n-y; we thus obtain
- a
which is
= (A, ...)(l, m, nf
yWt + (Jff, ...)(«, £ 7)]
+ (n -7) [W + m+(G, ...)(«, A 7)]}.
/, m, n) (a, & 7)
;, m, n)(a, /3, 7)
+ (^ .»)(«, A 7)2>
2(lU+mV+nW
which is
-(A, ...)(«, A 7)2-
+ 2 [(^ - a) a; + (m -/3)y+ (n - 7) 0] m.
Hence, rejecting the last line, which (as an even multiple of vn) leaves the exponential
unaltered, we see that *(Ult Flf W,) is =^(JT, F, TF) multiplied by the factor
. {-(A, ...)(«, /3, 7)2}.exp. {- 2 («^+ /9F+7F)},
which is the theorem in question.
In many cases a formula, which belongs to an indefinite number s of letters is
most easily intelligible when written out for three letters, but it is sometimes con
venient to speak of the s letters l,m,..., n, or even the s letters l,...,n, and to write
out the formulae accordingly.
C. XI.
50 [718
718.
ADDITION TO MR GENESE'S NOTE ON THE THEORY
OF ENVELOPES.
[From the Messenger of Mathematics, vol. vii. (1878), pp. 62, 63.]
THE example, although simple, is an instructive one. Introducing z, p for
homogeneity, the equation is
\2y (y — bz) + 2\/*xy + fj?x (x — az) = 0,
giving the envelope
xy [(x - az} (y - bz) - xy] = 0 ;
that is,
xy (bx + ay — abz) z = 0 ;
viz. we have thus the four lines
Writing these values successively in the equation of the curve, we find respectively
\-y (y - bz) = 0,
fj^ac (as — az) = 0,
(&X- a^l-0,
(\y + fjuK)2 = 0 ;
viz. in each case the equation in \, /j, has (as it should have) two equal roots; but
in the first three cases the values are constant ; viz. we find \ = 0, /u, = 0, 6\ — a/u, = 0,
dC \l
respectively ; and the curves x = 0, y = 0, - + r — z = 0, are for this reason not proper
envelopes.
718] ADDITION TO MR GENESE's NOTE ON THE THEORY OF ENVELOPES. 51
It is to be remarked that writing in the equation of the parabola these values
X = 0, /u = 0, b\-ap = 0 successively, we find respectively
x(x — az) = 0,
y(y- fa) = o,
(bx + ay) (&.',- + ay — abz) = 0 ;
viz. in each case the parabola reduces itself to a pair of lines, one of the given
lines and a line parallel thereto through the intersection of the other two lines; the
parabola thus becomes a curve having a dp on the line at infinity.
In the fourth case z = 0, the equation in X, /j, is (\y + /j,x)~ = 0, giving a variable
value \^-fju = — x-^y; hence 2 = 0, the line at infinity is a proper envelope.
The true geometrical result is that the envelope consists of the three points A, B, C,
and the line at infinity; a point qua curve of the order 0 and class 1 is not represent-
able by a single equation in point-coordinates, and hence the peculiarity in the form of
the analytical result.
7—2
52
[719
719.
SUGGESTION OF A MECHANICAL INTEGRATOR FOR THE
CALCULATION OF j(Xdx+Ydy) ALONG AN ARBITRARY
PATH'".
[From the Messenger of Mathematics, vol. vn. (1878), pp. 92—95 ; British Association
Report, 1877, pp. 18—20.]
I CONSIDER an integral f(Xdx+Ydy), where X, Y are each of them a given
function of the variables (as, y)- Xdx+Ydy is thus not in general an exact differential;
but assuming a relation between (x, y\ that is, a path of the integral, there is in
effect one variable only, and the integral becomes calculable. I wish to show how
for any given values of the functions X, Y, but for an arbitrary path, it is possible
to construct a mechanism for the calculation of the integral : viz. a mechanism such
that, a point D thereof being moved in a plane along a path chosen at pleasure, the
corresponding value of the integral shall be exhibited on a dial.
The mechanism (for convenience I speak of it as actually existing) consists of a
square block or inverted box, the upper horizontal face whereof is taken as the plane
of xy, the equations of its edges being y = 0, y=l, x=0> x = i respectively. In the
wall faces represented by these equations, we have the endless bands A, A', B, E'
respectively; and in the plane of xy, a driving point D, the coordinates of which' are
(x, y\ and a regulating point R, mechanically connected with D, in suchwise that
the coordinates of R are always the given functions X, Y of the coordinates of £f ;
the nature of the mechanical connexion will of course depend upon the particular
functions X, Y.
This being so, D drives the bands A and B in such manner that, to the given
motions dx, dy of D, correspond a motion dx of the band A and a motion dy of
* Bead at the British Association Meeting at Plymouth, August 20, 1877.
t It might be convenient to have as the coordinates of R, not X, Y 'but f, ,, determinate functions of
X, Y respectively.
719] SUGGESTION OF A MECHANICAL INTEGRATOR. 53
the band B; A drives A' with a velocity-ratio depending on the position of the
regulator R in suchwise that, the coordinates of R being X, Y, then to the motion
da; of A corresponds a motion Xdx of A'] and, similarly, B drives B' with a
velocity-ratio depending on the position of R, in suchwise that to the motion dy of
B corresponds a motion Ydy of B'. Hence, to the motions dx, dy of the driver D,
there correspond the motions Xdx and Ydy of the bands A' and B' respectively ;
the band A' drives a hand or index, and the band B' drives in the contrary sense
a graduated dial, the hand and dial rotating independently of each other about a
common centre ; the increased reading of the hand on the dial is thus = Xdx + Ydy ;
and supposing the original reading to be zero, and the driver D to be moved from its
original position along an arbitrary path to any other position whatever, the reading on
the dial will be the corresponding value of the integral I (Xdx + Ydy).
It is obvious that we might, by means of a combination of two such mechanisms,
calculate the value of an integral If (if) du along an arbitrary path of the complex
variable u, = x + iy ; in fact, writing f(x + iy) = P + iQ, the differential is
(P -(- iQ) (dx + idy), = Pdx - Qdy + i (Qdx + Pdy) ;
and we thus require the calculation of the two integrals
f(Pdx-Qdij) and ^(Qdx + Pdy),
each of which is an integral of the above form. Taking for the path a closed curve,
it would be very curious to see the machine giving a value zero or a value different
from zero, according as the path did not include or included within it a critical
point; it seems to me that this discontinuity would really exhibit itself without the
necessity of any change in the setting of the machine.
The ordinary modes of establishing a continuously-variable velocity-ratio between two
parts of a machine depend upon friction ; and, in particular, this is the case in Prof.
James Thomson's mechanical integrator — there is thus of course a limitation of the
driving power. It seems to me that a variable velocity-ratio, the variation of which is
practically although not strictly continuous, might be established by means of toothed
wheels (and so with unlimited driving power) in the following manner.
Consider a revolving wheel A, which by means of a link BC, pivoted to a point B
of the wheel A and a point C of a toothed wheel or arc D, communicates a reciprocating
motion to D; the extent of this reciprocating motion depending on the distance of B
from the centre of A, which distance, or say the half-throw, is assumed to be variable.
Here during a half-revolution of A, D moves in one direction, say upwards; and
during the other half-revolution of A, D moves in the other direction, say downwards ;
the extent of these equal and opposite motions varying with the throw. Suppose
then that D works a pinion E, the centre of which is not absolutely fixed but is so
connected with A that during the first half-revolution of A (or while D is moving
upwards), E is in gear with D, and during the second half-revolution of A, or while
54 SUGGESTION OF A MECHANICAL INTEGRATOR. [719
D is moving downwards, E is out of gear with D ; the continuous rotation of A
will communicate an intermittent rotation to E, in such manner nevertheless that, to
each entire revolution of A or rotation through the angle 2-rr, there will (the throw
remaining constant) correspond a rotation of E through the angle n.Zir, where the
coefficient n depends upon the throw*. And evidently if A be driven by a wheel
A', the angular velocity of which is - times that of A, then to a rotation of A'
2-7T
through each angle — , there will correspond an entire revolution of A, and therefore,
as before, a rotation of E through the determinate angle n . 2?r ; hence, X. being
sufficiently large to each increment of rotation of A', there corresponds in E an
increment of rotation which is ?i\ times the first-mentioned increment; viz. E moves
(intermittently and possibly also with some "loss of time" on E coming successively
in gear and out of gear with D, or in beats as explained) with an angular velocity
which is = n\ times the angular velocity of A'. And thus the throw (and therefore n)
being variable, the velocity-ratio n\ is also variable.
We may imagine the wheel A as carrying upon it a piece L sliding between guides,
which piece L carries the pivot B of the link EG, and works by a rack on a toothed
wheel a concentric with A, but capable of rotating independently thereof. Then if a
rotates along with A, as if forming one piece therewith, it will act as a clamp upon L,
keeping the distance of B from the centre of A, that is, the half-throw, constant ; whereas,
if a has given to it an angular velocity different from that of A, the effect will be to
vary the distance in question; that is, to vary the half-throw, and consequently the
velocity-ratio of A and E. And, in some such manner, substituting for A and E the
bands A and A' of the foregoing description, it might be possible to establish between
these bands the required variable velocity-ratio.
* If instead of the wheel or arc D with a reciprocating circular motion, we have a double rack D with a
reciprocating rectilinear motion, such that the wheel E is placed between the two racks, and is in gear on the
one side with one of them when the rack is moving upwards, and on the other side with the other of them
when the rack is moving downwards; then the continuous circular motion of A will communicate to £ a
continuous circular motion, not of course uniform, but such that to each entire revolution of A or rotation
through the angle 2ir, there will correspond a rotation of E through an angle n.lir as before. This is in
fact a mechanical arrangement made use of in a mangle, the double rack being there the follower instead of
the driver.
720] 55
720.
NOTE ON ARBOGAST'S METHOD OF DERIVATIONS.
[From the Messenger of Mathematics, vol. vn. (1878), p. 158.]
IT is an injustice to Arbogast to speak of his first method, as Arbogast's method*.
There is really nothing in this, it is the straightforward process of expanding
7 7Q 7«
by the differentiation of <f>u, writing a, b, c, d, ... in place of u, ^- , ^- , ^ , &c. or
say in place of u, u', u", u'", &c. respectively; thus
cfjft, (f>'a . b, ^ {fy'u ' c + <f)"a . b2}, ^ (4*'® • ^ ~^~ (f)"®' • ^
( +<f>"a.2bc + (f>'"a.b3
= £ {fia . d + <f>" a . 3bc + <f>'"a . b3}, &c.,
and in subsequent terms the number of additions necessary for obtaining the numerical
coefficients increases with great rapidity.
That which is specifically Arbogast's method, is his second method, viz. here the
coefficients of the successive powers of x in the expansion of <f> (a + bx + cx~ + da? +...),
are obtained by the rule of the last and the last but one ; thus we have
<j>a, <f)'a .b, <f>a.c + fi'a . ^b'2, <j>a.d + <f>"a . be + <j>'"a. ^b3, &c.,
where each numerical coefficient is found directly, without an addition in any case.
* See Messenger of Mathematics, vol. vn. (1878), pp. 142, 143.
56
[721
721.
FORMULA INVOLVING THE SEVENTH ROOTS OF UNITY.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 177 — 182.]
LET tw be an imaginary cube root of unity, &>2 + w + I = 0, or say w = |- {- 1 +i V(3)} ;
a3 = -7(l+3&>), /33 = - 7 (1 + 3<o2), values giving a3/33=343, and the cube roots a, ft
being such that a/3 = 7 ; then a + /3, = a + -, is a three-valued function (since changing
7\
the root &> we merely interchange a and -J ; and if r be an imaginary seventh root
of unity, then
3 (r + r6) = a + p-1,
3 (r4 + r3) = a>2a + a>/3 - 1.
Any one of these formulae gives the other two; for observe that we have «3 = -a/3(l + 3<w),
/33 = - a/3 (1 + 3<y2), that is, «2 = -/3(l +3w), /3- = -a(l + 3a>2); hence, starting for instance
with the first formula, we deduce
= - /3 (1 + 3(w) + 14 - a (1 + 3<y2) - 2a - 2/3 -f 1,
= - a (3 + 3a>2)-/3(3 + 3a>) + 15,
15,
that is,
3 (r- + r5) = too. + w2/3 - 1 ;
and in like manner by squaring each side of this we have the third formula
3 (r4 + r3) = &)2« + w/3 - 1.
721] FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. 57
The foregoing formulae apply to the combinations r + rs, r- + r\ r* + r3 of the seventh
roots of unity, but we may investigate the theory for the roots themselves r, r2, r3, r*, r5, r8.
These depend on the new radical V(- 7) or i V(7) ; introducing instead hereof X, Y,
where
then if
A3 = G + So) Z + (1 + 3&)2) F,
B* = 6 + 3&>2Z + (1 + 3o> ) Y,
where
we have (Lagrange, Equations Numeriques, p. 294),
3r = Z + 4 + 5.
I found that, in order to bring this into connexion with the foregoing formula,
3 (r + ?•«) = a + £ - 1, where as before a3 = - 7 (1 + 3a>), /33 = - 7 (1 + Sty2), a/3 = 7, it is
necessary that B, A should be linear multiples of a, j3 respectively, the coefficients
being rational functions of co, X ; and that the actual relations are
A = [a + o> + X (3 + 2<w)} ;
in verification of which, it may be remarked that these equations give
= {(20- *> - o>2) + Z (17 - 4o> - 4»») + Z2(3 - 4o> -
viz. in virtue of the equation w2 + &> + 1 = 0, the term in { } is =21
= 7(Z2 + 3Z + 3), or since Z2 + X + 2 = 0, this is =7(2Z+1), =7iV(7); the equation
thus is 1AB = a/3 . t'VCO, which is true in virtue of AB = i^/(tf) and a/3 = 7. The same
relations may also be written
- a = 5 (w2 + X),
- /3 = A (ai + Z).
I found in the first instance
3r = X + A + B,
8r»--l_X+ A(a>"-- X) + B(a>- X),
3r- = X + co-A + w B,
3?-r' = - 1 - X + A (o> - &)2Z) + B (w2 - «w Z),
3r« = Z + wA + tfB,
3?-3 = -l-Z+ ^1(1 - o>Z) + B (1 - ft,2Z),
C. XL
58 FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. [721
which in fact gave the foregoing formulae
3 (r + r6) = - 1 + a + j3,
3 (r2 + ^) = _ l + toa + 0,2/3,
3 (r4 + r3) = - 1 + w"a + o)/3.
But there is a want of symmetry in these expressions for r, r", &c., inasmuch as the
values of r, r2, r4 are of a different form from those of r6, r5, r3; to obtain the proper
forms, we must for A, B substitute their values in terms of a, /3, and we thus obtain
3r = X + ^{ 4- a> + X( l-2«)}+£{ 5+ o> + X ( 3 +
o> + X - 3 -
1 {- 3 - 2o> +
3r4 = X + y {- 5 - 4ft) + X (- 3 - to)} + ^ {- 1 + 4o, + X (- 2 + &>)},
3r^ = -l-Z + |{-2-3a,4-X( 3+ »)} + f { l + 3w + Z( 2- «)} ;
viz. each of the imaginary seventh roots is thus expressed as a linear function of the
cubic radicals a, /3 (involving &> under the radical signs) with coefficients which are
functions of &>, -X".
Recollecting the equations a2 = - £ (1 + 3a>), /32 = - a (1 + 3&>2), a/3 = 7 ; &>2 + a> + 1 = 0,
X2 + X + 2 = 0; it is clear that, starting for instance from the equation for 3r, and
squaring each side of the equation, we should, after proper reductions, obtain for 9r2
an expression of the like form ; viz. we thus in fact obtain the expression for 3r2 ;
then from the expressions of 3r and 3r2, multiplying together and reducing, we should
obtain the expression for Sr3 ; and so on ; viz. from any one of the six equations we
can in this manner obtain the remaining five equations.
At the time of writing what precedes I did not recollect Jacobi's paper "Ueber
die Kreistheilung und ihre Anwendung auf die Zahlentheorie," Berliner Monatsber.,
(1837) and Crelle, t. xxx. (1846), pp. 166—182 ; [Ges. Werke, t. vi. pp. 254—274]. The
yjp _ \
starting-point is the following theorem : if x be a root of the equation - — =- = 0,
CC ~~ J.
p a prime number, and if g is a prime root of p, and
F (a) = x + axv + aW* + . . . + a^-1 x°"~~,
a _
where a is any root of — y- = 0, we have
F(am) F(an) = ^ (a)
721] FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. 59
where ^ (a) is a rational and integral function of a with integral coefficients ; or, what
is the same thing, if a and ft be any two roots of the above-mentioned equation, then
F(a)F(/3) = + (*, ft) F (aft),
where ^(«, ft) is a rational and integral function of a, ft with integral coefficients.
As regards the proof of this, it may be remarked that, writing a? for x, F (a), F(ft),
and F(aft) become respectively arlF(a), ft^F(ft), (aft)~lF (aft) • hence, F(a)F(ft) + F(aft)
remains unaltered, and it thus appears that the function in question is expressible
rationally in terms of the adjoint quantities a and ft. With this explanation the
following extract will be easily intelligible :
" The true form (never yet given) of the roots of the equation x? — 1 = 0 is as
follows : The roots, as is known, can easily be expressed by mere addition of the
functions F(a). If X is a factor of p—l and aA = 1, then it is further known that
{JT(a)}A is a mere function of a. But it is only necessary to know those values of
F(a) for which \ is the power of a prime number. For suppose XX'X"... is a factor
of p — l', further let X, X', X", ... be powers of different prime numbers, and a, a', a", . . .
prime Xth, X'th, X"th, ... roots of unity, then
.. — — „
^(a, a , a , ...)
where ^(a, a', a",...) denotes a rational and integral function of a, a', a",... with
integral coefficients. Hence, considering always the (p — l)th roots of unity as given,
there are contained in the expression for x only radicals, the exponents of which are
powers of prime numbers, and products of such radicals. But if X is a power of a
prime number, = (Mn, suppose, the corresponding function F(a.) can be found as follows :
Assume
then
*» = <X{*i(«) *,(«) ...+*-•(•
. . . ^ (a^) F (a*)},
and so on, up to
P-I
a*1""1) +* («M""1) • • • *M-I («O (-) " Pi"
so that the formnlffi contain ultimately /ith roots only. It is remarked in a foot
note that, when n=l, the p— 1 functions can always be reduced to one-sixth part in
number, and that by an induction continued as far as /m — 31, Jacobi had found that
all the functions -v|r could be expressed by means of the values of a single one of
these functions.
" The p — I functions determine, not only the values of all the magnitudes under
the radical signs, but also the mutual dependence of the radicals themselves. For
replacing a. by the different powers of a, one can by means of the values so obtained
for these functions rationally express all the fin—l functions F (a1) by means of the
powers of F(a); since all the /j,n — I magnitudes [F(a)}i -4- F(o.{) are each of them
8—2
60 FORMULA INVOLVING THE SEVENTH ROOTS OF UNITY. [721
equal to a product of several of the functions ty (a). Herein consists one of the great
advantages of the method over that of Gauss, since in this the discovery of the
mutual dependency of the different radicals requires a special investigation, which, on
account of its laboriousness, is scarcely practicable for even small primes ; whereas the
introduction of the functions ty gives simultaneously the quantities under the radical
signs, and the mutual dependency of the radicals. The formation of the functions ty
is obtained by a very simple algorithm, which requires only that one should, from the
table for the residues of gm, form another table giving gm' = 1 + gm (mod. p), [see
Table IV. of the Memoir]. According to these rules one of my auditors [Rosenhain]
in a Prize-Essay of the [Berlin] Academy has completely solved the equations xp — 1 = 0
for all the prime numbers p up to 103."
I am endeavouring to procure the Prize-Essay just referred to. As an example —
which however is too simple a one to fully bring out Jacobi's method, and its difference
from that of Gauss — consider the equation for the fifth roots of unity, x4 + xs + x2+x + \ =0.
According to Gauss, we have x 4- .*4 and x2 + a3, the roots of the equation u? + u — 1 = 0 ;
say x + x* = % {— 1 + V(5)}, x2 + a? — ^ {— 1 — V(5)}- The first of these, combined with
x.x* = l, gives x-x* = V[- | {5 + \/(5)j] ; and thence 4# = - 1 + V(5) + V[- 2 {5 + V(5)}] ;
if from the second of them, combined with o?.x3 = l, we were in like manner to obtain
the values of a? and a?, it would be necessary to investigate the signs to be given
to the radicals, in order that the values so obtained for x2 and a? might be consistent
with the value just found for x. For the Jacobian process, observing that a prime
fourth root of unity is a = i, and writing for shortness F1} F2, F3, F4 to denote F(a),
), F(a.3), F(a.*) respectively, these functions are
Fl = x - x* + i (#2 - x3),
viz. we have F, = -l, F/=5, or say #, = V(5), Fi> = - (I + 2i) F,, = - (1 + 2i) V(5) ; and
similarly F<?= - (1 - 2i) F2> = - (1 - 2*) V(5) ; but also F1F3= — o, so that the values
F1 = ^/{-(l + 2t)V(5)}, F3 = V{-(1- 2t) V(5)}, must be taken consistently with this last
equation FtFs= V(5). The values of Flf F2, F3, Ft being thus known, the four equations
then give simultaneously x, x4, x2, x3, these values being of course consistent with each
other. It may be remarked that the form in which x presents itself is
with the before-mentioned condition as to the last two radicals; with this condition
we, in fact, have
V{- (1 + 2t) V(5)} + VI- (1 - 2i) V(5
as is at once verified by squaring the two sides.
722]
61
722.
A PEOBLEM IN PARTITIONS.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 187, 188.]
TAKE for instance 6 letters ; a partition into 3's, such as abc . def contains the 6
duads ab, ac, be, de, df, ef. A partition into 2's such as ab.cd .ef contains the 3
duads ab, cd, ef. Hence if there are a partitions into 3's, and /3 partitions into 2's,
and these contain all the duads each once and only once, 6a + 3/3 = 15, or 2a + /3=5.
The solutions of this last equation are (a = 0, /3=5), (a = l, /3 = 3), (a = 2, /3 = 1), and
it is at once seen that the first two sets give solutions of the partition problem, but that
the third set gives no solution; thus we have
x = 0, /3 = 5
a = 1, /3 = 3
ab . cd . ef
abc . def
ac .be . df
ad .be .cf
ad . bf. ce
ae . bf. cd
ae .bd .cf
af. bd . ce.
af. be . de
Similarly for any other number of letters, for instance 15 ; if we have a partitions
into 5's and yS partitions into 3's, then, if these contain all the duads, 4a + 2/3 = 14,
or what is the same 2a + (3 = 7 ; if a = 0, j3 = 7, the partition problem can be solved (this
is in fact the problem of the 15 school-girls) : but can it be solved for any other values
(and if so which values) of a, /3 ? Or again for 30 letters ; if we have a partitions into
5's, /3 partitions into 3's and 7 partitions into 2's ; then, if these contain all the duads,
4a + 2/3 + 7 = 29 ; and the question is for what values of a, /3, 7, does the partition-
problem admit of solution.
62 A PROBLEM IN PARTITIONS. [722
The question is important from its connexion with the theory of groups, but it
seems to be a very difficult one.
I take the opportunity of mentioning the following theorem : two non-commutative
symbols a, /3, which are such that fia = a?/32 cannot give rise to a group made up of
symbols of the form a*/S«. In fact, the assumed relation gives /Sa2 = a2£a2/32 ; and
hence, if /3a2 be of the form in question, =a?fty suppose, we have
= a- .
that is, l=«2/32, and thence /3a=l, that is, p = orl, viz. the symbols are commutative,
and the only group is that made up of the powers of a.
723] 63
723.
VARIOUS NOTES.
[From the Messenger of Mathematics, vol. viu. (1879), pp. 45—46, 126, 127.]
An Algebraical Identity: p. 45.
Let a, b, c, f, g, h be the differences of four quantities a, /3, y, 8, say
a, b, c, f, g, h = 0-y, y-a, a-/3, a - 8, /3 - 8, y - 8 ;
then
h — g + a = 0,
-h
— a — b — c . = 0.
Now Cauchy's identity
(a + 6)7 - a7 - 67 = 7a6 (a + b) (a2 + ai + ft2)3,
putting therein a + 6 = — c, becomes
a7 + b7 + c7 = lobe ( be + ca + ab)~ ;
hence we have
A7 - g7 + a7 = - 7agh (- ga + ah - hg)z,
+ b7 = - nhf (- hb + bf -fh)\
+ C7 = _ 7cfg (_/C + Cg - gff,
whence, adding,
(- ga + ah- hg)* + bhf(- hb + bf-fhf + cfg (-/c + cg- gff + abc (be + ca + ab)- = 0,
G4 VARIOUS NOTES. [723
or, as this may also be written,
agh (g* + A2 + a-')2 + bhf(h- +/2 + 62)2 + cfg (/2 + g* + c2)2 + abc (a2 + 62 + c2)2 = 0,
an identity if a, b, c, f, g, h denote their values in terms of a, ft, 7, 8.
Note on a Definite Integral : p. 126.
The integral
i
* frafdx
used by Weierstrass, is at once seen to be =K — E ; but the proof that the other integral
_
~
is = £" is not so immediate.
We have
d
and thence
viz. replacing the numerator by
-£+|a -**•>•.
this becomes
_^2 n rfy i
^2 Jo (l-2/2)*(l-%2)§ A?
that is,
f1
Jo (l-
or, writing k' for /;,
T- -^- =^'
Jo (1-/)*(1-^V)? k~
The integral J' writing therein x = -77^ -- ^7— - becomes
J' = k- i
' o
viz. its value is thus =E'.
1 dy
723] VARIOUS NOTES. 65
On a Formula in Elliptic Functions: p. 127.
cn ?/
Writing enM=B(jnM« tnen the formulae p. 63 of my Elliptic Functions give
sn(u , V\ = T~T> n(u •=& + &_.
and, substituting for T7, T', B, B', and (7, C" their values, we obtain
, sn u en ?; + sn v en M
sn(w + w) = ,-,-rr—
1 + A;- sn w en it sn v en v
, . en w en v — sn w sn v
en (w + v) = TT-
1 — & sn w en u sn v en v
formulae which, as regards their numerators, correspond precisely with the formula?,
sin (u- + v} = sin u cos v + sin v cos u
and
cos (u + v) = cos w cos v — sin w sin v,
of the circular functions, and which in fact reduce themselves to these on putting k = 0.
The foregoing formula?, putting therein &2 = -l, are the formula} given by Gauss,
Werke, t. in., p. 404, for the lemniscate functions sin lemn (a ±b) and cos lemn (a ± 6) ;
where it is to be observed that these notations do not represent a sine and a cosine,
but they are related as the sn and en, viz. that
cos lemn a = V(l — sin lemn2 a) -4- \/(l + sin lemn2 a).
C. XI.
66 [724
724
[From the Messenger of Mathematics, vol. vin. (1879), pp. 51, 52.]
THE following is a solution of Mr Greenhill's problem set in the Senate-House
Examination, January 14, 1878.
"Prove that, if a model of a hyperboloid of one sheet be constructed of rods
representing the generating lines, jointed at the points of crossing; then if the model
be deformed it will assume the form of a confocal hyperboloid, and prove that the
trajectory of a point on the model will be orthogonal to the system of confocal
hyperboloids."
Let (#j, y1} zj, (x.2, ?/2, z3) be points on the generating line of
••
— 4-2- — - -1
a2 T 62 c2
then
22°
a2 6- c2
/ - / / ~ " ~
a2 + T2 " "c2 =
4. 2 _ - 1 •
a2 " 62"~ c2 "
or, what is the same thing, if
^ ^ ?}- ^ 2/2 ^
a' b ' c ~^' 5l> ° a' 6 ' c
then
Pi + qi2 - r-c = 1,
724] ON THE DEFORMATION OF A MODEL OF A HYPERBOLOID. 67
Similarly, if (ft, r^l, ^), (£>, rj.,, £2) be points on generating line of
e+£-f.i
**& 72~
and if
& ^ &-D a r- ?2 ^ £?-D o r-
a' £' 7~P" q" '' a' £' 7~P2' q2' r2'
then
Pi2 + qi2 - I? = 1,
p22 + q22 - r22 = 1,
Pip2 + qiqa - ^r,, = 1.
Hence if (ss1, yl, Zj), (ft, ^l, £) be corresponding points on the two surfaces, that
is, if
*i yi *i_ft *?i. Si
i» Tf F «• 7P-71 ~^' 3l> n>
and similarly, if («?2, 7/2, ^2), (^, %, ^2) are corresponding points, that is, if
5? y» !»_& 5? ??_ a r.
a' bj c~ «' /3' y"1 ?2' 2'
then we have, as before, the system of three equations
Pi" + qi' - n2 = 1,
p** + qs - ^ = 1,
Pipa + qiqi-rfa = 1.
Then if the two surfaces are confocal, that is, if
«2> @~, — 72 = a2 + h, b- + k, — c- + h,
we shall have
(a?, - ^2)2 + (y, - ytf + (z, - ztf = (ft - ft)8 + (ih - ^ + (?i ~ r.)2-
For this equation is
«2 (Pi ~ Ptf + & (q, - q,)" + C- (r, - r.,)- = a2 (p, -p,)- + fr (q, - q,)* + ^ (TI - r2)2,
that is,
(pi -ptf + (qi - ^2)2 - (n - r2)2 = 0,
an equation which is obviously true in virtue of the above system of three equations.
Hence, if on confocal surfaces
we take two points P1} P2 on the first, and Qlt Q2 the corresponding points on the
second ; then P1? P, being on a generating line of the first surface, Qi, Q» will be
on a generating line of the second surface, and PjPa will be = Q^. The same
is evidently true for the quadrilaterals P1P,P3P^ and Q^QsQ* where P^,, P,P3>
, P*Pi are generating lines on the first surface : and therefore Q&, Q,Q3, Q3Q^,
are generating lines on the second surface, which proves the theorem.
9—2
68
725.
NEW FORMULAE FOR THE INTEGRATION OF
[From the Messenger of Mathematics, vol. vm. (1879), pp. 60 — 62.]
I HAVE found in regard to the differential equation
das dy
*J(a-x.b-x.c-x.d-x) \f(a-y.b-y.c-y.d-y)
= 0,
a system of formulae analogous to those given, p. 63, of my Treatise on Elliptic
Functions, for the values of sn (u + v), en (u + v), dn (u + v). Writing for shortness
a, b, c, d = a — x, b — x, c — x, d — x,
a,, blt c,, dj = a-y, b-y, c-y, d-y,
and (be, ad) to denote the determinant
1 , « + y, xy
1, b + c, be
1, a + d, ad
and (cd, ab), (bd, ac) to denote the like determinants; then the formulae are
fa — z \ \/(a — b.a — c) { VCadbjCj) +
/(a-z\-
V\d-z)
(be, ad)
(« — b . a - c) (x — y)
(a - b . a - c)
i) +
(o - b . a - c) { V(acb1d1)
dx dv
725] NEW FORMULAE FOR THE INTEGRATION OF -.^+ ^=0. 69
1 ,1 *
//" — * \ _ V ^
V U - f) ~
J 1
~ c) VCbdbA) + (b -d) V(acaA)}
(a — c) x/Cbdbidj) — (b —
a - d
(a — 6)
/ IB *
/ / v * i
V U - ,87 =
) - (6 - c> V(ada1d1)}
(a — c) VObdbjdi) — (b — d) ^(
/(a-c\n,
A / ~ j ("», ac)
V \a-d'^ '
_ _
(a — 6) V(cdcjdi) — (c — c?)V(aba1b1) '
The twelve equations are equivalent to each other, each giving z as one and the
same function of x, y ; and regarding z as a constant of integration, any one of the
equations is a form of the integral of the proposed differential equation.
Writing in the formulae x — a, b, c, d successively, the formulae become
x = a, x = b, x = c, x =d,
a — z &!
c —a I
>j 6 — a Cj
a — 6 . a — c QI
d-z dx '
d — b(.
>i ' d — c bx '
d — b.d — c &i
b — z bj
c — b i
ix 6 — a . b — c dj
a — b Cj
d — z dj '
rf — d (
3j ' d — a . d — c bj '
Oy ^~ C BJJ
C "~~ Z Cj
c — a . c — b c
1, 6 - c a,
a — c b:
d — z d/ d—a.d — bc^ d — a bj ' d — 6 aj '
viz. in the first case we have 2 = y, and in each of the other cases z equal to a
linear function - ^ of v.
w + s
Cambridge, July 3, 1878.
70 [726
726.
A FORMULA BY GAUSS FOR THE CALCULATION OF LOG 2
AND CERTAIN OTHER LOGARITHMS.
[From the Messenger of Mathematics, vol. vm. (1879), pp. 125, 126.]
GAUSS has given, Werke, t. IL, p. 501, a formula which is in effect as follows:
/1025V /1048576V /6560\3 /15624V /9801V
O196 — 1 A59 I II _ I I -- I I - ] I -- I
V1024J U048575/ \Q5Ql) U5625/ \9800J '
viz. this is
2*. 32. 7.31V / 34.112 V
2.3.11.31.4l
where on the right-hand side the several prime factors have the indices following, viz.
2, index is (59 + 160 + 15 + 24- 50- 12) = 196,
3 „ (16+16-8-24 ) = 0,
5 „ (59+10+3-16-48- 8) = 0,
7 „ (8-8 ) = 0,
11 „ (8-8 ) = 0,
31 „ ( 8- 8 ) = 0,
41 „ ( 5+ 3- 8 ) = 0,
or the right-hand side is = 2196 as it should be. The value of log 2 calculated from
2196=1039 is log 2 = ^- = '301020, viz. there is an error of a unit in fifth place of
decimals. The actual value of 2196 has been given me by Mr Glaisher :
2:96= 10043 36277 66186 89222 13726 30771
32266 26576 37687 11142 45522 06336.*
Supposing log 2 calculated by the form, we then have
41 - ({£f£) 212^ 102, giving log 41,
and
38 =10. MM- 24. 41, giving log 3;
and formulae may be obtained proper for the calculation of the logarithms of ty-, 11 . 31,
and 7.31.
* The value was deduced by Mr Glaisher from Mr Shanks's value of 2193 in his Rectification of the Circle,
(1853), p. 90.
727] 71
727.
EQUATION OF THE WAVE-SURFACE IN ELLIPTIC
COORDINATES.
[From the Messenger of Mathematics, vol. viu. (1879), pp. 190, 191.]
THE equation of the wave-surface
ax2 _ by2 cz2 =
' o i . .o i ».o 7, o i _ •> i ^«» J
a? -|- #2 + z* -a a? + y2 + z'2-b x- + y- + z
when transformed to coordinates p, q, r, such that
of- V* z-
i y. i
— a + p — b+p — c+p
x- y2 z'2
-i r \- - = 1»
— a + q — b + q —c + q
x~ y- z2
| t7 I . — J •
— a + r'—b+r — c+ r
x2 y2 z~
(that is, to the elliptic coordinates belonging to the quadric surface - -\ r + =1),
— ' Q — • C/ "~ 0
assumes the form
(q+ r— a — b— c)(r +p — a — b —c}(p +q — a — b — c) = 0,
(Senate-House Problem, January 14, 1879).
In fact, p, q, r are the roots of the equation
-a + u —b+u —c + u
we have therefore
(u-p) (u-q}(u- r) = (M - a) (u - b) (u - c)
- a;* (« - 6) (t< -c)-if (u - c) (u - a) - 22 (jt - a) (u - 6) ;
72 EQUATION OF THE WAVE-SURFACE IN ELLIPTIC COORDINATES. [727
whence, writing for shortness
A = a + b + c , P = p + q + r,
B = be + ca + ab, Q = qr + rp +pq,
C = abc , R = pqr,
we have
x- + y*+ z* = P-A,
(b + c) x* + (c + a) y- + (a + b)z- = Q - B,
bca? + cay- + abz~ — R — C,
and thence also
a(b+c)xn- + b(c+a)f + c(a+b)z2=B(P-A)-(R-C),
aa? + bf + cz- = A(P-A}-(Q-B).
The equation of the wave-surface is
ale - {a (b+c)tf + b(c + a) if + c (a + b) z-\ + (x- + y- + z-} (ax2 + by- + cz-) = 0.
By the formulae just obtained, this is
C-[B(P-A)-(R-C)] + (P-A)[A(P-A)-(Q-B)] = 0,
that is,
A3 - 2A-P + A(P«- + Q)-(PQ-R) = 0,
that is,
{A -(q + r)} [A -(r + p)} [A-(p + g) =0,
or, substituting for A its value a + b + c, and reversing the sign of each factor, we
have the formula in question.
It is easy to see that, taking a, b, c to be each positive, (a>b> c), and assuming
also p > q > r, we obtain the different real points of space by giving to these
coordinates respectively the different real values from oo to a, a to b, and b to c
respectively. Hence
greatest, least value, is
q + r, a + b, a + c,
r +p, oc , a + c,
p + q, x , a + b,
so that r+p, p + q, may be either of them = a + b + c, but q+r cannot be = a + b + c,
that is, q + r = a + b + c does not belong to any real point on the wave-surface. We
can only have r + p and p + q each = a + b + c, if p = a + c, q = r = b, and these values
belong as is easily shown to the nodes on the wave-surface ; hence, the equations
r + p = a + b + c and p +q = a + b + c being satisfied simultaneously only at the nodes
of the surface, must belong to the two sheets respectively. And it can be shown
that p + r = a + b + c belongs to the external sheet, and p + q = a + b + c belongs to the
internal sheet. In fact, for the point (0, 0, ^a), which is on the external sheet, we
have p = a + c, q = a, r = b, and therefore p+r=a+b + c: for the point (0, 0,
which is on* the internal sheet, either
(p = b + c, q = a, r = b) or (p = a, q = b + c, r — c),
according as b + c> a or b + c < a : but in each case
728] 73
728.
A THEOREM IN ELLIPTIC FUNCTIONS.
[From ,the Proceedings of the London Mathematical Society, vol. x. (1879), pp. 48 _ 48.
Read January 8, 1879.]
THE theorem is as follows :
If u + v + r + s = 0, then
1 k'2
— k 2 sn u sn v sn r sn s + en u en v en r en s — r dn u dn v dn r dn s = --
&2 #'
It is easy to see that, if a linear relation exists between the three products, then
it must be this relation: for the relation must be satisfied on writing therein
v=—u, s = — r, and the only linear relation connecting sn2 u sn2 r, en2 u en2 r, dn2 u dn2 r
is the relation in question
— k'2 sn2 u sn2 r + en2 u en2 r — j- dn- w dn'J r = — — .
# &-
A demonstration of the theorem was recently communicated to me by Mr Glaisher ;
and this led me to the somewhat more general theorem
- A/2 sn (a + /3) sn (a - /S) sn (7 + 8) sn (7 - 8)
+ en (a + @) en (a - y9) en (7 + 8) en (7 - 8}
- jp dri (a + /3) dn (a - £) dn (7 + S) do (7 - 8)
_k^ ^k'2 (sn2 a - sn2 7) (sn2 /3 - sn2 8)
~
A-2 1 - k2 sn2 a sn2 £ . 1 - &2 sn2 7 sn2 8 '
C. XT. 10
74 A THEOREM IN ELLIPTIC FUNCTIONS. [728
In fact, writing herein « + 7 = 0, that is, y = — ct, the right-hand side becomes =0;
-and the arcs on the left-hand side are a. + @, a. — /3, — a + S, — a — 8, which represent
any four arcs the sum of which is = 0.
Writing in the last-mentioned equation x, y, z, w for the sn's of a, /3, 7, &
respectively, also
Q = 1 - x2 - y2 + k2x2y2, Ql = 1 - z* - w2 + k2z2w2,
R = l-k2x2- k2y2 + k2x'~y2, Rl = l- k2z2 - k2w2 + k2z2w2,
D = l- k2x2y2, A = 1 ~ k2z2w2,
the equation is
% RRl_ k'2 2k'*(x*-z*)(y*-w*)
_
, DD, k2 DD,~ k2 DD,
that is,
- ^PPX + QQl - RR^ + DA + 2^'2 (a- - z*) (f - w2) = 0.
It is easy to verify that the terms of the orders 0, 1, 2, 3 and 4 in a;2, y2, z2, w-
separately destroy each other; for instance, for the terms of the order 2, we have
- k'2 (x2 - y2} (z- - w-) + {(a? + y2) (z- + w2) + k2 (x2y2 + z*w2)}
'
f
+ ^ |_ fc (x2y2 + zhu2)} + 2k'2 (x2 - z2) (y2 - w2) = 0,
fC
that is,
- k'2 (x2 - y2) (z2 -w2) + (l- k2) (x2 + y2) (z2 + w2)
+ (k2 -I- k'2) (x2if + z-w2) + 2k'2 (x2 - z2) (y2 - w2} = 0 ;
or, omitting the factor k'2, this is
- (x2 - y2) (z2 - w2) + (x2 + y2) (z2 + w2) - 2 (x2y2 + z-w2} + 2 (x2 - z2) (y2 - w2} = 0,
-as it should be.
The theorem in its original form was obtained by me as follows : using the elliptic
coordinates p, q, r, such that
y2 jL
« 1 1 „ — *- '
a +p b + p c +p
x2 y2 z2
j_ «7 I J
a + q b +q c + q
x2 y2 z2
- + rir— +r ~ = 1;
a+ r b + r c 4- r
728] A THEOREM IN ELLIPTIC FUNCTIONS. 75
or, what is the same thing,
— Pyx2 = a+p.a+q.a + r,
— yay* == b -if-p .b + q .b +r,
— aftz- = c +p .c+q.c+r,
where a, @, 7 denote b — c, c — a, a — b respectively ; then, treating r as a constant,
the coordinates x, y, z will belong to a point on the ellipsoid
a8 v2 &
• h r2 h — = 1,
a + r b +r c + r
and the differential equation of the right lines upon this surface is
dp dq
va+p.b+p.c + 2) Va + q. b + q.c + q
Take #0, y0, z0 the coordinates of a point on the surface, and p0, q0 the corresponding
values of p, q, so that
— ftyx0- = a + p0 . a + q0 . a + r,
— 7«2/o2 = b + p0 . b + qQ .b +r,
— aftz,? = c + p0 . c + q0 . c + r,
then the equation of the tangent plane at the point (ac0, y0, z0) is
— + --^- — \- — • — 1
a + r b + r c + r
or, substituting for #2, ar02, &c., their values, we have
0
. a + q0, &c.,
CL ~r~ *?*
and consequently the equation of the tangent plane is
.a + q.a+p0.a + q0 + @\/b+p.b + q.b+pQ.
the equation of a plane intersecting the ellipsoid in a pair of lines ; hence this
equation (containing in appearance the two arbitrary constants p0 and q0) is the integral
of the proposed differential equation.
Writing
sn2 u = A (a + p), crfu = B(b+p), dn?u = C(c+p),
the values of A, B, C, k are determined ; and, assuming for q, p0, q0 the like forms
with the arguments v, u0, v0, the differential equation becomes du=dv, having the
10—2
76 A THEOREM IN ELLIPTIC FUNCTIONS. [728
integral u — i(0 = v — v0', while the foregoing integral equation, on reducing the constant
coefficients contained therein, takes the form
— A;'2 sn u sn v sn u0 sn v0
+ en u en v en u0 en v0
— j dn u dn v dn u0 dn v0
_
A;2'
viz. this equation holds good if U — UO = V — VQ. And by a change of signs we have
the theorem.
If, as above, u + v + r+s = Q, the theorem gives a linear relation between the
three products sn u sn v sn r sn s, cnucnvcnr en s, dn u dnv dn r dn s, and regarding at
pleasure the sn's, the en's, or the dn's as rational, one of these products will be
rational while the other two will be each of them a quadric radical; and hence,
rationalising, we obtain an equation which contains the product in question linearly,
and contains besides only the squares of the sn's, en's, or dn's; that is, we have
three such equations containing the three products respectively. Bringing to one side
the terms which contain the product, and again squaring, we obtain an equation
involving only the squares of the sn's, en's, or dn's; but the three equations thus
obtained represent, it is clear, one and the same rational equation, which may be
expressed as an equation between the squares of the sn's, or of the en's, or of the
dn's, at pleasure. This equation may be obtained, as I will show, from the ordinary
addition-equations of the elliptic functions, but it is not obvious how to obtain from
them the three equations involving the products respectively, and these last have the
advantage of being of a degree which is the half of the equation which involves
only the squared functions.
Write x, y, z, w for sn u, sn v, sn r, sn s respectively ; then, writing
A = x Vl - f. 1 - &y, a = z Ji—
A ' = y Vl - «*Ti ~- k2a?, a = w V 1 '- z2 . 1 - fcz2,
P = x- — y-, TO = z- — w2,
D = 1 -
we have
sn (u + v) = - sn (r + s),
that is,
A+A' P a + a'
D A-A'~ S a -a.'
and consequently
£CT = _ (a _ a!) (A + A'),
P8 = - (a + a') (A - A') •
whence
728] A THEOREM IN ELLIPTIC FUNCTIONS. 77
that is, (** - w2) (1 - k2x2y2) - (& - 2/2) (1 - tfzW)
= 2 [xw Vl - y* . 1 - fcyTl - *2 . 1 - ^2 - ys Vl - #* . 1 - &V . 1 - w2 . 1 - Aftw*}.
Rationalising, we obtain, as mentioned above, an equation containing only the squares
x*, y2, z~, w2 ; it therefore is of a degree twice that of the equation containing
the product xyzw. I worked out in this way the equation in (#*, y2, z2, w2), but the
calculation was lost, and the easier way of obtaining it is obviously by means of the
equation involving xyzw.
We have, by the theorem,
— k'2 xyzw
K
that is,
k'2 (1 - k2xyzw) = k- Vl - a? . 1 - y2 . 1 - ^2 . 1 - w2
- Vl - k2x? . l~-k2y2 . 1 - k2z2 . 1 - k2w2 ;
and then, writing
71 ") i o i
p = x- + y2 +
Z? rffinflyfi _L r,
S\j J; U 6 T **
and using V^ to denote the rational function xyzw, we have
- 2k2 \(l-
or, if for a moment the radical is called VA, then the factor k2 divides out, and
the equation becomes
2 VA = 2 - (1 + &2) P + 2k2Q - (k2 + &4) R
whence
4 ( i _ p + Q - R + S ) ( 1 - k2P + k* Q - k«R
_ (2 - (1 + fc2) P + 2k2Q - (k2 + k4)R+ 2k*S\2 - 4>k'sS
= -2k*^S{2-(I+ k2) P + 2k2Q - (k2 + k*) R
The factor k'4 divides out; omitting it, we have
4Q _ p* _ 4 (i + k2) R + Wk*S + 2k2PR -4<(k2 + k4) PS - k*R2
= - 2 V# {2 - (1 + k2) P + 2k2Q - (k2
or, as this may also be written,
k2 (- 4>R + 2PR + US - 4PS) -f ^ (- E2 + 4QS - P>Sf)}
which is the required rational equation involving the product of the sn's.
78 [729
729.
ON A THEOREM RELATING TO CONFORMABLE FIGURES.
[From the Proceedings of the London Mathematical Society, vol. x. (1879), pp. 143 — 146.
Read May 8, 1879.]
CONSIDER two plane figures, say the figure of the points P referred to axes
Ox, Oy, and that of the points P' referred to axes Ox, Oy' ; and let x, y be the
coordinates of P, and x, y' those of P'. If the figures correspond to each other in
any manner whatever, P and P' being corresponding points, then we have x', y'
each of them a function of x, y\ and we may consider the second figure as derived
from the first by altering the distance OP in the ratio VV2 + y"1 -4- v x* + y1, and by
rotating it through the angle tan"1 — , — tan"1 - ; say by the Extension Va?'2 + y'- -4- *la? + yz,
x x
11' ij
and by the Rotation tan"1^-— tan"1 -; where the Extension and the Rotation are each
x x'
of them a determinate function of x, y, the coordinates of P.
Passing from the point P to a consecutive point Q, the coordinates of which
are x + dx, y + dy (the ratio dy -4- dx being arbitrary), then the coordinates of the
corresponding point Q' will be x' + dx', y + dy', where
7 . dx' , dx' , , . dif , dy' ,
Writing —-, and •— instead of dy' -4- dx and dy -4- dx, the expressions
CLvC \JLUU
-1 - - -1 -
dx'2 + dy'2 -4- d#2 + dy\ and tan-1 -, - tan-1 - ,
ax ax
will in general have values depending upon that of the arbitrary ratio dy : dx. But
they may be independent of this ratio ; viz. this is the case when x', y' are functions
of x, y such that
dx' _ dy' dy' __ dx'
dy dx ' dy dx"1
729] ON A THEOREM RELATING TO CONFORMABLE FIGURES. 79
and the two figures are then conformable (or conjugate) figures ; that is, figures similar
as regards corresponding infinitesimal elements of area. We have, in this case,
\/dx* + dy'~2 -T- •Jdaf + dy2, and tan"1 -/-t - tan"1 ^ ,
doc ct^c
each a determinate function of x, y, the coordinates of P; and we pass from the
element PQ to the corresponding element P'Q' by altering the length in the ratio
2 + dy'2 4- *Jdx- + dy2, and rotating the element through the angle tan"1 -^-t — tan"1 -^- ;
dx dx
say, this ratio and this angle are the Auxesis and the Streblosis respectively, these
being, as already mentioned, functions of x, y only.
Considering now any two conformable figures, say the figure of the points P,
and that of the points P'; we have the theorem that we can from the first figure
obtain a third conformable figure by means of an Auxesis and a Streblosis which
are respectively equal to the Extension and the Rotation by which the second figure
is derived from the first.
In fact, if in the three figures respectively we take x, y, x', y', and x", y", for
the coordinates of the corresponding points P, P', P", the first and second figures
are conformable : and we have therefore
dx _ dy' dy' _ dx'
dy dx' dy dx'
the third figure is to have the Auxesis V#'2 + y''2 H- V#2 + r/2, and the Streblosis
tan-1 y- - tan"1 ^ ;
x x
viz. writing r for V&-2 + y2, we ought to have
1 „ xx' + yy' 7 xy' — x'y ,
dx -- --?*- dx--^ -- ?-dy,
IY& )•- "
•, „ xy' — x'y i xx' + yy' ,
dy = s— r-1 dx + — f*~ dy ;
and it is therefore to be shown that there exist x", y" functions of x, y satisfying
these relations ; for, this being so, we have
dx" = _dy^ dy"_ = drf^,
dy dx ' dy dx '
and the third figure is thus conformable with the first.
Writing, for shortness,
_ xx' + yy' xy1 - x'y
——
80 ON A THEOREM RELATING TO CONFORMABLE FIGURES. [729
the equations are
dx" = Ada; — Bdy,
dy" = Bdx + Ady;
or the conditions for the existence of the functions x", y" are
dA dB dA dB
~T~ ~\~ ~f~~ == "> ~~ 7 -- J —
dy dx dx dy
We, in fact, have
and similarly
which proves the theorem.
The theorem is closely connected with the theory of the function of an imaginary
variable ; for, writing the conditions for the conformable figures in the form
_ _
dx dy dy dx
we have
dx = Fdx — Gdy,
dy' = Gdx - Fdy ;
that is,
dx' + idy = (F + iG) (dx + idy) :
whence F + iG is a function of x + iy, and then by integration x + iy' is also a
function of x + iy. In one point of view, any function such as $ (x, y) + ity (x, y) is
a function of x -I- iy, for the quantity x + iy is only known by means of its real
components x, y ; that is, knowing x + iy, we know x, y, and therefore also
<f>(x, y) + i^(x, y);
and Cauchy, adopting this definition, introduced the expression " fonction monogene "
of x + iy, to denote that which is in the more restricted (and the ordinary) sense
termed a function of x + iy. And MM. Briot and Bouquet, in their " Theorie des
fonctions elliptiques" (Paris, 1875), although not using Cauchy 's expression fonction
monogene, but the simple term fonction, do this under the qualification stated p. 3 : —
" Dans tout ce qui suit, nous ne nous occuperons que des fonctions qui admetteut
une derivee." Now, a function admitting of a derivative (that is, in the ordinary
729] ON A THEOREM RELATING TO CONFORMABLE FIGURES. 81
sense, a function) of the imaginary variable z, = x + iy, is a function such that, for a
consecutive value z', = x + iy + dx + idy, we have
z —z
= a quantity independent of the ratio of the real components dx, dy of the increment
dx + idy of the imaginary variable. Or, what is the same thing, writing f(z) = x+iy',
the condition in order that x + iy' may be a function of x + iy is
dx + idy' = (F + iG) (dx + idy),
where F and G are functions of x and y. It is not part of the condition that
F + iG shall be a function of x + iy, and it is only a long way further on that the
authors prove that this is the case (see the definition of a "function holomorphe,"
p. 14 ; and the proof, p. 137). The last-mentioned equation
dx' + idy' = (F+ iG) (dx + idy),
where F and G are only assumed to be functions of x and y, has, if we represent
x + iy by means of the point P with coordinates (x, y), and in like manner x' + iy' by
means of the point P' with coordinates (x, y'), the geometrical interpretation that the
figures of the points P and P' are conformable figures, that is, figures similar as
regards their infinitesimal elements. The foregoing theorem in regard to the Auxesis
and the Streblosis is that we can, by means of F and G, construct a third conformable
/nr
figure, — in fact, the Auxesis and the Streblosis are = VJ*2 + G2 and tan-1 -^ respectively ;
and, using these as an Extension and a Rotation, we have the third conformable figure
x" + iy" = (F + iG) (x+iy); that is, (F + iG) (x + iy), and therefore also F + iG, is a
function of x + iy, — and we have thus the derivative of a function of x + iy as itself
a function of x + iy.
It is to be remarked that, although the theorem of the Auxesis and the Streblosis,
considered as a property of conformable figures, is not by any means geometrically
self-evident, yet the foregoing analytical proof is only a proof conducted by means of
real quantities, of what (admitting the theory of imaginary quantities) is in fact
self-evident; viz. the analytical conclusion really is that, F, G denoting functions of
x, y, then, if dx + idy' = (F + iG) (dx + idy), that is, if (F + iG) (dx + idy) be a complete
differential, then F + iG is a function of x + iy.
C. XI. 11
82 [730
730.
[ADDITION TO ME SPOTTISWOODE'S PAPER "ON THE TWENTY-
ONE COORDINATES OF A CONIC IN SPACE."]
[From the Proceedings of the London Mathematical Society, vol. x. (1879),
pp. 194—196.]
WRITE
U=(a, b, c, d, f, g, h, I, m, n$x, y, z, t)2,
^o = ( „ $fc 77, £ co)2,
W=( „ fa, y,
P = (a, £, 7, 5$*, y, z, t),
PO=(«, & 7, s$£ 17, r, *>)•
Then the equation of the cone, having for its vertex the arbitrary point (£, 17, ^, &>), and
passing through the conic U=0, P — Q, is
Or if, to put the coefficients £, 77, ^, <u in evidence, we write for a moment
A = (a, A, 0, Z $>, y, z, t),
B=(h,b,f,m% „ ),
G= (g, f> o, w$ „ ),
D = (/, m, n, d% ),
and therefore
then the equation is
£7(a£ + y87; 4- 7^+5a))- - 2P (a| -H ^17
+ P2 (a, 6, c, d, /, 0, A, I, m, »$(?, 17, f, <w)2 = 0.
730]
ADDITION TO MR SPOTTISWOODE S PAPER.
83
And if we expand first in £, 77, £, <w, and then in x, y, z, t, the final result is
y? y* z2 P yz zx xy xt yt zt
e
+ rf
+ r
C
B
F
2Ar
2L
2L'
c
A
G
2B'
2M'
2M
£
A
H
20'
2N
2N'
F
G
H
IF'
2G'
2ir
2A'
2F'
-2A
-20'
-2B'
2(Q-R)
-2M
-2N'
2B'
2G'
-20'
-2B
-2A'
-2L'
2(R-P)
-2N
20'
211'
-IB
-2A'
-20
-2L
-2M>
2(P-<?)
2M'
2N
2(Q-JR)
-2L'
-2L
- 2F
-2H'
-2G'
2L
2jy
-2M
-2(R-P)
-2M'
-2H'
-2G
-2F'
2L'
2M
-'2N'
-2N
2(P-Q)
- 2G'
-2F'
-2H
= 0.
In particular, if t] = 0, £ = 0, &> = 0, then we have the foregoing equation X = 0 ; and the
like for the equations F=0, Z=0, and TF=0 respectively.
Take a, b, c, f, g, h for the six coordinates of the line through the points
that is, write
where, of course,
x, y, z, t ;
a = y% — zrj, f = xw — t%,
h = zw —
af+bg + ch = 0.
Then the foregoing equation of the cone is
Atf + £b2 + Cc2 + Ft* + Gg* +
- ZA'bc - 25'ca - 2(7'ab + 2^'gh + 2£'hf + ZH'fg
+ 2Paf + 2Mag -
2Qbg +
= 0.
And this may be regarded as the equation of the conic in terms of the twenty-one
coordinates of the conic, and of the six coordinates of an arbitrary line meeting the
conic. It is, in fact, the general form of the equation given in the paper — Cayley,
" On a new Analytical Representation of a Curve in Space," Quart. Math. Jour.,
vol. in. (1860), [284; this Collection, vol. iv. p. 453].
11—2
84 [731
731.
ON THE BINOMIAL EQUATION a!"- 1=0; TRISECTION AND
QUAKTISECTION.
[From the Proceedings of the London Mathematical Society, vol. xi. (1880), pp. 4 — 17.
Read November 13, 1879.]
THE solution of the binomial equation xp — 1 = 0, p a prime number, or, say rather,
the equation
XP~l+ £cP-2+ ...+# + 1 =0,
depends upon the Jacobian function
Fa. = x1 + 0x9 + . . . + a?-*acav~'2 ,
where g is a prime root of p, a. any root whatever of the equation up~l — 1 = 0. Taking
e a factor of p — 1, and / for the complementary factor (that is, p — 1 = ef), then, if for
a we write of, or, what is the same thing, taking a/, = /3, a root of ue — 1 = 0, we have
F/5 = X0 + /3Z, + . . . + /9-'Z_lf
p — 1
where X0, X1} ..., Xe^ denote each of them a period or sum of f, = — — , roots, viz.
e
X0 =(1, g\ ...,<7<'-n
(read X0 = x1 + a&* + . . . -f xP(f 1]e, and so for the other functions).
We have, of course, F(l), = X0 + Xl + ... +Xe-lt the sum of all the roots = — 1;
and, further, the general property that any rational and integral function of these
periods is expressible as a sum
^o-^-o + QiXi + ... + Ct>e—iXe—i
with known coefficients
731] ON THE BINOMIAL EQUATION Xp — 1=0. 85
The several cases e = 2, 3, 4, ... may be termed those of the bisection, trisection,
quartisection, &c., of the equation ; viz.
e = 2, there are two periods, X, Y, and F(— l) = X — F;
e = 3, three periods, X, Y, Z, and Fj = X + yY+y-Z, if 7 is a root of w3- 1 = 0;
e = 4, four periods, X, Y, Z, W, and F8 = X + 8Y+8°-Z + 83W, if 8 be a root of w4-l=0.
It is sufficient to attend to the prime roots 7 and 8 of the equations
u3 - 1 = 0, w4 - 1 = 0,
respectively; for, if 7 or 8 be = 1, we have simply -^(1), = — 1; and if 8 be = — 1,
then the function is F(-I), = X + Z-(Y+W), where X+ Z and Y+ W are the
periods for the bisection. The prime roots 8 are of course i and —i, and we have
F(i) = X + iY-Z-iW,
F(-i} = X-iY-Z + iW,
respectively.
g-i
As regards the bisection, it is known that (X — F)2 = (— ) 4 p, which is +p or — p,
according as p is = 1 or 3, mod. 4 ; and the values of X, Y are thus determined.
In what follows, I consider the cases e = 3 and e = 4 of the trisection and the
quartisection respectively.
It is to be remembered that, not the division into periods, but the order of the
periods, depends on the choice of g, a prime root at pleasure of p ; and, in what
follows, I select the prime root used in Reuschle's Tafeln complexer Primzahlen
welche cms Wurzeln der Einheit gebildet sind (4to, Berlin, 1875): viz. these are
p = 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89, 97,
0 = 2,2,5, 2, 2, 3, 2, -2, 2, 3, 2, 6, 3,10, 2,
2, 2, 2, 62, 5, 3, 2, 30, 10,
where I quote the whole series, although I am here only concerned with the values
of p which are = 1 (mod. 3), or = 1 (mod. 4).
The periods are consequently those of Reuschle, viz. X, Y, Z are his T/O, 771, ijz, and
X, Y, Z, W his r)Q, i}lt 772, 7;3 : they can of course, without referring to his work, be
easily recalculated, but it is, I think, convenient to have for his values of g the
series of residues such as are given (for differently selected values of g) in Jacobi's
Canon Arithmetics (4to, Berlin, 1839); and I have accordingly taken out of Reuschle,
and annex, such a table.
For instance, p=13, the powers of g are 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7;
and, by writing these down in order in columns of 3 or of 4,
1 8 12 5 139
2 3 11 10 265
4697 4 12 10
8 11 7
86 ON THE BINOMIAL EQUATION Xp — 1=0. [731
we have the periods X, Y, Z or X, Y, Z, W, belonging to the trisection and the
quartisection of p = 13.
I further remark that the equations which I am concerned with are all given in
Reuschle, but in a somewhat different form ; thus, p = 13, quartisection (see p. 13), he has
(where observe that here and in every case the value of 7/0773 is at once obtained
from that of 770771 by a mere cyclical interchange of the suffixes, so that the last
equation is in fact superfluous) ; the other equations, using 770 + t]l + 772 + 773 = — 1 to
eliminate any constant term which occurs, give my values
X2 = ( 0, 1, 2, 0) (X, Y, Z, W),
XY=( 1, 1, 0, 1)( „ ),
XZ = (- 3, - 2, - 3, - 2) ( „ ).
Similarly, in the case of a trisection, the equation for 770773 is superfluous, and the
other equations give my values of X2 and X Y.
Reuschle gives also, and I take from him, the cubic and the quartic equations (such
as £>=13, 77s + 772 — 477+ 1 =0, 77" + T?3 + 2772 — 477 + 3 = 0), which determine the periods in
the trisections and the quartisections respectively.
Many of the results obtained accord with, and furnish exemplifications of general
theorems contained in Jacobi's memoir, "Ueber die Kreistheilung und ihre Anwendung
auf die Zahlentheorie," Crelle, t. xxx. (1846), pp. 166 — 189 ; [Ges. Werke, t. vi. pp.
254—274].
Trisection, e = 3; p = l (mod. 3).
We have three periods X, Y, Z; and we thence obtain
X9 =(a, b, c)(X, Y, Z),
XY=(f,g,h)( „ ),
the coefficients a, b, c, f, g, h being determinate integers. And, by cyclical inter
changes, we obtain equations which may be written
X" = a, b, c,
F2 = c, a, b,
Z- — b, c, a,
XY=f, g, h,
YZ=h,f,g,
ZX=g,h,f;
viz. here and elsewhere the coefficients a, b, c are written to denote the sum
aX + bY+cZ.
It is easy to see that
731] ON THE BINOMIAL EQUATION Xp — 1 = 0. 87
in fact, a period contains %(p — 1) terms, and in two consecutive periods X, Y, there
are no terms the product of which is unity; hence XY contains ^(p — I)2 terms,
each a power of x, and the sum XY+YZ+ZX contains ^(p — I)2 such terms, being
in fact the sum X+Y+Z taken ^(p — 1) times; whence the relation in question.
Hence also
YZ + ZX + XY=-±(p-l\
From the equation X+Y+Z=-l, multiplying by X, and for X2, XY, XZ
substituting their values, we obtain an expression
which must identically vanish ; viz. the three coefficients must be each of them = 0 ; or
we must have
a = -f-g-l,
b = — g — h,
'--A-/;
so that, taking /, g, h as known, the other coefficients a, b, c are given in terms of them.
The equations give
We have X.YZ=Y.ZX; that is, X (h, f, g)=Y(g, h,f): or, substituting for X2,
XY, &c. their values,
h (a, b, c) = g (f, g, h)
,V +h(c, a, b)
,/) +f(h,f, g);
that is,
ah+f* +0* = gf + ch+fh,
ch+fh +fg=gh + bh +fg,
equations which reduce themselves to the single equation
and this is the only relation obtainable by consideration of the three equal values
X.YZ, Y.ZX, Z.XY.
Moreover, this equation being satisfied, the six functions in the three equations become
each of them =fg — h? ; or we have
that is,
XYZ-K-fy.
We have
Fy.Fy*= X°+Y> + Z--YZ-ZX-XY
+ c-f-g-h)(X+ Y+Z)
88 ON THE BINOMIAL EQUATION Xp-l=0. [731
that is,
Fy.Fy-=p.
We have, moreover,
(Fy)"- = X*+2YZ + y(Z* 4- 2ZF) + 72(F2
[(a, b, c) + 2 (A, / #)]
+ 7 [(6, c, o) + 2(/>flf, A)]
+ 72[(c, a, 6) + 2(flr, A./)],
which is
= {(a + 2A) + 7 (6 + 2/) + 72 (c + 20)} (X + 72F +
as is at once seen by comparing the coefficients of X, Y, Z respectively.
Hence, writing
a + 2h + 7 (b + 2/) + 72 (c + 20)
we have
4 = a + 2A - c - 20 = 3A - 3^r - 1,
5 = b + 2/ - c - 2^r = 3/ - 3$r.
We have
and thence, writing 72 for 7,
equations which give
Fy.F<f, =p, =
or, say p = A- — AB + B2 ; viz. p has the complex factor
A+By, =3/i-3<7-
Hence also
(Fy)*=p(A+B7)>
and, as before,
Fy.Fy-=p;
which equations determine Fy, Fy-, and from these and ^(1) = — ! we obtain the
periods X, Y, Z\ we have thus, in fact, the solution of the cubic equation which gives
these periods. We have already found the coefficients of this cubic equation, viz.
X+Y+Z=-l, YZ+ZX + XY=-^(p-l), XYZ = h2-fg;
the equation thus is
<+tf-|<p-i)*+0&-*)-ft
As already remarked, the values of a, b, c ; /, g, h, and the equations in 77, are in effect
given in Reuschle ; the complex factors of p, as given p. 1 (7 = 2y — 3y2, &c.), when
reduced to the form A + By, are not identical with the A +By of the foregoing theory;
viz. this A + By is not Reuschle's selected primary form. I give, in the annexed table
731]
ON THE BINOMIAL EQUATION Xp — I = 0.
89
for the primes 7, 13, ..., to 97, the values from Reuschle of a, b, c; f, g, h, and of the
coefficients of the ^-equation ; also the values of A and B derived from /, g, h by the
foregoing formulae. It will be seen that all the values are consistent with the theory.
TABLE FOR THE TRISECTION.
f
a, b, c
/. g, h
/+%
A B
Page in
Reuschle
7
2 1-2
1 0 1
- 2 1
2 3
p. 6
13
- 4 - 3 - 2
1 2 1
4 - 1
4 -3
P- 15
19
4 .5-4
1 2 3
6 - 7
2 -3
p. 26
31
• 7 - 6 - 8
424
- 10 - 8
5 6
P- 45
37
- 8 - 10 - 7
543
- 12 11
- 4 3
P- 54
43
- 11 8 -10
644
- 14 8
1 6
p. 69
61
- 14 - 13 - 15
587
-20 - 9
4 -9
P- 97
67
- 16 - 13 - 16
967
— 22 5
2 9
p. 105
73
- 16 - 18 - 15
699
- 24 - 27
- 1 -9
p. 128
79
-20 -17 -16
9 10 7
-26 41
- 10 -3
p. 138
97
- 20 - 23 - 22
10 9 13
- 32 - 79
11 3
p. 1 68
C. XI.
12
90 ON THE BINOMIAL EQUATION 2^—1 = 0. [731
Quartisection, e = 4 ; p = 1 (worf. 4).
We have four periods X, Y, Z, W ; and we obtain
Z2 =(a, 6, c, d)(Z, F, £, F),
XY=(f,g,h,k)( ),
XZ=(l, m, l,m)( „ ),
the coefficients being determinate integers. It can be shown that Z + m = ^(p-l) or
i) according as /> = 1 or 5 (mod. 8). And then, by cyclical interchanges,
X" = a, b, c, d,
F- = d, a, b, c,
Z- = c, d, a, b,
W'2 = b, c, d, a,
X7=f, g, h, k,
YZ =k,f,g, ht
ZX =h, k,f, g,
XW = g, h, k,f,
XZ = I, m, I, m,
YW = m, I, m, I.
We have, in like manner as for the trisection,
and so also the expression for
2XY, =XY+XZ + XW+YZ+YW + ZW
is
h+k + l + m) = -i(p-l)-l-m;
nnd, in virtue of the foregoing value of l + m, this is = — f(p — 1) or £(_p + 3) according
as p = 1 or 5 (mod. 8).
Again, from the equation X+Y+Z+ W= — l, multiplying by X and reducing,
a = - 1 -/- g-l,
b= —g — h — m,
c= —h — k — l,
d= —k—f—m,
and thence
a + 6 + c + d = -l-2 (/+ g + h + k)-2(l+ m),
and
a-b + c-d = -l + 2 (m - 1).
731] ON THE BINOMIAL EQUATION ^—1=0. 91
We have
X.YZ=Y.ZX = Z.XY,
that is,
X(k,f, g, h}=Y(l, m, I, m) = Z(f, g, h, k),
and thence
k (a, b, c, d) = I (f, g, h, k) = / (I, m, I, m)
+f(f>9> h k) +m(d, a, b, c) +g(k,f, g, h)
+ g(l, m, I, m) +1 (k, f, g, h) +h(c, d, a, b )
+ h(g, h, k,f) +m(m, I, m, I) + k(h, k, f, g),
that is,
ka+f2 + gl + gh=lf + md + lk + m2 = If +gk + ch + kh,
kb+fg + gm + h2 = Ig + am + If + ml =fm +fg + hd + k2,
kc +fh + gl + hk = Ih + mb + Ig + m2 =fl + g* + ah + kf,
kd+fk+ gm + fh = kl + me + Ih + Im =fm +gh + bh + gk,
in which equations a, b, c, d may be regarded as having their foregoing values.
One of these equations is
kc +fh +gl + hk = lf+g" + ah + kf,
that is,
-k(h + k + 1) +fh+gl + hk = lf+g* - h (/+ g + I + 1) + kf,
or, reducing,
l(ff + h ~f- k) = g°- + k*- 2hf- hg -h + kf,
which gives I.
Again, another equation is
^ +/9 +gm + h- =fm +fg + hd + k2,
that is,
-k(g + h +» +fg + gm + A2 =fm +fg - h (k +f+ m) + k\
or, reducing,
m (g + h —f— k) = k- — h- + gk — hf,
which gives m.
And we have also
md + Ik + m2 = gk + ch -f kh,
that is,
- m (k+f+ m) + lk + m2 = gk + kh -h(h + k + I),
or, reducing,
I (k + h) -m(f+ k) = gk - k\
Substituting herein for I, m their values, we have
(A- + li) [g2 + k2- 2hf- hg + kf- h] - (/+ k) [k2 - h2 + gk - hf] + (h2 - gk) [g + h -f- k] = 0.
12—2
92 ON THE BINOMIAL EQUATION Xp — 1=0. [731
In this equation the only terms of the second order are -h(h+ k), which contain the
factor h; the terms of the third order contain this same factor h, and throwing it out,
and reducing, the equation is found to be
or, as it may also be written,
gt + fc- 2hf- h + (h* +f*-2gk-k) = 0 ;
and the foregoing values of I, m are
(0. + jfc. -2hf-h) - (gh-kf)
g + h-k-f
&-h* + gk- hf
— -. •
- 77 /• ?
g+h-k-f
and by means of these three equations all the foregoing equations are satisfied.
We have
FiFi3 = (X - Zy + (Y - Wy
' W--2(XZ+YW)
or, substituting for a, b, c, d, this is
= 1+2 (/+ g + h + k) + 4 (I + m),
viz. it is
l) + 4(1 •*-«);
or, substituting for l + m its before- mentioned value, then, according as p = l or 5 (mod. 8),
the value is =p or — p ; that is, we have
i»-i
FiFi3 = (-) * p.
Again, we have
)2 = (.Y + iY - Z - i W)*
- YZ+ZW- WX)
- W)
where
A = a — b+c — d + 2(m — l\ = - 1 + 4 (m - I),
or, since X— Y+Z- W = F (— 1), this equation is
(FiY = (A
and similarly
731] ON THE BINOMIAL EQUATION Xp — 1=0. 93
Moreover
[j-(-i)]»=(-)Vp> =P-
and we have therefore
(±py = (A* + &-)p,
that is,
A- + B2=p;
or the expression A + Bi determined as above is a complex factor of p.
We may investigate the quartic equation for the determination of the periods X, Y,
Z, W. The values of X + Y+ Z + W and XY+ XZ+ XW + YZ + YW + ZW are already
known: for the next coefficient XYZ + XYW + XZW + YZW, we have XYZ = (a, /3, 7, &),
where each of the coefficients a, fi, 7, B is given under three different forms : the values
of YZW, ZWX, WXY are (8, a, & 7), (7, 8, «, £), (/?. 7, 8} a); and the required sum
therefore is
W\ = -(a
Taking the first expressions of these coefficients respectively, we have
2g(l + m)
We find XYZW most readily as the product of XZ and YW '; we thus obtain
XYZW=lm(X*-+ Y* + Z*+ W* + 2XZ+2YW) + (l* + m*)(XY+ XW+ YZ+ZW),
= lm(-a-b-c-d-2l- 2m) - (I'2 + m2) (f+g + h + k\
= Im {1 + 2 (/+ g + h+k)}-(l* + m?)(f+g + h + k);
or, substituting forf+g+h + k its value i (/)-!), this is
Im -\(l- my (p - 1), = ^ {(« + m)3 -(I- rn^p}.
Hence the required equation, having roots X, Y, Z, W, is
where, for the sake of having a single formula, I have retained I + m in place of its
value =-»(p_l) or ^(p+'3) according as p=l or 5 (mod. 8).
94 ON THE BINOMIAL EQUATION 0^—1=0.
We thus have the following : —
TABLE FOR THE QUARTISECTION.
[731
p
abed
f 9 1> *
I in
7?4 + 77;! +
7T T,1 7,0
A B
Page in
Reuschle
5
0100
0001
1 1
1 1 1
- 1 -2
p. 2
13
0120
1101
3 2
24 3
3-2
P- 13
17
4 2 3 4
2011
1 1
6 1 1
- 1 4
p. I9
29
2302
1123
5 - 6
4 20 23
-5 - 2
P- 36
37
2124
2241
7 — 7
5 7 49
- 1 6
P- 53
41
- 10 6 7-8
4222
3 2
- 15 18 4
-5 4
p. 6 1
53
2362
4423
- 11 9
7-43 47
7-2
p. 80
61
4326
3363
-11 -12
8 42 117
-5 6
p. 96
73
- 16 - 13 - 12 - 14
6552
4 5
- 27 - 41 2
3 8
p. 126
89
- 19 - 18 - 16 - 14
4855
6 5
- 33 39 8
-5 -8
p. 152
97
- 22 - 16 - 17 - 18
8655
7 5
- 36 91 - 61
-9 4
p. 167
731] ON THE BINOMIAL EQUATION Xp—l=Q. 95
TABLE OF THE POWERS OF REUSCHLE'S SELECTED PRIME ROOTS.
3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
2252232
21
2
3 2
6 3
10 2
2
2 2
62 5
3 2
30 10
1
444494
4
4
9 4
36 9
6 4
4
4 4
10 25
9 4
10 3
2
... 3 6 8 8 10 8
15
8
27 8
11 27
13 8
8
8 8
52 52
27 8
33 30
3
2 5 3 13 16
16
16
19 16
25 38
36 16
16
16 16
29 41
2 16
11 9
4
3 10 6 5 13
14
3
26 32
27 28
31 32
32
32 32
23 59
6 32
63 90
5
9 12 15 7
18
6
16 27
39 41
28 11
5
3 64
6 3
18 64
21 27
6
7 11 11 14
10
12
17 17
29 37
45 22
10
6 61
17 15
54 45
7 76
7
3 9 16 9
3
24
20 34
10 25
27 44
20
12 55
60 2
4 7
32 81
8
6 5 14 18
17
19
29 31
19 32
35 35
40
24 43
28 10
12 14
70 34
9
10 8 17
12
9
25 25
32 10
21 17
21
48 19
32 50
36 28
53 49
10
... 7 7 15
22
18
13 13
28 30
22 34
42
35 38
67 31
29 56
77 5
11
4 11
2
7
8 26
4 4
32 15
25
9 9
36 9
8 29
85 50
12
12 3
19
14
24 15
24 12
38 30
50
18 18
31 45
24 58
58 15
13
2 6
8
28
10 30
21 36
4 7
41
36 36
5 6
72 33
49 53
14
6 12
7
27
30 23
3 22
40 14
23
11 5
26 30
58 66
46 45
15
5
9
25
28 9
18 23
24 28
46
22 10
50 4
16 49
45 62
16
10
5
21
22 18
26 26
5 3
33
44 20
47 20
48 15
15 38
17
13
13
4 36
33 35
3 6
7
27 40
3 27
65 30
5 89
18
20
26
12 35
34 19
30 12
14
54 13
44 62
37 60
61 17
19
6
23
5 33
40 14
18 24
28
47 26
30 18
32 37
50 73
20
11
17
15 29
35 42
39 48
56
33 52
14 17
17 74
76 51
21
5
14 21
5 40
14 43
53
5 37
16 12
51 65
55 25
22
10
11 5
30 34
46 33
47
10 7
69 60
74 47
48 56
23
20
2 10
16 16
37 13
35
20 14
18 8
64 11
16 75
24
11
6 20
14 5
41 26
11
40 28
51 40
34 22
35 71
25
22
18 3
2 15
34 52
22
19 56
38 54
23 44
71 31
26
15
23 6
12 2
11 51
44
38 45
13 51
69 5
83 19
27
7 12
31 6
16 49
29
15 23
25 36
48 10
87 93
28
21 24
22 18
19 45
58
30 46
59 34
68 20
29 57
29
11
9 11
2 37
57
60 25
37 24
46 40
69 85
30
22
13 33
20 21
55
59 50
22 47
59 80
23 74
31
7
37 13
12 42
51
57 33
15 16
19 77
67 61
32
14
17 39
26 31
43
53 66
7 7
57 71
52 28
33
28
20 31
25 9
27
45 65
8 35
13 59
47 86
34
19
38 7
15 18
54
29 63
70 29
39 35
75 84
35
23 21
9 36
49
58 59
9 72
38 70
25 64
36
15 20
43 19
39
55 51
61 68
35 57
38 58
37
8 17
7 38
19
49 35
19 48
26 31
72 95
38
7 8
23 23
38
37 3
42 21
78 62
24 77
39
24
42 46
17
13 6
48 32
76 41
8 91
40
29
44 39
34
26 12
65 14
70 82
62 37
41
11 25
9
52 24
45 70
52 81
80 79
42
29 50
18
43 28
11 58
77 79
86 14
43
8 47
36
25 29
43 71
73 75
88 43
44
33 41
13
50 58
39 63
61 67
59 42
45
29
26
39 49
4 23
25 51
79 32
46
5
52
17 31
35 42
75 19
56 29
47
10
45
34 62
40 64
67 38
78 96
48
20
31
7 57
66 28
43 76
26 87
49
40
3
14 47
45 67
50 69
68 94
50
96 ON THE BINOMIAL EQUATION Xp — 1 = 0. [731
TABLE (continued).
53 59 61 67 71 73 79 83 89 97
51
27 6 28 27
21
43
71
55
82
67
51
52
12 56 54
24
69
55
27
57
88
52
53
24 51 41
68
53
7
54
19
7
53
54
48 41 15
27
46
21
25
36
70
54
55
37 21 30
41
11
63
50
12
21
55
56
15 42 60
57
55
31
17
4
16
56
57
30 23 53
55
56
14
34
31
63
57
58
46 39
2
61
42
68
40
48
58
59
31 11
53
13
47
53
43
92
59
60
22
20
65
62
23
44
47
60
61
44
33
33
28
46
74
82
61
62
21
58
19
5
9
84
44
62
63
42
46
22
15
18
28
52
63
64
17
12
37
45
36
39
35
64
65
34
34
39
56
72
13
59
65
66
49
49
10
61
34
8
66
67
56
26
30
39
41
80
67
68
64
57
11
78
73
24
68
69
63
66
33
73
54
46
69
70
38
20
63
18
72
70
71
44
60
43
6
41
71
72
22
3
2
22
72
73
26
6
60
26
73
74
40
12
20
66
74
75
41
24
66
78
75
76
44
48
22
4
76
77
53
13
37
40
77
78
26
42
12
78
79
52
14
23
79
80
21
64
36
80
81
42
51
69
81
82
17
11
82
83
65
13
83
84
81
33
84
85
27
39
85
86
9
2
86
87
3
20
87
88
6
88
89
60
89
90
18
90
91
83
91
92
54
92
93
55
93
94
65
94
95
68
95
732] 97
732.
A THEOREM IN SPHERICAL TRIGONOMETRY.
[From the Proceedings of the London Mathematical Society, vol. XL (1880), pp. 48 — 50.
Read January 8, 1880.]
IN a spherical triangle, where a, b, c are the sides, and A, B, C the opposite
angles, we have
— tan \c tan \ a tan \~b sin (A — B) = tan ^b sin A — tan \a sin B,
tan |c {1 — tan^a tan |6 cos (/I — B)} = tan ^b cos A + tan
which are both included in the form
„. tan^c — tani& (cosJ. +i sin .
tan \a (cosB — i sinB) = -= — -. — - — =Ln-; -. —
1 + tan£c tan £6 (cos A +tsin
For the first of the two identities : from
cos A + cos B cos C
cos a —
cos 6 =
sin B sin C
cos B + cos A cos C
sin -4 sin C
we deduce
cos a —
1 /cos J. cos 5\ cos G /cos 5 cos A
A — . -- _ I _ — ----
y — -y^l-T* • 4 J '/"Vl'Tl ' A
sm U \sm 5 sin A) sin (7 Vsm B sin .4
1 ^ (sin 2^ -sin 2^) cos C sin (4 - B)
sin C sin J. sin B sin C' sin J. sin B
^ —r/Ar.. „<,
— ,—.-7, cos (A+B)+ cos C}
l
,
smCsmA
sin A —
c. xi.
98 A THEOREM IN SPHERICAL TRIGONOMETRY. [732
that is,
f A r>\ sin C , L x
— sin (A - B) = — - (cos a — cos 6)
' 1 — cos c
sm G sine , 7 ,
= -s — — (cos a — cos 6) ;
sm c 1 — cos c
or, what is the same thing,
— tan ic sin ( A — B} = — — (cos a — cos b).
sine v
Here cos a — cos & is =(1 + cosa) — (1 + CGS&); substituting for — — successively —
sm c ' sm a
and - — y , the right-hand side is
sm b
1 + cos a . 1 + cos b . ,,
= — -: — - sin A . — =— sm B,
sm a sm b
= cot ^a sin A — cot ^b sin B ;
whence, multiplying each side by tan \ a tan ^b, we have the relation in question.
For the second identity which is
tan \c {1 — tan \a tan \ b cos (.4 — J5)} = tan | b cos .4 + tan ^a cos 5 ;
if on the right-hand side we substitute for cos A, cos B their values
cos a — cos b cos c , cos b — cos a cos c
: — ; — : — and — : : — - ,
sm 6 sm c sm a sm c
the right-hand side becomes
1 (cos a — cos b cos c cos 6 — cos a cos c)
J I I *
sin c ( 1 + cos b 1 + cos a ) '
whence, multiplying the whole equation by sin c (1 + cos a) (1 + cos b), it becomes
(1 - cos c) {(1 + cos a) (1 + cos b) — sin a sin b cos ( A — B)}
= (1 + cos a) (cos a — cos b cos c) + (1 + cos b) (cos b — cos c cos a).
We have here
( * m , A D , • i • T> (cos a — cos & cos c) (cos & — C°S C COS a) + D
cos (J. — B) = cos J. cos B + sm -d sm B = - — — : — ~. : —
sm2 c sm a sm b
by substituting for cos .4, cos B their foregoing values, and for sin -4, sin B their values
vn vn
where
sin 6 sin c ' sin a sin c '
D = 1 — cos2 a — cos2 6 — cos2 c + 2 cos a cos 6 cos c.
732] A THEOREM IN SPHERICAL TRIGONOMETRY. 9<>
The numerator is
cos a cos b — cos c (cos2 a + cos2 6) + cos a cos b cos2 c
+ 1 — cos2 c — (cos2 a + cos2 6) + cos a cos 6 . 2 cos c ;
viz. this is
= cos a cos b (1 + cos c)2 — (cos2 a + cos2 b) (1 + cos c) + 1 — cos2 c,
having the factor 1 + cos c, which is also a factor of sin2 c, = 1 - cos2 c, in the
denominator. We have, therefore,
, A m_ cos a cos 6(1+ cos c) — (cos2 a + cos2 6) + 1 — cos c
COS Lfl — Jj) — — — - -. ; •
(1 — cos c) sin a sin b
and the equation thus is
(1 - cos c) (1 + cos a) (1 + cos b) - {cos a cos 6(1 + cos c) - (cos2 a + cos2 6) + 1 - cos c}
= (1 + cos a) (cos a — cos 6 cos c) + (1 + cos 6) (cos 6 — cos c cos a),,
where each side is in fact
= cos a + cos2 a + cos 6 + cos2 6 — cos c (cos a + cos 6) — 2 cos a cos 6 cos c ;
and the second identity is thus proved.
13—2
100 [733
733.
ON A FOKMULA OF ELIMINATION.
[From the Proceedings of the London Mathematical Society, vol. XL (1880), pp. 139 — 141.
Read June 10, 1880.]
CONSIDER the equations
(a, ...10, 1)»=0,
(A,... 10, i)» = o,
where a,..., A,... are functions of coordinates. To fix the ideas, suppose that each
of these coefficients is a linear function of the four coordinates x, y, z, w. Then,
eliminating 6, we obtain V = 0, the equation of a surface ; and (as is known) this
surface has a nodal curve.
It is easy to obtain the equations of the nodal curve in the case where one of
the equations, say the second, is a quadric : the process is substantially the same
whatever may be the order of the other equation, and I take it to be a cubic ;
the two equations therefore are
(a, b, c, d^e, 1)3 = 0,
(A, B, CIO, 1)2 = 0;
giving rise to an equation
V, = (a, b, c, dy(A, B, C)\ =0.
And it is required to perform the elimination so as to put in evidence the nodal
line of this surface.
Take 01} #2 the roots of the second equation, or write
(A, B, cie, i)*=A(0-01)(0-0,)>,
that is,
^ 0
733]
then, if
we have
ON A FORMULA OF ELIMINATION.
101
®, = («, b, c, d^0lt I)3,
©2 = (a, b, c, d$0.2) I)3,
V = A3®^
viz. on the right-hand side, replacing the symmetrical functions of #1( 02 by their
values in terms of A, B, C, we have the expression of V in its known form
V = a*
&c.
Form now the expressions
_ # (S) # 20 — /? 30 d 3(S) _ d 3(S)
l *^1^^2j •-'2 vj/j ^~ l/i *— '2» "2 ^-^1 "l ^-^2j
each divided by Q^ — 6.2. These are evidently symmetrical functions of 01} 0.2, the
values being given by the successive lines of the expression
(
0,
-i,
1,
o,
l + 02, e,2 + 6 A + e.^d, Sc, 36, a) ;
0,e,, 0&(0i + 0
o, ^,»
Q?Q-?, o
and, consequently, these same quantities, each multiplied by A2, are given by the
successive lines of
( 0,
A*, -2AB, -
-43, 0, AC, -2BC
2AB, -AC, 0, C3
-4&, 2BC, -C2, 0
, 3c, 36, a).
Calling these Jf, F, ^, W, that is, writing
+ (- A C + 4B2) a, &c.,
then X, F, ^, W are the values of
each multiplied by J.2 -=- (0! — #2) ; and the functions all four of them vanish if only
®1 = 0, 02 = 0; or, what is the same thing, the equations X = 0, F=0, Z=0, W=Q
constitute only a twofold system.
The functions
( X, Y, Z )
F, Z, W
102 ON A FORMULA OF ELIMINATION. [733
contain each of them the factor ®!®2, that is, V ; they, in fact, each of them vanish
if (S^ = o, and they also vanish if B2 = 0 ; or, by a direct substitution, we have
XZ - Y* = A • - (ft - ^)2 ©i©- - - A*®!®,,
XW-YZ = „ -(tfi-Wfli + tfo)®!©*, = -A*®l®s(8l + 03),
YW-Z- = „ - (0i - 0^ 0A®i®3, = - AW&S0&.
Or, what is the same thing, these are = — J.V, 25V, —(7V, respectively; thus the
first equation is
- QABb + (- AC + 4£2) a} [2ABd - 3ACc + C*a]
_ (_ A'-d + 3ACb- 25(7a)2 = -A (A3d2 + &c.), = - A V ;
and similarly for the other two equations. The nodal curve is thus given by the
twofold system Z = 0, F=0, Z=Q, W=0.
The method may be extended to the case where, instead of the quadric equation
(A, B, C~§6, 1)2 = 0, we have an equation of any higher order, but the formulae are
less simple.
734]
103
734.
ON THE KINEMATICS OF A PLANE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879),
pp. 1-8.]
IT seems desirable to bring together under this title various questions which
have been, or may be, proposed or discussed. We consider two planes in relative
motion one upon the other, but, for convenience, they may be distinguished as a
moving plane and a fixed plane, the first moving upon the second. Any point of
the moving plane traces out on the fixed plane a curve, and any line of the mo vino-
plane envelopes on the fixed plane a curve; similarly, any point of the fixed plane
traces out on the moving plane a curve, and any line of the fixed plane envelopes
on the moving plane a curve. More generally, any curve of the moving plane envelopes
on the fixed plane a curve, and any curve of the fixed plane envelopes on the
moving plane a curve. There is, moreover, in the moving plane a curve which rolls
upon a curve in the fixed plane, and these two curves (a single relative position
being given) determine the motion.
Fig. 1.
0 MX
The analytical theory presents no difficulty. Taking in the fixed plane the fixed
axes Ox, Oy (fig. 1), and, fixed in the moveable plane so as to move with it, the
axes O^j, 0,3^; then the position of the axes 0,0?^ may be determined, say by
104 ON THE KINEMATICS OF A PLANE. [734
a, 13, the coordinates of Oj in regard to Oxy; and by 6, the inclination of Olxl to
Ox. And denoting by x, y, xl, y^ the coordinates of a point P in regard to the two
sets of axes respectively, then
x = a + x-i cos 0 — yl sin 0,
y = ft + xl sin 6 + 3/3 cos 6 ;
or, what is the same thing,
a-j = (x — at) cos 6 + (y — /3) sin 0,
yl = — (x — a) sin 6 + (y — (3) cos 0 ;
or, as these last equations may be written,
#j = otj -f # cos (— 0) — y sin (— 0),
7/j = & + ar sin (- 0) + y cos (- 0),
where a1} &, = — a cos 6 — /Ssin 0, a sin 6 — ft cos 0, are the coordinates of 0 referred to
the axes OiX^, and — 6 is the inclination of Ox to O^.
When the motion is given, a, /3, 0 are given functions of a single variable
parameter, say of t* ; or, if we please, a, /3 are given functions of 6.
The velocities of a given point (x, y) are determined by the equations
x — CL — (%i sin 0 + yl cos 6} 6',
y' = j3' + (#! cos 6 — yl sin 6} & :
that is,
or, as these equations may also be written,
- (of - a!) sin 6 + (;/' - /3') cos 9 = x^',
- (x' - a') cos 6 - (y - j3') sin 0 = y^ff.
Hence if x =0, y' = 0, we have
#!#' = a' sin 6 — /3' cos 0, or a' = (y — /3) 0',
2A0' = a' cos 0 + £' sin (9, _£' = (#_ a) 0',
which equations determine in terms of t, xl and ^ the coordinates in regard to the
axes 0^x^yl, and x and y the coordinates in regard to the axes Oxy, of /, the centre
of instantaneous rotation.
If from the expressions of xl} yl we eliminate t, we obtain an equation between
(#!, 7/j), which is that of the rolling curve in the moveable plane ; and, similarly, if
* t may be regarded as denoting the time, and then the derived functions of x, y in regard to t will
denote velocities ; and, to simplify the expression of the theorems, it is convenient to do this.
734]
ON THE KINEMATICS OF A PLANE.
105
from the expressions of x, y we eliminate t, we obtain a relation between (x, y),
which is that of the rolled-on curve in the fixed plane.
The system may be written
a' . a /3' /3'
xl = sin 6 - , cos 0, X = CL- -,
,= , cos 0
or, if we take 0 as the independent variable,
X-L — a' sin 6 — $ cos 0, x = a. — /3',
yl = a.' cos 6 + /3' sin 6, y = @ + «'.
To find the variations of /, we have
a?/ = a" sin 0 - /3" cos 6 + a.' cos 0 + £' sin 0, = a" sin 0 - fi" cos 0 + yl ,
yi = a" cos 0 + /3" sin 0 - a' sin 0 + /3' cos 0, = a" cos ^ + /8" sin ^ - a;:,
=a -
Hence
'/ = x cos ^ + y' sin 0, or #' = #/ cos 0 — y-[ sin ^,
/ = — x' sin 0 + y' cos 0, ?/' = #/ sin 0 + y/ cos 0,
values which give x'2 + y'2 = a;/2 + y^, which equation expresses that the motion is in
fact a rolling one.
Imagine the two curves, and the initial relative position given ; say the two
points A, Al (fig. 2) were originally in contact, then the arcs AI, A^I are equal, and,
calling each of these s, and X, Y, Xl} Fj the coordinates of / in regard to the two
sets of axes respectively, we have X, Y, X1} Yl given functions of s, such that
X'2+ Y'2 = 1, X1'2+Y1'* = 1, the accents now denoting differentiation in regard to s.
We have, from the figure,
Y' Y'
0 = tan"1 -^ — tan"1 -^, ;
.A JL i
C. XI.
14
106
ON THE KINEMATICS OF A PLANE.
[734
or, what is the same thing,
tan 0 = ( FX' - F/A') + (*'*/ + FT/),
say
sin 0, cos 0 = FX' - FXX ZX' + F IV ;
and then, as before,
x = a + #1 cos 0 - 2/1 sin 0,
2/ = /9 + #1 sin 0 + 2/1 cos 0 ;
or, what is the same thing,
x - X = cos 6 (xl — Xj) - sin 0 (yx - Fj),
2/ - F = sin 0 (Xl - X,} + cos 6 (yl - Ft),
where X, Y, Xl} F1} and therefore also 6, denote given functions of s. The formulae
will be of a like form if X, Y, X1} Y1 are given functions of a parameter t.
A well known but very interesting case is when two points of the moving plane
describe right lines on the fixed plane. This may be discussed geometrically as
follows: Suppose that we have the points A, G (fig. 3) describing the lines OA0,
OC0, which meet in 0; through A, G, 0 describe a circle, centre On and with centre
Fig. 3.
0 and radius =200i, describe a circle touching the first circle in a point /; and suppose
that A0, CQ denote points on the second circle. Then it is at once seen that, considering
the first or small circle as belonging to the moving plane, and the second or large
circle as belonging to the fixed plane, the motion is in fact the rolling motion of
the small upon the large circle ; and, moreover, that each point of the small circle
describes a right line, which is a diameter of the large circle. In fact, the angle
IOiG at the centre is the double of the angle IOC at the circumference; that is,
734] ON THE KINEMATICS OF A PLANE. 107
it is the double of the angle /0(70; and therefore (the radius of the small circle
being half that of the large circle) the arcs 1C, IC0 are equal, so that the rolling
motion will carry the point C along the radius OC0, and will, in like manner, carry
the point A along the radius OA0, or the motion will be as originally assumed.
And, in like manner, for any other point B of the small circle the motion will be
along the radius OB0 ; in particular, taking AB a diameter, the angle A0OB0 will be
a right angle ; and the motion is determined by means of the two points A, B
describing respectively the two lines OA 0, OB0 at right angles to each other, viz.
there is no loss of generality in assuming that the two fixed lines are at right
angles to each other. It thence at once follows, as will presently appear, that each
point of the moving plane describes an ellipse (but we have the special case already
referred to, each point on the small circle describes a right line, and also the special
case, the centre Ol of the small circle describes a circle). Considering any point Q
of the moving plane, let the line Q0l meet the small circle in the points E, F (or,
what is the same thing, let E, F be the extremities of the diameter which passes
through Q); then the points E, F describe the lines OE, OF at right angles to
each other, and Q is a point on EF or on this line produced; clearly the locus is
an ellipse having the lines OE, OF for the directions of its axes, and having the
lengths of the semi-axes = QF, QE respectively.
Taking the points to be A, B moving along the two lines OB0, OA, at right
angles to each other, these lines may be taken for the axes Ox, Oy; the point Ol
for the origin of the coordinates xlt y,, the axes 01x1 being in the direction 0,B
and 0^ at right angles to it; calling the length AB=2c, we have 01A = OlB = c,
and the angle ABO may be called 0 (but this angle was previously taken with a
contrary sign). We have then for the point P, having in regard to O,A\ and O.y, the
coordinates (x1} y^,
x = a + XT_ cos 0 — i/! sin 01
y = /3-x1s'm0-yl cos 0} '
where the sign of y, has been changed, and a = ccos0, /3 = csin0: the equations thus
become
x = (c + Xj) cos 0 — y1 sin 0,
y = (c — #1) sin 0 — y1 cos 0,
where observe that c + xly c-a\ are the distances M.A, M,B respectively. And we
have, conversely,
xl = x cos 0 — y sin 0 — c cos 20,
yl = — x sin 0 — y cos 0 + c sin 20.
If, in particular, yl = 0, then
. a?, y = (c + O cos 0, (c-#,)sm0;
or we have
of yn-
14—2
108 ON THE KINEMATICS OF A PLANE. [734
viz. the curve on the first plane is an ellipse, the semi-axes of which are ± (c + #0,
±(c-xl\ each taken positively; if a?1* + yi8 = cs, viz. if P be on the circle having AB
for its diameter, then y? = (c + #1) (c — %i), and we have
y + x = -(c-a;1)(smd -- ^— cos 6 ) -=- yx ( sin 0 - °- — OOitfj, =- (c-x,) + ylt
\ c — Xi j \ y\ i
viz. as mentioned above, the curve on the fixed plane is a right line.
In the general case, we have
x(c — a?0 + 2/2/i = (c2 - #i2 - 2/i2) cos 0,
xyi + y (c + a?i) = (c2 - #i2 - £/i2) sin 0,
and thence
{x (c - x,) + yytf + [xy, + y(c + x,)}2 = (c2 - x? - 2A2)2 ;
or, what is the same thing,
oc- {(c - xtf + y:-} + bxycyl + f {(c + serf + y?} = (c- - x? - yff.
Considering (x1, y^) as given, the curve traced out by P on the fixed plane is
of the second order; it would be easy to verify from the equation that it is an
ellipse, and to obtain for the position and magnitude of the axes the construction
already found geometrically.
The same equation, considering therein (x, y) as constant and (xlt y±} as current
coordinates, gives the curve traced out on the moving plane ; the curve is obviously
of the fourth order. Transferring the origin to A, we must in place of xl write
xl — d ; the equation thus becomes
x2 {(x, - 2c)2 + 2/r} + toyixy + y- (x,2 + y?) = (x? + y? - 2c^)2 ;
or, what is the same thing,
Oi2 + yi - 2c#02 - (x2 + y2) (x,2 + 2/j2) + 4c# (xxl - yy,} - 4c2^- = 0 ;
and if we suppose herein x = 0, it becomes
(#i2 + 2/i2 - 2c^)2 - y- (x? + y/) = 0 ;
or, writing #1 = 7*1 cos 01} y1 = r1sin61, where 0a = angle QAB, this is
(r, - 2c cos 002 - 2/2 = 0,
or say it is
TI = 2c cos 0! - y,
which is the polar equation of the curve described on the moveable plane by the
point 8, whose coordinates in respect to Ox and Oy are (0, y).
There is no loss of generality in assuming x = 0. In fact, starting with any point
8 whatever of the fixed plane, if we draw OS meeting the small circle in A, and
734] ON THE KINEMATICS OF A PLANE. 109
through 0 draw at right angles to this a line meeting the same circle in B, then,
as before, the points A and B move along the fixed lines OA0, 050; or as regards
the relative motion, taking A, B as fixed points, we have the originally fixed plane
now moving in such wise that the two lines OA0, OB0 thereof (at right angles to
each other) pass always through the points A and jB respectively, and the curve is
that described by the point S on the line OA ; the point 0 describes the circle on
the diameter AB (the small circle), equation rz = 2c cos 6l ; and OQ having a given
constant value =y, we have for the curve described by the point 8 the foregoing
equation rt = 2c cos 6l — y; or writing y = — /, that is, taking S on the other side of
0 at a distance OS=f, the equation is rx = 2c cos &i+f', viz. this is a nodal Cartesian
or Lima^on, the origin being an acnode or a crunode according as /> or <2c; and
if f=2c, then we have the cuspidal curve or cardioid rx = 2c (1 + cos #1), =4ccos2£01.
The general conclusion is that the centre 0 of the large circle describes on the
moving plane a small circle (centre Oi), and that every other point of the fixed plane
describes on the moving plane a Limac,on having for its node a point of the small
circle, and being, in fact, the curve obtained by measuring off along the radius vector
of the small circle from its extremity a constant distance.
Considering in connexion with the point, coordinates (a?1} y^, (x, y), a second
point, coordinates (Xlt F,), (X, Y), in regard to the two sets of axes respectively,
we have
x=(c + ocj cos 6 — yl sin 6, X = (c + XJ cos 0 — Yl sin 6,
y = (c — XT) sin B — y1 cos 6, Y=(c — Xj) sin 6 — F: cos 6 ;
from the first two equations we have
cos 6 : sin 9 : 1 = a? (c — a?j) + yy-i : xy1 + y(c + x1) : c2 — x? — yf ;
and substituting these values in the second set, we find
X : Y : 1
= x {c2 + c (X, - x,) - X& -Y1y1}+y{ c (yl - F,) + y,X, - ^ Y,}
:x{ C(y1-Y1)- y,X, + x, Y,}
or the points (x, y), (X, Y), considered as each of them moving on the fixed plane,
are homographically related to each other.
To find the curve enveloped on the fixed plane by a given curve of the moving
plane, we have only in the equation f(xl} 2/0 = 0 of the curve in the moving plane
to substitute for xl, yt their values in terms of x, y, 6, and then considering 6 as
a variable parameter, to find the envelope of the curve represented by this equation.
And, similarly, we find the curve enveloped on the moving plane by a given curve
of the fixed plane.
110 ON THE KINEMATICS OF A PLANE. [734
Thus, in the particular case of motion above considered, writing, as before,
x = (c + Xj) cos 0 - y1 sin 6,
y = (c- n\) sin 6 — 7/1 cos 6 ;
or conversely
xl= x cos 6 — y sin 0 — c cos 26,
yl = — x sin 6 — y cos 6 + c sin 20 ;
the envelope on the moving plane of the line
Ax + By + C = 0
of the fixed plane is given as the envelope of the line
[A (c + #0 - Byj\ cos0+{-A+B(c- xj} sin 8 + C= 0 ;
viz. this is
{A (c + x,) - By,Y + {Ay, -B(c- atf? - C* = 0 ;
that is,
(A2 + B?) (x* + y* + c2) + 2 (A* - #0 Gxl - ^ABcy, = 0,
a circle.
But the envelope on the fixed plane of the line
A^ + Byi + C-Q
of the moving plane is given as the envelope of the line
C + (Aas + By) cos 6-(Ay + Bx) sin d - AC cos W + BO sin 20 = 0,
which can be obtained by equating to zero the discriminant of a qtiartic function,
and is apparently a sextic curve.
735] 111
735.
NOTE ON THE THEORY OF APSIDAL SURFACES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879),
pp. 109—112.]
I OBTAIN in the present Note a system of formulae which lead very simply to
the known theorem, that the apsidals of reciprocal surfaces are reciprocal ; or, what is
the same thing, that the reciprocal of the apsidal of a given surface is the apsidal
of its reciprocal; the surfaces are referred to the same axes, and by the reciprocal is
meant the reciprocal surface in regard to a sphere radius unity, having for its centre
a determinate point, say the origin ; and it is this same point which is used in the
construction of the apsidal surfaces. The apsidal of a given surface is constructed as
follows; considering the section by any plane through the fixed point, and in this
section the apsidal radii from the fixed point (that is, the radii which meet the curve
at right angles), then drawing a line through the fixed point at right angles to the
plane, and on this line measuring off from the fixed point distances equal to the
apsidal radii respectively, the locus of the extremities of these distances is the apsidal
surface. We have the surface, its reciprocal, the apsidal of the surface, the apsidal of
the reciprocal ; and I take
(a, y, z\ (x>, y', z'\ (X, Y, Z\ (X', Y', Z'}
for the coordinates of corresponding points on the four surfaces respectively.
The condition of reciprocity gives xx' + yy' + zz' — 1 = 0, and (the equations being
U = 0, V = 0) x', y', z' proportional to dx U, dy U, dz U, and x, y, z proportional to
dx' U', dy> U', d? U' ; or, what is the same thing, we must have
x'dx 4- y'dy + z'dz = 0 and xdx' + ydy + zdz = 0 ;
one of these is implied in the other, as appears at once by differentiating the equation
xx' 4- yy + zz' — 1 = 0.
112 NOTE ON THE THEORY OF APSIDAL SURFACES. [735
The other two surfaces will therefore be reciprocal if only we have the like
relations between the coordinates (X, Y, Z) and (X1, Y', Z'}; that is, if
XX' + YY' + ZZ' -1 = 0,
X'dX+ Y'dY+Z'dZ=0,
XdX' + YdY' + ZdZ' = 0.
To find the apsidal surface, we consider an arbitrary section as cos a + y cos /3 + z cos 7 = 0
of the surface U = 0, and seek to determine the apsidal radii thereof, that is, the
maximum or minimum values of R- = x2 + y2 + z- when x, y, z vary subject to these
two conditions. Writing x', y', z' to denote functions proportional to dxll, dyU, dzll,
we thus have the set of equations
x + \x' + fj> cos a = 0,
y + \y' + fi cos /3 = 0,
z +\z' + p cos 7 = 0,
where X, p are indeterminate coefficients ; taking then X, Y, Z as the coordinates of
the extremity of the line drawn at right angles to the plane, we have R- = X- -f Y- + Z-,
X Y Z
and cos a, cos /3, cos 7 = -p- , -p , -p ; substituting these values in the equation
x cos a + y cos $ + z cos 7 = 0,
we have Xx + Yy+Zz = 0, and substituting in the other equations, and instead of
\, fj, introducing the new indeterminate coefficients p, a-, we obtain
X , Y, Z = px + <rx', py + a-y', pz + crz'.
Hence these last equations, together with
R2 = X2 + Y2 + Z2 = a? + y2 + z\
and
Xx+ Yy + Zz=l,
contain the solution of the problem. If for convenience we introduce R'2 to denote
aP + y'^ + z'*, and imagine the absolute values of x', y , z' determined so that xx' + yy' + zz = 1,
then substituting for X, Y, Z their values in the equations X- + Y2 + Z2 = R2 and
Xx + Yy + Zz= 1, we find
R* = p*R-
and thence
^-
or, finally assuming
p~
we have
X, 7, Z = x- J2V, y - R2y', z - R-z,
each divided by
* - 1),
735] NOTE ON THE THEORY OF APSIDAL SURFACES. 113
where I recall that x, y', z' are proportional to dxU, dyU, dzU, and are such that
xx + yy' + zz = 1 : they in fact denote
dxU, dyU, dzll, each divided by xdx U + ydy U + zdz U ;
and that R- and R- denote x2 + y* + z2 and <v'a + y'a + z'2 respectively. The coordinates
X, Y, Z of the point of the apsidal surface are thus determined as functions of x, y, z.
For the apsidal of the reciprocal surface, we have in like manner
X1, Y, Z' = x'-R'*x, y'-R'*y, z-R'*z,
each divided by
- */(R2R2 - 1),
and then the two sets of values give, not only
as is obvious, but also
X'dX + Y'dY+ Z'dZ = 0, and XdX' + YdY' + ZdZ' = 0.
In fact, writing for a moment p, p instead of R-, R'2, and *J(R-R'2 — 1) = \/(pp' — I), = CD,
then
X'dX +Y'dY+ Z'dZ
ft) ft)
_ x' — xp (dx — pdx' — x'dp (x — x'p) dco]
~ f "T* OCCt
CO ( CO CO' )
= — { x'dx + y'dy + z'dz
ft)2
— p (x'dx' + y'dy' + z'dz)
(x'z + y'2 +z'2 )dp
— p (xdx + ydy + zdz )
+ pp' (xdx -f ydy' + zdz' )
+ p (xx' + yy' 4- zz' ) dp}
dco ( . . .
xx +yy + zz
&)
- p (®'2 + y'2 + z'- )
+ pp (xx + yy' + zz' )},
or, since the terms in { } are
0 - p . | dp' - p'dp -p.^dp + 0+ p'dp, = - % (pdp + p'dp),
and
this is
= — {— ^(pdp +p'dp)+ codco}, =0,
in virtue of ar=pp'-l. And similarly the other equation XdX' + YdY' + ZdZ' = 0
might be directly verified.
c. xi. 15
114 [736
736.
APPLICATION OF THE NEWTON-FOURIER METHOD TO AN
IMAGINARY ROOT OF AN EQUATION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879),
pp. 179—185.]
I CONSIDER only the most simple case, that of a quadric equation xz = n2, where
7i2 is a given imaginary quantity, having the square roots n, and — n ; starting from
an assumed approximate (imaginary) value x = a, we have (a + h)2 = n2, that is,
= rc2, h = --, and a + h =
that is, the successive values are
_ a- + n- _ a? + n2
®1 — o > ^2 — ri > — i
2a 2ttj
and the question is, under what conditions do we thus approximate to one determinate
root (selected out of the two roots at pleasure), say n, of the given equation.
The nearness of two values is measured by the modulus of their difference ;
thus a nearer to n, than al is to n, means mod. (a — n) < mod. (ax — n), and so in
other cases; in the course of the approximation a, alf a«, ... to n, any step, for
instance a to a1} is regular if a-i is nearer to n than a is, but otherwise it is
irregular; the approximation is regular if all the steps are regular, and if (after one
or more irregular steps) all the subsequent steps are regular, then the approximation
becomes regular at the step which is the first of the unbroken series of regular
steps.
We do by an approximation, which is ultimately regular, obtain the value n, if
only the assumed value a is nearer to n than it is to — n ; or, say, if the condition
mod. (a — n) < mod. (a 4 n) is satisfied, and the approximation is regular from the beginning
736]
APPLICATION OF THE NEWTON-FOURIER METHOD.
115
if mod. (a — n) < f mod. n, viz. this condition is a sufficient one*; the first step a to c^
will moreover be regular under a less stringent condition imposed upon a; and it would
seem that, without the condition mod. (a — n) < § mod. n being satisfied, the subsequent
steps will in some cases be also regular; that is, that the last-mentioned condition
is not a necessary condition in order to the approximation being regular from the
beginning; it is, however, the necessary and sufficient condition, to be satisfied by the
modulus of a — n, in order that the approximation may be regular from the beginning.
All this will clearly appear from the geometry.
We take N, N' (fig. 1) to represent the values n, —n; and similarly A, A1} &c.
to represent the quantities a, alt ... ; we have then
AN = mod. (a— n),
= mod. (^ — n) ...,
so that the approximation is measured by the approach of the points A, A-^ to N.
The line NN' joining the points N, N' passes through, and is bisected at, the origin
0 ; drawing then QQ' through 0 at right angles to NN' the condition
mod. (a — n) < mod. (a + n)
means that the point A, which represents the imaginary quantity a, lies on the
-ZVr-side of QQ', and it will be assumed throughout that this is so. Take now on the
line ON, OM = ±ON, and on N'M as diameter, describe a circle, which may be called
the "circle of unfitness"; regarding as an area the segment hereof which lies on
the JV-side of QQ', say this is the "segment of unfitness." It will be shown that
if according as A is situate inside, on the boundary of, or outside the segment of
* In the Smith's Prize Examination, Jan. 28, 1879, I gave the theorem under the following form : "If a, n
are imaginary quantities, the latter of them given, and the former assumed at pleasure, subject only to the con
dition mod.(a-?j) <f mod. n; then if «i=a-^. ^^t^"' &C>? sh°W that the terms a' Ul' a-""' wil1 converge
to the limit ?i." This is strictly true, but it would have been better to say "will converge regularly."
15—2
116 APPLICATION OF THE NEWTON-FOURIER METHOD [736
unfitness, A^N will be greater than, equal to, or less than AN. It may be added
that, if A be within or upon the boundary of the segment of unfitness, then A^
will be outside it, but this by no means hinders that the next point A.,, or some
later point, shall be within the segment of unfitness; arid, further, that when A is
outside the segment of unfitness, then the next point Aa, or some later point, may
very well be within the segment of unfitness; the conclusion is, that A being inside
the segment of unfitness, A^ is less than AN, but that it does not thence follow
that A,N is less than A,N, A,N than A*N,...; the approximation although regular
at the first step, may then, or afterwards, for a step or steps, cease to be regular.
If, however, AN be less than $ON, that is, if the condition mod. (a - n) < f mod. n
be satisfied, then the point A lies within the circle centre N and radius NM, and
is consequently outside the segment of unfitness ; A^N being less than AN, the point
Al is a fortiori outside the segment of unfitness, and the like for all the subsequent
points Az, A3,..., that is, in this case, the approximation is regular throughout. The
circle, centre N, and radius NM, = f mod. n, may be called the "safe circle"; and
the conclusion is that, if the point A or any subsequent point be within the safe
circle, then every subsequent point will be within the safe circle, and the approximation
will be regular.
The successive points A, Alt A», ... (or, as it will be convenient to call them,
Alt A.-,, ...) may be obtained each from the preceding one by a simple geometrical
construction.
I recall that any circle through the two (imaginary) antipoints of N, N' is a
circle having its centre on the indefinite line NN' ; it is such that the ratio of the
distances of a point thereof from the points N, N' respectively has a certain constant
value, viz. for the circles with which we are here alone concerned, those which lie
on the N-side of QQ', the centres lie beyond the point N (further away, that is, from
0), and the values of the ratio, distance from N to distance from N', are less unity.
Starting then from the given point AI, for which this ratio Afl : A^r/ has a
given value, suppose A1N = kA1N', we describe a first circle (passing of course through
_4j) for each point of which this ratio has the value k; let the diameter of this
circle be V^W-^ Fj being the extremity between 0 and N, W^ (not shown in the
figure), that beyond N ; we then describe a second circle, for which the ratio is
= &2; let its diameter be F2Tf2, F2 being the extremity between 0 and N (or say
between Vl and N), W2, that beyond N (or say between N and Tfj); the point
An lies on this second circle, and is determined as the single intersection of the line
FoAj with the second circle. And of course drawing a third circle, for which the
ratio is =fc4, on the diameter V3W3, then A3 lies on the third circle, and is the
intersection with it of the line V3A2, and so on ; the radii of the successive circles
diminish very rapidly, their centres, in like manner, continually approaching the point
N; hence, the points Alt A2, A3, ..., which lie on the several circles respectively
approximate, and that very rapidly, to the point 0. But by what precedes, if, for
instance, the point Al be within the segment of unfitness, then also some of the
subsequent points may be within the segment of unfitness, and for each point Ap,
736] TO AN IMAGINARY ROOT OF AN EQUATION. 117
for which this is the case, the next point Ap+1 is at a greater distance, so that
NAP+1>NAP; it is, however, clear that we always arrive at a point Aq, such that
NAq<$ON, and so soon as such a point is arrived at the approximation becomes regular.
The point A2 determined from Alt as above, is a point such that the subtended
angle NA«N' is = twice the subtended angle NA^' ; or calling the latter angle </>,
the former is = 2$. It is, in fact, this property which gives rise to the construction ;
for let the values of A^, A^7', regarded as imaginary quantities, be called for a
moment
P! (cos 0j + i sin 0^, p/ (cos 0/ + i sin 0/) ;
and, similarly, those of A.2N, A.2N' be called
p» (cos 02 + i sin 02), p2' (cos 02' + i sin 0,') ;
then these are the values of «j — n, a-^ + n, a» — n, a.2 + n respectively, or we have
= A [cos (0j — 0/) + i sin (0X — 0/)| = k (cos 0 + i sin <£),
Cvj "T" *' P\ •
- = ^ {cos (0, - 02') + i sin (02 - 02')} = ^2 (cos 2<f> + i sin
that is,
tto — ?l
a2 + n
which relation between a.2t ax is in fact the original relation
and, conversely, alt a.2 being thus connected, then the representative Aa is obtained
from the representative point Al by the foregoing geometrical construction.
I give the analytical proofs; we may without loss of generality take, and it is
convenient to do so, the axis of x as coinciding with the line ON, and to put also
ON = 1. We then in place of the original coordinates x, y of any point take the
new coordinates k, </> which are such that
x + iy - 1
T-^-TT =
x + iy + 1
x — iy — 1
- ^L __ _ l-p—ify •
• • i "'^ ?
x — %y + 1
equations which may also be written
- if = e-** [x2 +(y
or, what is the same thing,
x- + f- - 1 - 2y cot <f> = 0,
118
APPLICATION OF THE NEWTON-FOURIER METHOD
[736
where of course the equation with k shows that k is equal to the ratio of the
distances of the point from the points N, N' respectively, and the equation in <£,
taken in the second form, shows that <f> is the angle subtended at the point by N, N'.
It is sometimes convenient to write ke^, ke^'^^p, q respectively; we then have
. l+p . 1+q
~ ~~
Suppose for a moment that we have (pl9 qj, (p2) <?2), (ps, q3) as the (p, q) coordinates
of any three points, the condition that these three points may lie in a line, is given
in the form, determinant = 0, where each line of the determinant is of the form
l+p l+q
1,
1-p' 1-q'
or, what is the same thing, it is
l—pq+p — q, l-pq-p + q, 1 + pq — j
pq-l, p-q, p + q-2,
or, again
viz. the condition is
p3q3 -I, p3- q3, p3 + q3-
Suppose the (k, <£) coordinates of the three points are (I, a), (m, ft), (n, 7) respectively;
then this equation is
I2 — 1 , I sin a , I cos a — 1 =0,
m2 — !, m sin ft, m cos ft— 1
n2 — 1, n sin 7, n cos 7 — 1
I2 — 1, I sin a, 1 =0,
m2— 1, m sin ft, 1
n2 — 1, n sin 7, 1
viz. it is
I2 — 1, I sin a, I cos a
m2 — 1 , m sin ft, m cos ft
n" — 1, n sin 7, n cosy
or, what is the same thing, it is
[(I- - 1) mn sin (ft - 7) + (m2 - 1) nl sin (7 - a) + (n2 - 1) foi sin (a - ft)]
+ [(m2 - n2) I sin a + (n2 - I2) m sin ft + (I- - m-) n sin 7] = 0.
If in this equation 7 is put = TT, and ft = 2a, so that sin (a — ft) = — sin a, the equation
will contain only terms in sin a, and sin 2a, viz. it will be
that is,
[ (m2 — ri*)l + (m2 — 1) nl — (n- — 1) lm] sin a
+ [- ( I" - 1 ) m n + m ( n- - 1-) ] sin 2a = 0,
I (m — 1) (n + 1) (m — n) sin a + m (m + 1) (n — I2) sin 2a = 0,
736] TO AX IMAGINARY ROOT OF AN EQUATION. 119
or, what is the same thing,
(m + 1) sin a {I (n + 1) (m -n)+2m(n- lz} cos a} = 0,
which is satisfied for any values whatever of I, m, n, by a proper value of cos a ;
and is also satisfied irrespectively of the value of a. if only m = n = I2 ; or, writing
k instead of I, say if I = k, m = n = k* ; that is, writing also </> in place of a, the
three points
(k, </>), (k\ 20) and (*- TT)
are in a right line; viz. the point A1} circle k, subtended angle <f> ; the point A2,
circle k2, subtended angle 20 ; and the point V2, same circle, subtended angle IT ;
are in a right line.
The equation of the circle of unfitness can be obtained more easily in a different
manner ; but I have thought it worth while to give the investigation by means of
the foregoing (p, q) coordinates.
Suppose that pit ql refer to the point A^ : then we have
(AW = (x, - I)2 + y,» = fa + tyx -!)(%- iyi - 1), = (\^ - l) f ]±& - l
\i PI / \ i — jj
that is,
Similarly, if pa, q2 refer to the point A2, then
?12>
since ^2, q.2=pi*, q^. The two are equal if
(l+p1)(l
that is,
Pi + qi + 1 = 0.
Writing for a moment xl + iyl = %, x^ — iyl = y, we have
£ + 1' -n
and the equation is
that, is,
or substituting for £, 77 their values, the equation is
that is,
120 APPLICATION OF THE NEWTON-FOURIER METHOD [736
the equation of a circle on the diameter N'M, which is, in fact, the before-mentioned
circle of unfitness; viz. A-L being on the circumference of this circle, or say on the
boundary of the segment of unfitness, then A-^N = A»N; whence also, according as
Al is inside or outside the segment, A1N<A^N or >A»N.
Suppose A1 to be on the circle, that is, p1+q1+l =0; it is easy to show that
the locus of A» is also a circle. We have in fact (p^ + q^ — 1 = 0, that is,
pz + qz + 2^-1=0,
or say
viz. this is
that is,
or finally
3 — 2£2
Measuring off from 0 in the direction of ON, a distance OS = ^,2 (always >£,
-L \" Z/V
since &2<1), the circle in question is that on the diameter N'S; this is a circle
touching at N', and containing within it the circle of unfitness ; if k = 1 (that is, for
Al on the line QQ') it becomes identical with the circle of unfitness, but except
in this limiting case it does not meet the circle of unfitness in any point on the
JV-side of QQ', that is, A^ being on the boundary of the segment of unfitness A,, is
never on this boundary ; and it thus appears that A1 being inside the segment, A., is
always outside the segment.
It is to be further noticed, that we have
or
that is,
"7 A » V \ _, """" -I- ^~" In./ * HT\n>
where T is the tangential distance of A1 from the circle of unfitness; there should,
it appears to me, be some more elegant formula for the ratio A^ + A2N which
determines whether the step is regular or irregular.
736] TO AN IMAGINARY ROOT OF AN EQUATION. 121
It is worth noticing how the conditions
mod. (a — n) < mod. (a + n) and mod. (a — ri) < f mod. n,
present themselves in the real theory. Making the usual construction by means of
the parabola y = xi, the first condition means that the point A must be taken on
the ^-side of 0 (fig. 2) ; the second that, in order to the regularity of the approxi-
Fig. 2.
N'
mation, A must be taken at a distance from 0 >%ON; in fact, if (as in the figure)
OA = ±ON, then AN=NAl} or the point A1 is at an equal distance with A from
N; and thence, according as OA is greater or less than ^ON, the point Al is
nearer or further than A to or from N.
C. XI.
16
122 [737
737.
ON A COVARIANT FORMULA.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879),
pp. 224—226.]
STARTING from the equation
/*
/>>
which presents itself in the Newton-Fourier problem, it is easy to see that, if a be
a root of the equation fx = 0, then
_ ^(x-a)f'x-fx
contains the factor (#-a)2, that is, the equation (x - x^f'x -fx = 0, considered as an
equation in x containing the parameter xlt will have a twofold root, if xl is equal
to any root a of the equation /# = 0; and, consequently, the discriminant in regard
to x of the function (x-x^f'x-fx will contain the factor fx^ But if fx be of the
order n, then the discriminant is of the order 2?i-2 in #1; and there is consequently
a remaining factor <j>xl of the order n — 2.
The like theorem applies to the homogeneous form
(xy, - x,y} a + ft /(*, y) - (^ -
which reduces itself to the foregoing on writing a = I, ft = Q, y = yl = 1 ; or, changing
the notation, say to the form
737] ON A CO VARIANT FORMULA. 123
viz. the discriminant hereof in regard to £, ij, being a function, homogeneous of the
order 2n — 2 in regard to x, y, to a, ft, and to the coefficients of /(£, 77), will
contain the factor f(x, y), and there will be consequently a remaining factor of the
order n — 2 in (x, y}, 2n — 2 in (a, fi) and 2?i — 3 in the coefficients of /(£, 17).
The most simple case is when /(%, 77) is the quadric function (a, b, C$1, T?)2.
The form here is
-(ay-fix)(a, b, c$f, ^)2 = (a, b,
where the coefficients are
a = 2y (a* + bfi) - a (ay - fix), = a fa + (aa + 2bfi) y,
b = y(ba + cfi)-x (aa + bfi) -b(ay- fix},
= — aouc + cfty ,
c = - 2# (&a + c/3) — c (ay — fix), = — (26a + cfi) x — cay ;
and we then have
ac - b2 = - (26a/3 + cfi2) ax"
- {2a6aa + (2ac + 462) afi + 26c/32} «y - (««2 + 26«yS) c?/2
- a^2 . a«2 - {- 2aca/3} a;y - c/32 . cy2,
which is
= - (aa2 + 26a/3 + c/32) (ax- + 2bxy + cy").
The discriminant is in this case
= — (a, b, c$a, fi)2.(a, b, cQx, y)*.
In the case of the cubic function (a, b, c, dj[j~, rff, the form is
(&/ - a»i) {3 (aa + bfi, ba + cfi, ca + d&$g, ^}
-(ay -fa) (a, b, c, d$f, ^ = (a, b, c,
the values of the coefficients being
a= a fix +(2aa+3bfi)y,
b = — aax +( ba + 2cfi) y,
c = - (2ba + cfi) x + dfi y,
d = - (3ca + 2dfi) x- da y.
Attending only to the terms in x2, we have
ac - b2 = - (aa2 + 26a/3 + cfi-) ax",
ad - be = - 2 (6a2 + 2cafi + dfi2) ax2,
bd-c2 = {(3ac-462)a2 + (2arf-46c)a/S-c2/32}a;2.
16—2
124
ON A COVARIANT FORMULA.
[737
And hence, in
a2d2 + 4ac3 + 4b3d - 3b2c2 - 6abcd, = (ad - be)2 - 4 (ac - b2) (bd - c2),
we have the term
400* . x [a (bof + 2ca/3 + d/32)2 + (aa2 + 26a/3 + c^2) {(3ac - 462) a2 + (2ad - 46c) a/3 - c2/?2}] ;
then, forming the analogous term in y4, and assuming that the whole divides by
(a, b, c, d^x, y)s, and also expanding the ayS-functions within the square brackets, we
find
Discriminant = 4 (a, b, c, dQx, y)3 multiplied by
3a2c -3a62
2a*d + 6abc - 8b3
6abd+6ac2 -I2b2c
Qacd - Qbc2
ad2 - c3
Writing down the Hessian of (a, b, c, dQa, /3)3,
H = (ac — b2, ad — be, bd — c
and the cubicovariant
a2d - Babe + 2b3
abd - lac2 + b2c
- acd + 2b2d - be2
'. -ad2 + 3bcd-2c3
it is easy to see that the coefficient of x is
= 3 (a, b, c$a, @)2.(H-
hence also that of y is
a2d - b3
Qabd - Qb2c
6acd + Qb-d — I2bc2
'2 +6bcd-8c3
/3)2,
= 3(6, c, dJia, /3)
and the final result is that the discriminant = 4 (a, b, c, d^jx, y)3 multiplied by
{3 (a, b, c, d$a, 0)3(*, y)fT+ (ay -#&)*}.
It would be interesting to calculate the result for the quartic (a, b, c, d, e^, rj)4.
March 14, 1879.
738] 125
738.
NOTE ON A HYPERGEOMETRIC SERIES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879),
pp. 268—270.]
IN the memoir on hypergeometric series, Schwarz, "Ueber diejenigen Falle, &c.,"
Crelle, t. LXXV. (1873), pp. 292—335, the author shows, as part of his general theory,
that the equation
x .\— x dx x.\—x
which belongs to the hypergeometric series F(±, — -j^, f, a?), is algebraically integrable,
having in fact the two particular integrals
± V(- a5
where a is a prime sixth root of -1, «6+l=0, or say a4-a2+l=0 (see p. 326,
a being for greater simplicity written instead of S2, and the form being somewhat
simplified).
It is interesting to verify this directly ; writing first y = ^(Y) and then x = X3,
the equation between Y, X is easily found to be
F.__F_1
1 dX2 l-X3* dX 2(dXj+I-X**
and the theorem in effect is that that equation has the two particular integrals
P and Q being linear functions of X : in fact,
P = a - a.*X,
126 NOTE ON A HYPERGEOMETRIC SERIES. [738
Starting say from the equation
or, as it is convenient to write it,
F = p4 + $,
where P and Q are assumed to be linear functions of X, we have
d*Y -> 3
±A = - jp-ip'2 - iQ'i Q'-\
and thence
dX"
Yd? = | (P' + Q') + iP-^P' +
where P', Q' are written to denote the derived functions of P, Q respectively.
Substituting these values, the resulting equation contains on the left-hand side
a rational part, and a part with the factor P~%Q~*, and it is clear the equation
can only be true if these two parts are separately = 0. We have thus two equations
which ought to be verified ; viz. after a slight reduction these are found to be
1 9Y- Y
(QP* + PQ*) + j^ (P' + Q') - f-^3 (P + Q) = 0,
P'Q'-1 + Qr-P'°- + PQP'Q' + PQ (PQf + P'Q) - P2^2 = 0,
and it is very interesting to observe the manner in which these equations are, in
fact, verified by the foregoing values of P, Q.
We have
p + Q = (a-tf)(i+X), p' + y = *-(?,
and hence
2X(P' + Q')-
or, in the first equation, the second part
is
738] NOTE ON A HYPERGEOMETRIC SERIES. 127
viz. this is
We have
QP'* + PQ'2 = a10 (- a5 + aX) + a2 (a - a5Z),
= a3 - a15 - (a7 - a11) X, = (a - a5) X ;
and
hence
and the sum of the two parts is = 0.
Similarly as regards the second equation, the second part
s
Here PQ' + P'Q is a (a - a5X) - a5 (- a5 + aX), which is = 1 + 2Z ; and PQ being
= 1+X + X-, the term in { } is
hence, outside the { } writing for PQ its value = 1 + X + X'2, the term is
which is the value of the second part in question ; the first part is
(PQ' + QPJ-PQP'Q', =(1
and the sum of the two terms is thus = 0.
128 [739
739.
NOTE ON THE OCTAHEDEON FUNCTION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XVI. (1879),
pp. 280, 281.]
A SEXTIC function
U=(a, b, c, d, e,f, g$x, y)\
such that its fourth derivative
(U, If?, = (ae-4,bd + '3c*)x*
+ 2 (af - 3be + 2cd) a?y
+ (ag - 9ce + 8d2) afy8
+ 2 (bg - 3cf + 2de) xy3
is identically =0, is considered by Dr Klein, and is called by him the octahedron
function. Supposing that by a linear transformation the function is made to contain
the factors x, y, or what is the same thing assuming a = 0, g = 0, then the equations
to be satisfied become
which are all satisfied if only c = d = e = 0 ; and then assuming, as is allowable,
b = -/=!,
we have his canonical form xy (x* - y4) of the octahedron function.
But the equations may be satisfied in a different manner; viz. the first and last
equations give
7_3c2 ,_3e2
' S~'
739] NOTE ON THE OCTAHEDRON FUNCTION. 129
and, substituting these in the remaining equations, they become
^ (- 9ce + 8d2) = 0, - 9ce + 8d- = 0, ^ (- 9ce + 8d2) = 0,
all satisfied if only — 9ce + 8d* = 0. Assuming b =/= 2, the values are
b, c, d, e,f=2, 2^(2), 3, 2V(2), 2,
and the form is
xy
This is, in fact, a linear transformation of the foregoing form XY(X*— F4) ; for
writing
we have
and therefore
or finally
and the two forms are thus identical.
c. XL 17
130
740.
ON CERTAIN ALGEBRAICAL IDENTITIES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879),
pp. 281, 282.]
IF P0, Plt P2 are points on a circle, say the circle «2 + ?/2 = l, then it is possible
to find functions of (P0, PO and of (Plt P2) respectively, which are really independent
of Pj, and consequently functions of only P0 and P2: the expression "function of
a point or points" being here used to mean algebraical function of the coordinates of
the point or points. Thus the functions of (P0, Pj) and of (P1; P2) being
#03/1 — #i2/<» and #i#2 + y\y^ %\y* — %»y\> we have
and another like equation. This depends obviously on the circumstance that the
coordinates of a point of the circle are expressible by means of the functions sin,
cos, x = cos u, y = sin u ; and the identity written down is obtained by expressing the
cosine of uz — u0, = (w2 — u^) + (i^ — w0)» m terms of the cosines and sines of ua— i^
and U^ — UQ.
Evidently the like property holds good for a curve, such that the coordinates of
any point of it can be expressed by means of " additive " functions of a parameter
u ; where, by an additive function f(u), is meant a function such that f(u + v) is
an algebraical function of f(u), f(v) ; the sine and cosine are each of them an additive
function, because
sin (u + v) = sin u V(l — sin2 v) + sin v V(l — sin2 u),
and, similarly, for the cosine. But it is convenient to consider pairs or groups f(u),
<j>(u),..., where f(u + v), <f>(u + v\... are each of them an algebraical (rational) function
of f(u}, 0 (u), . . . , f(v), <f>(v), ...; the sine and cosine are such a group, and so also are
the elliptic functions sn, en, dn ; but the H and ®, or say the ^--functions generally,
are not additive.
740] ON CERTAIN ALGEBRAICAL IDENTITIES. 131
In the case of the elliptic functions, we may consider the quadriquadric curve
so that the coordinates of a point on the curve are sn u, en u, dn u. Taking then
P0, P1} P2, points on the curve, and (#„, y0, 20), (xlt ylt z^), (#2, y2, za), the coordinates of
these points respectively, we have in the same way, from u2 — UQ = (u2 — v^) + (MI — MO),
three equations, of which the first is
1 - teicfxf (1 - texfxff (1 - l&xfxff - A2 (d^yo^o - x»y\Zif (o^y^i -
The form of the right-hand side is
A + Bx^z-i
C + Dx^y^z-i '
where A, B, C, D are each of them rational as regards a^2; and it is easy to see
that the equation can only subsist under the condition that we have separately
0 — ^0^2 _ -4 _ -5
'~
implying of course the identity AD- 5(7=0. The values of B and D are found
without difficulty; we, in fact, have
B = 2k2 (x/ - x2 ) (xfyjsQj&i +
D = 2A8
so that, comparing the left-hand side with B + D, we have the identity
«a*yoV ~ ^o22/22^2 = (^22 - tfo2) (1 - ^a?aV),
which is right. The comparison with A + C would be somewhat more difficult to effect.
17—2
132 [741
741.
ON A THEOREM OF ABEL'S RELATING TO A QUINTIC
EQUATION.
[From the Proceedings of the Cambridge Philosophical Society, vol. III. (1880),
pp. 155—159.]
THE theorem in question is given, (Euvres Completes, [Christiania, 1881], t. II.,
p. 266, as an extract from a letter to Crelle dated 14th March, 1826, as follows:
" Si une equation du cinquieme degre dont les coefficients sont des nombres
rationnels est resoluble algebriquement, on peut donner aux racines la forme suivante :
x = c + caaa
ou
a = m + n V(l + e-) + V|> (1 + e3 + \/(l + e-))],
a^ = m- n V(l + e-) + >J[h (1 + e2 - V(l + e2))],
a2 = m + n V(l + e-} - \/[h (1 + e2 + V(l + e2))],
a3 = TO - n V(l + e2) - VO (1+ e- - V(l + e2))],
A = K + K'a + K"a, + K'"aa,, A1 = K+ JK'a, + K"a3
A2 = K + K'a, + K"a + K'"aa, , A3 = K + K'a,
Les quantitds c, ^, e, m, n, K, K', K", K'" sont des nombres rationnels. Mais de
cette maniere 1'equation of + ax + b = 0 n'est pas resoluble tant que a et b sont des
quantites quelconques. J'ai trouve de pareils theoremes pour les equations du 7fcme,
It is easy to see that a; is the root of a quintic equation, the coefficients of
which are rational and integral functions of a, a1} a2, as: these coefficients are not
symmetrical functions of a, a1} a.2y a3, but they are functions which remain unaltered
741] ON A THEOREM OF ABEI/S RELATING TO A QUINTIC EQUATION. 133
by the cyclical change a into a1} a^ into a2, «2 into a:i, a3 into a. But the coefficients
of the quintic equation must be rational functions of c, h, e, m, n, K, K ', K", K'" :
hence regarding a, al, a.2, a3, as the roots of a quartic equation, the coefficients of
this equation being rational functions of m, n, e, h, this equation must be such that
every rational function of the roots, unchanged by the aforesaid cyclical change of
the roots, shall be rationally expressible in terms of these quantities m, n, e, h: or,
what is the same thing, the group of the quartic equation, using the term "group
of the equation" in the sense assigned to it by Galois, must be aa^a^, a^^a,
Oya.Aaal} a-faa-fi*. And conversely, the quartic equation being of this form, x will be
the root of a quintic equation, the coefficients whereof are rational and integral
functions of c, h, e, m, n, K, K', K", K'".
To investigate the form of a quartic equation having the property just referred
to, let it be proposed to find 7, 7' functions of e, h, such that 72 + 7/2 is a rational
function of e, h, but that y2 — y''2, 77' are rational multiples of the same quadric radical
\/#. Assume that we have
72 _ 7'2 = ^p V0, 77' = g V0 ;
then
(7* +y?
that 72 + 7'2 may be rational, we must have p2 + q2 = \"0, or say p2 + q3 = h26 ; hence,
«2 <f
6 = j- + ^2 must be a sum of two squares, or, assuming one of these equal to unity
and the other of them equal to e2, say 6 = 1 + e", we satisfy the required equation by
taking p — h, q = he : viz. we thus have
77' = he Vl + e2, 72 + y'- = 2h (1 +e2);
and thence also
r = h (1 + e2 + VlTe2), 7/a = h (1 + e* - Vl + e2),
the roots of these expressions, or values of 7, 7', being such that
77' = he Vl + e2.
Taking now a rational, =m suppose, and /3 a rational multiple of
VI + e2, = A Vl + e\
suppose ; it is easy to see that the quartic equation which has for its roots
a, «!, «2, a3 = a + /3 + 7, a -
has the property in question, viz. that every rational function of the roots unchangeable
by the cyclical change a into a1} ax into a2, a., into a3, a3 into a, is rationally
expressible in terms of e, h, m, n.
It will be sufficient to give the proof in the case of a rational and integral
function; such a function, unchangeable as aforesaid, is of the form
<£(tt, alf a2, a3) + ^(a1} a.2, a;{, a)+<£(a2, a3, a, a1) + ^)(a3, a, a1} a2):
134 ON A THEOREM OF ABEL'S RELATING TO A QUINTIC EQUATION. [741
and if </>(«, c^, a«, a3) contains a term a.mfinffy'v, then the other three functions will
contain respectively the terms
viz. the sum of the four terms is
n [{1 + (-
This obviously vanishes unless p and q are both even, or both odd ; and the
cases to be considered are 1°, n even, p and q even ; 2°, n odd, p and q even ;
3", n even, p and q odd; 4°, n odd, JD and q odd. Writing, for greater distinctness,
2n or 2w + 1 for n, according as n is even or odd, and similarly for p and q, the
term is, in the four cases respectively,
= 2a'B/S2n+1
The second, third, and fourth expressions contain the factors
£(72-7/a)> ry'(7a-7/2)» ^77^
respectively; and the first expression as it stands, and the other three divested of
these factors respectively are rational functions of a, /S2, 72, 7'*, that is, they are
rational functions of m, n, e, h. But the omitted factors fi (j2 - <y'2), 77' (7- — 7/2),
fiyy', =2wA(l + e2), 2h-e(l+ez), nhe(l + e2) are rational functions of n, Jt, e; hence
each of the original four expressions is a rational function of m, n, h, e; and the
entire function
<£(«, a1} a2, a3) + ^>(a1, a,, a3, a) + (f>(a2, a3, a, a1) + ^(a3, a, alt ««)
is a rational function of m, n, h, e.
Replacing a, 0, y, y by their values, the roots of the quartic equation are
m + n V(l + e2) + J[h (1 + e2 + V(l + e2))],
m - n V(l + e2) + »J\h (1 + e2 - V(l + e2))],
m + n V(l + e2) - v'O (1 + e2 + V(l + e2))],
m - w V(l + e2) - V[A (1+ e2 - V(l + e2))].
And I stop to remark that taking m, n, e, h = -±t +|, 2, -1 respectively, the
roots are
741] ON A THEOREM OF ABEL'S RELATING TO A QUINTIC EQUATION. 135
viz. these are the imaginary fifth roots of unity, or roots r, r2, r1, r3 of the quartic
equation #4 + #3 + #2+#+l = 0; which equation, as is well known, has the group
Reverting to Abel's expression for x, and writing this for a moment in the
form
the quintic equation in a; is
0 = (x - c)5
+ (x — cf . — 5 (pr + qs)
+ (# — c)2 . — 5 (p-s + q*p + r-q + s'2r)
+ (x — c) . — 5 (p3q + q*r + i^s + ssp) + 5 (p-r- + q-s'-) — opqrs
+ (x — c)° . — (p5 +q5 + r5 + s5)
+ 5 (psrs + q3sp + r^pq + ssqr)
— 5 (p2q2r + q2r*s + r-s-p + s~p'2q).
If we substitute herein for p, q, r, s their values, then, altering the order of the
terms, the final result is found to be
+ (x — c)3 . — 5 (A A» + -4 1^.3)
+ (x — cf . — 5 (A- A !«,«.., + AfAzdyft, + A.fAzaa-L + A^Aa
+ (x — c) . — 5 ( A^A-fi^a-^ -f A^AaM^a + A^A^a2^ + A
+ (x - c)° . - (^
+ 5 (A'A^Aja&s + A?AiA&& + A^A^Aaa^ + A/AA^aJ (aa1a2a3)a
- 5 (A^-A^AM^ + Ai-A2A3ciM3 + AfA?Aa& + AfAfA&a^ (aa^a^2 ;
viz. considering herein A, Al> A.,, As as standing for their values
K -f K'a + K"a, + K'"aa», &c.
respectively, each coefficient is a function of a, alt a», a3, which is unaltered by the
cyclical change of these values and therefore is a rational function of
m, n, e, h, K, K', K", K'".
136
[742
742.
ON THE TRANSFORMATION OF COORDINATES.
[From the Proceedings of the Cambridge Philosophical Society, vol. in. (1880),
pp. 178—184.]
THE formulae for the transformation between two sets of oblique coordinates assume
a very elegant form when presented in the notation of matrices. I call to mind that a
matrix denotes a system of quantities arranged in a square form
( « , ft , 7 )-
«' , & , y
a", (3", y"
see my "Memoir on the Theory of Matrices," Phil. Trans, t. CXLVIII. (1858), pp. 17 —
37, [152]; moreover (a, ft, yQx, y, z) denotes ax + @y + yz, and so
( a , 0 , 7 ]fo 2/> z)
*', £', 7
a", 0", y"
denotes
(ax + /3y + yz, OL'X + $y + y'z, a"x + ft'y + y"z},
and again
( a , (3 , y J>, y, *$£, 1?, S) denotes £(a#+/3y + 72)
of Q ry'
*", &', y
Consequently
( a >
«",
7
7'
7'
y, z~$%, 17, (^ = ( a , a' , a
7,
742]
ON THE TRANSFORMATION OF COORDINATES.
137
In the case of a symmetrical matrix
( a, h, g ),
h, b, f
the equal expressions
(a, h, g $a?, y,
h, b, f
ff> f> c
.are also written
(a, b, c, f, g,
In particular, if
then
( a, h, g $#, y, zf is written (a, b, c, f, g, h\x, y, zf.
h, b, f
Two matrices are compounded together according to the law
> i?, £)> =( a> ;i' # $f 17. £fe y, z\
h, b, f
g, f, c
f, f£f V, £)> or («. •••££ *?,
(f, *7, £) = (*, y, *),
(a, a', a"), (/3, £', /S"), (7, 7> 7")
a , 6 , c
a' , b' , c'
a", b", c"
5
a , /3 , 7
«', £', 7
a", yS", 7"
) = (a , 6 , c )
(a', b', c')
(a", b", c")
)> !» »
» >» ;>
>' )> »
viz. in the compound matrix, the top-line is
(a, b, c$a, a', a"), (a, 6, c$/3, /3', £"), (a, 6, c$7, 7, 7"),
and the other two lines are the like functions with (a', b', c'), and (a", b", c"), re
spectively, in the place of (a, 6, c).
The inverse matrix is the matrix the terms of which are the minors of the
determinant formed out of the original matrix, each minor being divided by this
determinant, viz.
*', P, 7
«", £", 7"
where V is the determinant
7' a" — 7"a' , 7"a — 7«", 70' — 7'a
C. XI.
18
138 ON THE TRANSFORMATION OF COORDINATES.
Coming now to the question of transformation, write
[742
x y
x y z Xi 2/x zl
X
1 v fj, I a
a' a" = x
y
v 1 X j /3
ft' ft" y
a w
z
/i X 1 | 7
y 7" z
*l
a /3 7 i 1
if // *>•
"i /*l *^1
2/i
a' £' 7' | "i
1 Xj yx
v n,
ti
a" £" 7" ! Mi
Xj 1 ^
viz. the axes of x, y, z are inclined to each other at angles the cosines whereof are
X, p, v: those of xl, yl, zl are inclined to each other at angles the cosines whereof
are Xj , ^ , vl: and the cosines of the inclinations of the two sets of axes to each
other are a, ft, 7 ; a', ft', 7' ; a", ft", 7" : as is more clearly indicated in the diagram,
the top-line showing that cosine-inclinations of x to
are
«, y, z, ®i, 2/i> zi>
1, v, p, a, a, a",
respectively, and the like for the other lines of the diagram. The letters H, Oj, F,
W are used to denote matrices, viz. as appearing by the diagram, these are
( 1, V, /* ), ( 1 , W1,
i;, 1, X i/j, 1,
U., X, 1 | /*!, Aa,
), ( a , ft , 7 ), ( a, a', a"),
A P, P
i -jt
y, y , 7
respectively.
The coordinates (x, y, z) and (^x, yly z^) form each set a broken line extending
from the origin to the point ; hence projecting on the axes of x, y, z and on those
of #i> 2/i' 2i respectively, we have two sets, each of three equations, which may be
written
y, z} = (W^Lxl, y,, z,},
where of course each set implies the other set.
We have
(x , y , z) =
ylt z,), =
y, z), =
, ylf z,},
; , y , *),
the first giving in two forms («, ?/, ^) as linear functions of (X, yly z^, and the
second giving in two forms (x1} yl, z^ as linear functions of (x, y, z}; comparing
the two forms for each set, we have
n-1 w = F-1 H! , w~i n = nr1 F,
742] ON THE TRANSFORMATION OF COORDINATES.
or, what is the same thing,
139
where in each equation the two sides are matrices which must be equal term by
term to each other; but, the matrices being symmetrical, the equation thus gives (not
nine but only) six equations. Writing
and
we have
(a, b, c, f, g, h) = (l-A,2, l-/u2, 1 - v-, fjuv-\, V\-/JL, \fj,-v),
h, b, f
g. f> c
The first equation, written in the form
V( a, h, g )W =
h, b, f
denotes the six equations
(a, b, c, f, g, h) (a , 0 , 7 )2
(«', P, yy
(«"» fi", 7")2
(a', £', 7')(«", r,
(a,
and
then
And, similarly, writing
1
hi, bj, fj
and the second equation, written in the form
=K ,
=K ,
gi,
18—2
140 ON THE TRANSFORMATION OF COORDINATES. [742
denotes the six equations
(a,, D!, cx, f1} gj, hx][a, a', a")2 =K^ ,
08, ff, PJ = #! ,
(y> y> 7")2 = Ki »
03, /3', r$7> y, 7")
The two sets each of six equations are, in fact, equivalent to a single set of six
equations, and serve to express the relations between the nine cosines
(a, 0, 7, «', P, 7', «"< £". 7"),
and the cosines (X, //., y) and (X^ ^, ^). Observe that the nine cosines are not
(as in the rectangular transformation) the coefficients of transformation between the
two sets of coordinates.
From the original linear relations between the coordinates, multiplying the
equations of the first set by x, y, z and adding, and again multiplying the equations
of the second set by (x±, y1} ^) and adding, we have
(fl $> , y , z)*
(rVft^, ylt z1)2 = (VQx, y
But
(W^, ?/!, ^$a;, y , z}
and
denote one and the same function ; hence
(n$>, y, <gr)2 = (ni$a?J, ?/!, ^)a,
that is,
(1, 1, 1, X, fjL, v^x, y, 42=(1, 1, 1, X1? A*!, ^$«i, y^ ^)2,
or the linear relations between (x, y, z) and (a?!, y1} ^) are such as to transform
one of these quadric functions into the other: the two quadrics, in fact, denote the
squared distance from the origin expressed in terms of the coordinates (x, y, z) and
Oi, 2/i, *i) respectively.
Since the nine cosines are connected by six equations, there should exist values
containing three arbitrary constants, and satisfying these equations identically : but,
by what just precedes, it appears that the problem of determining these values is, in
fact, that of finding the linear transformation between two given quadric functions:
the problem of the linear transformation of a quadric function into itself has an
elegant solution; but it would seem that this is not the case for the transformation
between two different functions.
742] ON THE TRANSFORMATION OF COORDINATES. 141
The foregoing equation
# = (a, b, c, f, g, h$a, 0, 7)2,
is a relation between X, /j,, v, the cosines of the sides of a spherical triangle, and
(a, 0, 7) the cosines of the distances of a point P from the three vertices: it can
be at once verified by means of the relation A + B + C = 2?r, and thence
1 — cos- A — cos2 B — cos2 C + 2 cos A cos B cos (7=0,
which connects the angles A, B, C which the sides subtend at P. Writing a, b, c
for X, fi, v, and /, g, h for a, /3, 7, the relation is
+ 2 (be — a) gh -f 2 (ca — b) hf+ 2 (ab — c)fg,
viz. this is
1 - a- - b'2 - c2 -/2 - g2 - h? + 2abc + 2agh + 2bhf+ 2cfg
- a2/- - &y - c*h? + 2bcgh + 2cahf + 2abfg = 0 ;
where (a, b, c, f, g, h) are the cosines of the sides of a spherical quadrangle ;
(a, b, c), (a, h, g), (h, b, f), (g, f, c) belong respectively to sides forming a triangle, and
the remaining sides (/, g, h), (b, c, f), (c, a, g), (a, b, A) are sides meeting in a vertex.
The equation
Jftri/1 = (a, b, c, f, g, h][a, /3, 7) (a', /3', 7')
is a relation between X, /j,, v, the cosines of the sides of a spherical triangle ; a, /&, 7, the
cosines of the distances of a point P from the three vertices ; of, /3', 7', the cosines of
the distances of a point Q from the three vertices ; and vl , the cosine of the distance
PQ.
Drawing a figure, it is at once seen that
Vl = oa + VlT-^o2 Vl - a'2 cos (0 - 0'),
where
cos 0 = — — . ,
and therefore
sin 0 = —
- _
V 1 - a2 V 1 - v*
also
- — _ _ -
Vl - a'2 Vl - v*
and therefore
Vv~'
sm ff - . ____ . ___ ,
Vl - a'2 Vl - iX
i/, = aa! + j J— {(/3 - avljtft' - ccV) + Vv V'j.
A' = (a, b, c, f, g, hja, /8, 7?. K" = (ft, ...Jo', £', 7')'-',
give
142 ON THE TRANSFORMATION OF COORDINATES. [742
the values of V, V being
V = l _ a3 - /8s - i/a + 2a/9i/,
V ' = 1 - a'" - ff* -v*
the resulting value of j/, is therefore
The equations
A' = (a, b, c, f,
(ga + f/3 + 07 )8 =
(ga' + f/3' + e7')2 = 7vV':
and we therefore have
(ga + f/8 + c7$ga' + f/9' -f cy') = ^ V V V ' ;
recollecting that 1 - i/2 = c, the formula thus is
"i = ««' + c |(/3 - o»W ~ a'v) + g (g« + W + cy$g«' + f/?' 4- cy')| ,
or say,
Kvt = K*a' + c {A" (ft - av-%/3' - ct'v) + (ga + f/3Jga' + f/3')} + g (ay' + a'y) + f (fty + /3'y) + cyy.
The sum of the first and second terms is readily found to be
and the equation thus becomes
Kv, = (a, b, c, f, g, h$a, ft, 7$a', /3', 7'),
as it should do.
743] 143
743.
ON THE NEWTON-FOURIER IMAGINARY PROBLEM.
[From the Proceedings of the Cambridge Philosophical Society, vol. in. (1880),
pp. 231, 232.]
THE Newtonian process of approximation to the root of a numerical equation
y(w) = 0, consists in deriving from an assumed approximate root £ a new value
/"(£)
£ = £ — f,--j-A • which should be a closer approximation to the root sought for: taking
the coefficients of f(u} to be real, and also the root sought for, and the assumed
value £, to be each of them real, Fourier investigated the conditions under which
£ is in fact a closer approximation. But the question may be looked at in a more
general manner: £ may be any real or imaginary value, and we have to inquire in
what cases the series of derived values
J: =*_ .-*
/'(£)'
converge to a root, real or imaginary, of the equation f(u) = 0. Representing as usual
the imaginary value £, =x + iy, by means of the point whose coordinates are x, y,
and in like manner £,, = #i + ty1, &c., then we have a problem relating to an infinite
plane; the roots of the equation are represented by points A, B, (7,...; the value
| is represented by an arbitrary point P; and from this by a determinate geometrical
construction we obtain the point Px, and thence in like manner the points P2, P3, ...
which represent the values £,, £,, £:!,... respectively. And the problem is to divide
the plane into regions, such that, starting with a point PI anywhere in one region,
we arrive ultimately at the root A ; anywhere in another region we arrive ultimately
at the root jB; and so on for the several roots of the equation. The division into
regions is made without difficulty in the case of a quadric equation ; but in the next
succeeding case, that of a cubic equation, it is anything but obvious what the division
is : and the author had not succeeded in finding it.
144 [744
744.
TABLE OF A^O71 - n (m) UP TO ra = n = 20.
[From the Transactions of the Cambridge Philosophical Society, vol. xin. Part I. (1881),
pp. i_4. Read October 27, 1879.]
THE differences of the powers of zero, AmOn, present themselves in the Calculus
of Finite Differences, and especially in the applications of Herschel's theorem,
for the expansion of the function of an exponential. A small Table up to A10010 is
given in Herschel's Examples (Camb. 1820), and is reproduced in the treatise on
Finite Differences (1843) in the Encyclopedia Metropolitana. But, as is known, the
successive differences A0n, A20W, A30n, ... are divisible by 1, 1.2, 1.2.3,... and
generally A^O" is divisible by 1.2.3...W, = II (ra) ; these quotients are much smaller
numbers, and it is therefore desirable to tabulate them rather than the undivided
differences A™0n : moreover, it is easier to calculate them. A table of the quotients
AwOn -r- II (m), up to m = n = 12 is in fact given by Grunert, Crelle, t. xxv. (1843),
p. 279, but without any explanation in the heading of the meaning of the tabulated
numbers Cnk, = AwOfc -f- II (n), and without using for their determination the convenient
formula Cnk+l = nCnk -f Cn_f given by Bjorling in a paper, Crelle, t. xxvni. (1844),
p. 284. The formula in question, say
U(m)
= m
is given in the second edition (by Moulton) of Boole's Calculus of Finite Differences,
(London, 1872), p. 28, under the form
AmOn = m (^m-iQn-l + AmQn-l).
It occurred to me that it would be desirable to extend the table of the quotients
AmOn -f- II (m), up to m = n = 20. The calculation is effected very readily by means
744]
TABLE OF AmO"-7-n(m) UP TO m = w = 2
145
of the foregoing theorem, which is used in the following form ; viz. any column of
the table for instance the fifth, being
A, then the following column is A,
B, ... 25 + 4,
C, ... 3C+B,
D, ... 4D + C,
E, ... oE + D,
+ E-
and then we obtain a good verification by taking the sum of the terms in the new
column, and comparing it with the value as calculated from the formula,
Sum = 2A + SB + 4(7 + 5D + QE.
Observe that, in the two calculations, we take successive multiples such as 4.D and
5D of each term of the preceding column, and that the verification is thus a safe
guard against any error of multiplication or addition.
TABLE, No. 1, OF AW0" + II (m\
<1
•«
a
O1
O2
O3
O4
O5
O6
O7
O8
O9
O10
O11
O12
O13
O14
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
o
1
3
7
15
31
63
127
255
511
1 023
2047
4095
8 191
3
1
6
25
90
301
966
3025
9330
28 501
86526
261 625
788 970
4
1
10
65
350
1 701
7 770
34 105
145 750
611 501
2 532 530
10 391 745
5
1
15
140
1 050
6 951
42 525
246 730
1 379 400
7 508 501
40 075 035
6
1
21
266
2 646
22 827
179 487
1 323 652
9321 312
63 436 373
7
1
28
462
5 880
63 987
627 396
5 715 424
49 329 280
8
1
36
750
11 880
159 027
1 899 612
20 912 320
9
1
45
1 155
22 275
359 502
5 135 130
10
1
55
1 705
39 325
752 752
11
1
66
2 431
66 066
12
1
78
3 367
13
1
91
14
1
15
16
17
18
19
20
C. XI.
19
146
TABLE OF AmOM -r II (m) UP TO «l = U = 20.
[744
<I
13
a
i— i
O15
Ql* O17
O18 O19
O20
1
1
1
1
1
1
2
16 383
32 767
65 535
131 071
262 143
52428
3
2 375 101
7 141 686
21 457 825
64 439 010
193 448 101
580 606 44
4
42 355 950
171 798 901
694 337 290
2 798 806 985 11 259 666 950
45 232 115 90
5
210 766 920
1 096 190 550
5 652 751 651
28 958 095 545
147 589 284 710
749 206 090 50
6
420 693 273
2 734 926 558
17 505 749 898
110 687 251 039
693 081 601 779
4 306 078 895 38
7
408 741 333
3 281 882 604
25 708 104 786
197 462 483 400
1 492 924 634 839
11 143 554 045 65
8
216 627 840
2 141 764 053
20 415 995 028
189 036 065 010
1 709 751 003 480
15 170 932 662 67
9
67 128 490
820 784 250
9 528 822 303
106 175 395 755
1 144 614 626 805
12 Oil 282 644 72
10
12 662 650
193 754 990
2 758 334 150
37 112 163 803
477 297 033 785
5 917 584 964 65
11
1 479 478
28 936 908
512 060 978
8 391 004 908
129 413 217 791
1 900 842 429 48
12
106 470
2 757 118
62 022 324
1 256 328 866
23 466 951 300
411 016 633 39
13
4 550
165 620
4 910 178
125 854 638
2 892 439 160
61 068 660 38
14
105
6020
249 900
8 408 778
243 577 530
6 302 524 58
15
1
120
7 820
367 200
13 916 778
452 329 20
16
1
136
9 996
527 136
22 350 95
17
1
153
12 597
741 28
18
1
171
15 67
19
1
19
20
Writing down the
1 2 3
(0) (2) (4)
sloping lines as columns thus:
45 6
(6) (8) (10)
7
(12)
8 etc.
(14) etc.
1
1
1
1
3
1
1
6
7
1
1
10
25
15
1
1
15
65
90
31
1
1
21
140
350
301
63
1
1
28
266
1 050
1 701
966
127
1
36
462
2 646
6 951
7 770
3 025
1
45
750
5 880
22 827
42525
34 105
1
55
1 155
11 880
63 987
179 487
246 730
1
66
1 705
22 275
159 027
627 396
1 323 652
1
78
2431
39 325
359 502
1 899 612
5 715 424
1
91
3367
66 066
752 752
5 135 130
20 912 320
1
105
4550
106 470
1 479 478
12 662 650
67 128 490
1
120
6 020
165 620
2 757 118
28 936 908
193 754 990
1
136
7 820
249 900
4 910 178
62 022 324
512 060 978
1
153
9 996
367 200
8 408 778
125 854 638
1 256 328 866
1
171
12 597
527 136
13 916 778
243 577 530
2 892 439 160
1
190
15 675
741 285
22 350 954
452 329 200
6 302 524 580
20 19
17
16
15
14
13 etc.
744]
TABLE OF AwOn -5- II (w) UP TO m = 7l =
147
it appears by inspection that, in the second column the second differences, are constant,
in the third column the fourth differences, in the fourth column the sixth differences,
and so on, are constant; and we thence deduce the law of the numbers in the
successive columns : viz. this can be done up to column 7, in which we have 14
numbers in order to find the 12th differences : but in column 8 we have only 13
numbers, and therefore cannot find the 14th differences. The differences are given in
the following
TABLE, No. 2 (explanation infra).
<
"O
3
1
2
3
4
5
6
7
0
1
1
1
1
1
1
1
1
9
6
14
30
62
126
2
1
12
61
240
841
2 772
3
10
124
890
5 060
25410
4
3
131
1 830
16 990
127 953
5
70
2 226
35 216
401 436
6
15
1 600
47 062
836 976
7
630
40796
1 196 532
8
105
21 225
1 182 195
9
10 930
795 718
10
945
349 020
11
90090
12
10395
We have, by means of this Table, the general expressions of ArOr,
up to A1'"6^, viz. the formulae are
A1K -5- n (r) = 1,
/r _ o
A»->0>-- n (r-l)-l+l
+1
Ar-2Qr + n (r - 2) = 1 + 6 (T 3Y + 12
10
r-3\3
table ; and where for shortness
is written to denote the binomial coefficient
-3y
2 y "V 3 ) "\ 4
&c., &c.,
where the numerical coefficients are the numbers in the successive columns of the
/ '/¥* i '
k
rjfTj. . For instance, r = 10, we have
A8010 -r- n (8) = 1+ 6 . 7 + 12 . 21 + 10 . 35 + 3 . 35, = 750,
agreeing with the principal Table. It will be observed that, in the successive columns
of the Table, the last terms are 1, 1, 1.3, 1.3.5, 1.3.5.7, 1.3.5.7.9, and
1.3.5.7.9.11. This is itself a good verification: I further verified the last column
by calculating from it the value of A14020 -r- II (14), = 6 302 524 580 as above. The
Table shows that we have Ar~»zOr -f- II (r - m) given as an algebraical rational and
integral function of r, of the degree 2m. But the terms from the top of a column,
A0''=l, A20'' - 1 . 2 = 2r~1 - 1, &c., are not algebraical functions of r.
22 October, 1879.
19—2
148 [745
745.
ON THE SCHWAEZIAN DERIVATIVE, AND THE POLYHEDRAL
FUNCTIONS.
[From the Transactions of the Cambridge Philosophical Society, vol. xin. Part I. (1881),
pp. 5—68. Read March 8, 1880.]
dii
the second order, ~r^2+p -f^ + qy = Q, is determined by a differential equation of the
THE quotient s of any two solutions of a linear partial differential equation of
second
third order
tfo /d*s\n-
da? 0 / da? \ l ( dp
&-fl* ;r -KF+2e£
dx \dx/
where the function on the left-hand is what I call the Schwarzian Derivative; or
say this derivative is
where the accents denote differentiations in regard to the second variable x of the
symbol.
Writing in general (a, b, c ,\\X, Y, ZJ- to denote a quadric function
(a, b, c, £(a-b-c), i(-a+b-c), \ (- a - b + c)$X, F, Z}\
then, if the equation of the second order be that of the hypergeometric series,
generalised by a homographic transformation upon the variable x, the resulting differ
ential equation of the third order is of the form
{s, x] =(a, b, c .'.)(
v
111
x — a' x — b ' a — cj '
745] ON THE SCHWARZIAN DERIVATIVE. 149
and, presenting themselves in connexion with the algebraically integrable cases of this
equation, we have rational and integral functions of s, derived from the polygon, the
double pyramid, and the five regular solids. They are called Polyhedral Functions.
The Schwarzian Derivative occurs implicitly in Jacobi's differential equation of the
third order for the modulus in the transformation of an elliptic function (Fund. Nova,
1829, p. 79, [Ges. Werke, t. I., p. 133]) and in Rummer's fundamental equation for the
transformation of a hypergeometric series (Kummer, 1836 : see list of Memoirs) : but it
was first explicitly considered and brought into notice in the two Memoirs of Schwarz*,
1869 and 1873. The latter of these, relating to the algebraic integration of the
differential equation for the hypergeometric series, is the fundamental Memoir upon the
subject, but the theory is in some material points completed in the Memoirs by Klein
and Brioschi.
The following list of Memoirs, relating as well to the Polyhedral Functions as to
the Schwarzian Derivative, is arranged nearly in chronological order.
a /•?
Kummer, Ueber die hypergeometrische Reihe 1 + ——«+... Crelle, t. xv. (1836),
pp. 39—83 and 127—172.
Schwarz, Ueber einige Abbildungsaufgaben. Crelle-Borchardt, t. LXX. (1869), pp. 105 — 120.
Ueber diejenigen Falle in welchen die Gaussische hypergeometrische Reihe
eine algebraische Function ihres vierten Elementes darstellt. Do. t. LXXV. (1873),
pp. 292—335.
Cayley, Notes on Polyhedra. Quart. Math. Jour. t. vn. (1866), pp. 304—316; [375].
On the Regular Solids. Do. t. xv. (1878), pp. 127—131 ; [679].
Fuchs, Ueber diejenigen Differentialgleichungen zweiter Ordnung welche algebraische
Integralen besitzen, und eine Amvendung der Invariantentheorie. Crelle-Borchardt,
t. LXXXI. (1875), pp. 97—142.
Klein, Ueber binare Formeii mit linearen Transformationen in sich selbst. Math. Ann.
t. ix. (1875), pp. 183—209.
Brioschi, Extrait d'une lettre a M. Klein. Math, Ann, t. XL (1877), pp. Ill 114.
Klein, Ueber lineare Differentialgleichungen. Math. Ann. t. XL (1877), pp. 115 — 118.
Brioschi, La theorie des formes dans 1'integration des Equations differentielles lineaires
du second ordre. Math. Ann. t. XL (1877), pp. 401—411.
Gordan, Ueber endliche Gruppen linearer Transformationen einer Veranderlichen.
Math. Ann. t. xn. (1877), pp. 23—46.
Binare Formen mit verschwindenden Covarianten. Math. Ann. t. XII. (1877),
pp. 147—166.
[* Schwarz, Ges. Werke, t. n., p. 351, remarks that the Derivative occurs implicitly in a memoir by
Lagrange, " Sur la construction des cartes ggographiques," (1779), (Euvres, t. iv., p. 651.]
150 ON THE SCHWARZIAN DERIVATIVE [745
Klein, Ueber lineare Differentialgleichungen. Math. Ann. t. xn. (1877), pp. 167—179.
Weitere Untersuchungen liber das Icosaeder. Math. Ann. t. XII. (1877),
pp. 503—560.
Cayley, On the Correspondence of Homographies and Rotations. Math. Ann. t. xv.
(1879), pp. 238—240; [660].
On the finite Groups of linear transformations of a Variable. Math. Ann.
t. xvi. (1880), pp. 260—263, and pp. 439—440; [752],
I propose in the present Memoir to consider the whole theory: and, in particular,
to give some additional developments in regard to the Polyhedral Functions.
I remark that Schwarz starts with the foregoing differential equation of the third
order
\s, x\ = (a, b, c .*.) I , — f , - ) i
' \x-a x-b x — c)
and he shows (by very refined reasoning founded on the theory of conformable figures,
which will be in part reproduced) that this equation is, in fact, algebraically integrable
for 16 different sets of values of the coefficients a, b, c. It may I think be taken
to be part of his theory, although not very clearly brought out by him, that these
integrals are some of them of the form, x = rational function of s ; others of the form,
rational function of x = rational function of s ; the rational functions of s being in fact
the same in the last as in the first set of solutions: they are quotients of Polyhedral
functions.
But as regards the second set of cases, the solution of these, introducing for con
venience a new variable z in place of s, may be made to depend upon the solution
in the form, x = rational function of z, of an equation of a somewhat similar form, but
involving two quadric functions of x and z respectively, viz. the equation
and we have the theorem that the solution of this equation depends upon the determ
ination of P, Q, R rational and integral functions of z, containing each of them
multiple factors, which are such that P + Q + R = 0. Using accents to denote differ
entiation in regard to z, this implies P' + Q' + R = 0, and consequently
QR - Q'R = RP' - R'P = PQ' - P'Q.
Further, they are such that the equal functions QR' - Q'R, RF - RP, PQ' - P'Q contain
only factors which are factors of P, Q or R.
In fact, writing /, g, h = b — c, c — a, a — b, the required relation between x, z is
then expressed in the symmetrical form f(x — a) : g (x — b) : h(x — c) = P : Q : R.
745] AND THE POLYHEDRAL FUNCTIONS. 151
The last-mentioned differential equation is considered by Klein and Brioschi: the
solutions in 13 cases, or such of them as had not been given by Schwarz, were
obtained by Brioschi : and those of the remaining 3 cases, subject to a correction in
one of them, were afterwards obtained by Klein.
The first part of the present Memoir relates, say to the foregoing equation
f ) / K \f l l l V
[g, x\ = (a, b, c .'.) , • — , , ,
\ae - a x — b x — cj
although the other form in {x, 2} may equally well be regarded as the fundamental
form.
We consider in the theory:
A. The Derivative {s, x}, meaning as above explained.
B. Quadric functions of any three or more inverts — -. .
SO *"** 6
C. Rational and integral functions P, Q, R having a sum = 0, and which are
such that QR'-Q'R, =RP'-R'P, =PQ'-P'Q, contains only the factors of P, Q, R.
D. The differential equation of the third order.
E. The Schwarzian theory in regard to conformable figures and the corresponding
values of the imaginary variables s and x.
F. Connexion with the differential equation for the hypergeometric series.
The second part of the Memoir relates to the Polyhedral Functions.
The paragraphs of the whole Memoir are numbered consecutively.
PART I.
The Derivative {s, x}. Art. Nos. 1 to 7.
., T, s" d (, ds\ , , , dp i ,
1. It p = — = ^r { loff -=- , then Is, # = -^ — i»-.
s dx \ 6 dx) ' dx 2/^
2. The derivative {s, x} may be transformed in regard to either or both of the
variables.
Suppose, first, that s is a function of the new variable S, (hence also S is a
function of x) : using subscript numbers to denote differentiations in regard to S, and
the accents as before for differentiations in regard to x, we have
s' = S's1}
whence, differentiating the logarithms,
15:
. >-
-I ::
i If m 1
-
-7
.- -.-
*fcr *.
:-
raioe
-.-.-.
: -,:-_
m fiKt, mdadec Ifce
= 0= >, xr we hare t
._- -:-
-V = • V T
•-• -
V — • .^ *• 2 •
"
5. Wiis«g£=
-,. : v
-:-.:-
~ -r ' T - " ^ _ --. ~. '.
•
-if •: --.-'
- &
•4 — *f-
745] AND THE POLYHEDRAL FUNCTIONS. 153
Hence in the first formula {S, x} = {s, x}, that is,
(as + b } , .
viz. we may, in the derivative {s, x], write for s any homographic function (as + b) -r- (cs + d)
of s.
6. Again, if X — — — ^ , then from the second formula
that is,
and here, changing s into (as + b) -=- (c-s + d), we have finally
(as + b ax + 0} _ (yx + S)4
[cs + d ' 7tf
which is the formula for the homographic transformation of the two variables s, x.
7. Let s be a given function of x, the equation {8, x} = {s, x} is a differential
equation of the third order in S, and by what precedes, its general integral is S=aS+ .
CS -p Ct
The direct process is as follows : we have a first integral -™ = ^7 -^— 7 ; a second
S s cs + d
integral log S' = log s - 2 log (cs + d) + const., that is, &--*£*• and thence a final
(cs + d)-
A
integral 8 = B — ——~j , which is equivalent to the foregoing value of S.
The Quadric Function of three or more Inverts. Art. Nos. 8 to 15.
8. We consider a quadric function of any number of inverts — - — all
x — OL x - /3 ' '
of them different: it is assumed that the constant term is =0, and also that the
sum of the coefficients of the linear terms is =0. We have therefore square terms
TaT^ttS' ' Pro(luct terms - — - , and linear terms , where the sum of the
\" w/ X — a . X — p x — Ct
h
coefficients A is =0. Any product term - a is expressible in the form of a
x — ct . x — p
_--^ —--___ _
difference ^ _--^ —--___ _ of two linear terms, and (the coefficients of these
C. XI.
154 ON THE SCHWARZIAN DERIVATIVE [745
being equal), after it is thus expressed, the sum of the coefficients of the linear terms
is still = 0. The function is thus always expressible in the form
where the sum A + B + ... is = 0 : this may be called the reduced form.
9. Observe that any particular invert - may disappear altogether from the
00 "— OL
reduced form : this will be the case if a = 0, that is, if the original form contains no
term in , and if also A = 0. An invert thus disappearing from the reduced
(x - a)2
form is said to be non-essential: and the inverts which do not disappear are said to
be essential. The original form contains in appearance the non-essential inverts, but
it is really a quadric function of the essential inverts only.
10. Imagine the original function expressed as a rational fraction, the denominator
being the product (x -a)2 (x- /S)2(^~7)2 ••• of tne squared factors corresponding to all
the inverts (non-essential as well as essential): the numerator will be in general of a
degree less by 2 than that of the denominator, but the coefficients of any one or
more of the higher powers of x may vanish, and the numerator will then be of a
lower degree. But this numerator will for any non-essential invert — '- - contain the
factor (x — 7)2, or, dividing the numerator and denominator each by this factor, the
difference of the degrees of the numerator and denominator will remain unaltered;
that is, the difference will have the same value whether we do or do not attend to
the non-essential inverts; or say it will have the same value for the original form and
for the reduced form.
A 7? C1
11. It is to be remarked that the linear terms 1 -5 + - -+..., where
x — a. x—p x — 7
A+B + C + ... =0, can be (and that in a variety of ways) expressed as a sum of
differences n > tnat is. as a sum °f product-terms -5 . Hence the
x— a. x — p x — a.x — p
quadric function can be (and that in a variety of ways) expressed as a homogeneous
/ Y 1 1 \2
function (a, ... 0 — , - —5 >•••); w^ must have in the form all the essential inverts,
\ S^X — Of X — P /
and we need have these only. Supposing that this is so, and that the number of
the essential inverts is =w, then the number of constants is = ^n(n + 1), whereas the
number of constants in the reduced form is only = 2n — 1 : hence the coefficients are
not determinate; or, what is the same thing, we may have different quadric functions
having each of them the same reduced function; these quadric functions, as having
the same reduced function, can only differ by multiples of the evanescent expressions
5 H - - H — —5 , &c.
x—p.x — y x — ry.x — a. x—OL.x — p
745] AND THE POLYHEDRAL FUNCTIONS. 155
In particular, if the number of essential inverts is = 3, then the quadric function is
of the form
/ Y 1 1 1 \2
(a. b, c. f. er, hO , ~ , I ,
V ' *' X*-« ' x-ft' x-j '
0
which contains one superfluous constant, and equivalent functions differ only by a
multiple of
_j3-7 _,_ y-ct + a- ft
x — ft.x — y x—y.x — a x — a. as — ft'
12. A quadric function such that the degree of the numerator is less by 4 than
that of the denominator is said to be "curtate."
The conditions, in order that the function
( v. f uX 1 1 1 \2
a, b, c. f, Q, h - , -- = , — — 1
V X*-« x-ft' x-yj
may be curtate, are easily found to be
a + b -f c + 2f + 2g + 2h = 0,
and by reason of the superfluous constant we are at liberty to assume a third condition :
the three conditions may be taken to be a + h + g, h + b + f, g + f+c each =0; and
this being so the values of f, g, h are =£(a-b-c), £(-a + b-c), H~a-b + c)
respectively. Hence the form is
1 1 1 \2
a, b, c, £ (a - b - c), £ (- a + b - c), £ (- a - b + c)
A#-« #-/* # -ry
which, as already mentioned, we denote by
,
— a x —
We have thus the theorem that a curtate function of any number of inverts, but with
only the three essential inverts
1 1 ^_
x — a' x — ft' x — y'
is always expressible in the foregoing form
1
,,..-_, , __
AX - a x - ft ' x - y
13. It may be remarked that the function (a, b, c .-.$X, Y, Zf is a function of
the differences of the variables Z, Y, Z; and similarly, in the case of four variables
a function (a, b, c, d, f, g, h, 1, m, n£X, Y, Z, W)\ for which
a+h+g + 1, h + b + f+m, g + f+c + n, 1+m + n + d,
20—2
156 OX THE SCHWARZIAN DERIVATIVE [745
are each =0, is a function of the differences of the variables X, Y, Z, W: and so in
general. Any such function is said to be " diaphoric " : and it is easy to see that,
taking for the variables any inverts whatever, a diaphoric function is always curtate.
14. The function
a b c
a b c 2
+ — + ~ +—
~
a? - a 8 — p x-y
where the coefficients a, b, c, . . . satisfy the relation a + b + c+... = — 2, is diaphoric,
and therefore curtate. In fact, forming the sum, coeff. . - ^ + 1 coeff. - - -5 + . . . ,
(x — a)2 x — a . x — p
this is — a— £a2 — £ab — £ac — ..., = — £a(2 + a + b + c +...), which is =0; and similarly
the other conditions are satisfied.
15. The function
/ , Y a Oi b &i c GI Y
a, b, c /.A- -+—-+..., ~z + -:La+-~> - + -—+...,
\ Ax -««-«! « — fi x — & x — yx — y! J
regarded as a function of the inverts
11 1
where
a + tti + . ..=6 + 6j +... = 0-1-0!+..., = & suppose,
is diaphoric, and therefore curtate. In fact, the condition in regard to - - is
CC ^~ CL
a (a2 + aax + aa2 + . . .) + 1(- a + b - c) (a6 + a6x + . . .) + |(- a - b + c) (ac + acj + . . .) = 0 ;
that is,
ak {a + i (- a + b - c) + £ (- a - b + c)} = 0,
which is satisfied. And similarly the other conditions are satisfied.
The functions P, Q, R. Art. Nos. 16 to 20.
16. We consider P, Q, R, rational and integral functions of z, such that P + Q + R = 0 :
hence, using the accent to denote differentiation in regard to z, we have also P' + Q + R' = 0 ;
and therefore QR - Q'R = RPf - RP = PQ' - P'Q, = @ suppose : and we require to find
P, Q, R such that the function @ contains only the factors of P, Q, R.
17. It is to be observed that, effecting upon a solution P, Q, R any linear sub
stitution (az + /3) H- (yz + 8), and omitting the common denominator, we have a solution ;
but this is regarded as identical with the original solution. The three functions, if
745] AND THE POLYHEDRAL FUNCTIONS. 157
not originally of the same order, can thus be made to be of the same order ; or by
taking account of the root z= oo , we may in the original case regard them as being
of the same order, and it is convenient so to regard them : say they are taken to
be of the same order 8. And there is clearly no loss of generality in taking the
three functions to be prime to each other ; for any common factor of two of them
would divide the third, and might therefore be struck out.
18. We may therefore write
-iy?, Q = GTl(z-m)<i, R
where (z — l)p is taken to denote the distinct simple or multiple factors of P, and
the like as regards Q and R ; the factors z — I, z — m, z — n are thus all of them different.
And we have 8 = ^p, = ~2q, = 2r.
19. It is at once seen that ® is of the degree 28 — 2, and moreover that it
contains the factors II (z — iy~l, II (z — m)9"1, II (z — n)r~^ ; hence it contains the factor
n (z - ly-1 (z - m)?-1 (z - nf-\
Suppose the number of distinct indices p is = <r1, that of distinct indices q is <r2, and
that of distinct indices r is <r3 ; then the degree of the factor is = 3S — o-j — <r.2 — a-3 ;
and if this be = 28 — 2, then ® can have no other variable factor : viz. if the numbers
o-j, <72, 0-3 of the distinct indices p, q, r respectively are such that o-j + o-2 + cr3 = 8 + 2,
a relation which is henceforth taken to be satisfied, then we have
© = Kli (z - I)?-1 (z - m)?-1 (z - n)r~\
As already in effect remarked, the conclusion extends to the case where P, Q, R are
not of the same degree ; the equation P + Q + R = 0 here implies that two functions,
say P, Q, are of the same degree, and the third function R of an inferior degree ;
(z \'
1 -- j of
the degree t proper for raising its degree up to that of P or Q.
20. Solutions are given in the following PQR-Tsible : in which, where required,
the proper factor (1 -- J has been added; the first column headed Ref. No. (Reference
Number) will be explained further on. The Annex to the same Table will also be
explained.
158
ON THE SCHWARZIAN DERIVATIVE
^ +
>
-r +
OJ +
+ 1
+
a
745]
AND THE POLYHEDRAL FUNCTIONS.
159
1 i 1 II
£ | S « H *
11 1 l|l|
•o o -S o -^ o M
CH ft H 0 ft
^0
£ — C n ^ ^ n *^ * «
— o O O
C .^C £H -"*
•i ^ « 1 3
,3 § S * ^
I jlll
H O ft
o
55
^VN^^J^VX-^S^^^^VN
- a £ £
a
8 S^88^®^**^
^ ^ *^ ** ^ M
t^ ** N isi
*"* rH CO rH. O3
*& — ^5 r^ (jq |iQ ^**rH C<11O CM CMiOrHGM ^* |ut) W 25 SI
S co loo co loo co ;oo co 100
I
^,^«.M»ii.ii»».i
U U -
1 | •* IO5 -^ IOS ••* OS
1-1 Ji-
^ 25 ^ sis «» ^ 53 ^ 53 ^» ^
k
O
.£_
O
a
.0
la
"3
O O O O O
II II II II II
rH O3 CO ^^ W5
o o w o o
1-10 OS °
^ u"5 n"5 t
rHOcnWirHO (NU5 ml1*
(M»CCD(MC>)'0 O rHIN O °°lc<)
II II II II II II II
<M CO <N IO rH IO CO
O O U O O O O
yj 00 ^ 00 rH |O ^jOOWUSC^WrHOfMlOCd'Oi— lOrH IO
u n u u H • i u u i \F ^^
(MW(N<NrHrH(MrHrHIMlM
OOOOC5OOOOOO
3
o
0 0 O 0 O
II II II II II
.a ,a ,a ^3 ja
O 0
II II
0 0 O OCOIOOO O 0 OCOIOOO
II II II II II II II II II II II
<N<M<NC<«rHC<)(N(M(Mi-ie<I
o o o o o
II II II II II
O
II
™ ' rH ICO ^^ ^^ W O rH <O ^5
II II II II II 11 II II
r-irHrHi-l CO O3 COCO
c8 c3 c3 c3 ci c3 c3 c3
**^ iCft ^^ JA ^* |Oi ^j («o ^ **^ 35 ^^ M^ IO5 ^^ ^* IOi ^?J* ICTi
u II H II u u u II il u II
c3 03 *§ 'rt 'S
cacSc3cSo3c3e3e3c3c8c3
*
"o
JO
3.
O ^ ICO ^ IO ^ ^-, ,_) |(-g
r ...
1 co loo co 100 co oo co 100
rH
ft---
| | •* ICS -* ICS -# IO5
eo loo eo 100 eo loo co loo co i» E E : E r r
^^rnls^i^SS^io5 " " - = - "
6
<*-.'
rH « =0 ^ U5
n^pa^^^aS^g
t? o
•- ^
*• — -x
o \<y\ a
nlco g
T3 -3
a
dS %_i
XI
160 ON THE SCHWARZIAN DERIVATIVE [745
The Differential Equations [x, z] and {s, x}. Art. Nos. 21 to 45.
21. In reference to what follows, it is convenient to put P=XP0, P'^X^,
where P0 is written for IT (z — l)p~l, the G.C.M. of P and P' ; and X is consequently
= F multiplied by the product II (z — 1) of the several factors taken each with the index
unity; and so for Q and R: viz. we write
P, Q, R = XP0, YQ0, ZR0,
P', Q', R' = X1P0, FjQo, Z^RQ,
and the foregoing value of © then is
We come now to the investigation of the leading theorem. Take a, b, c arbitrary,
f, g, h = b — c, c — a, a — b ; P, Q, R functions of z as above ; and write
f(x-a) : g(x-b) : h(x-c) = P : Q : R,
equations, which are consistent with each other and determine x as a rational function
of z. Using, as before, the accent to denote differentiation in regard to z, and taking
the coefficients (a, b, c) arbitrary, it is required to find the value of
{x, z}+x'2(a, b, c .-.I) - , - j, - ) .
V Ax — ax — bx — cJ
22. Calculation of the first term {x, z}.
t p \ / p \ (P }
We have x = & function \an + fi) + (y^p + &}, and thence {x, z} = \^, z\ , = {£, z}
\ JK J \ £i ] \1\ J
for a moment; then
RP'-R'P P0Q0R0 P0Q,
T}O y r/*) T> f) y F7-i D *
M2 Z2M0- Z 2M0
Substituting the values
we have
£" _p-l q-\ r + l
-£-, — z* r T *-i ~ — ft~ — ,
z — I z — m z — n
and thence
-2
z — I z — m z — n
or say
p— 1 PI— I q — 1 <?i — 1 r+l n 4- 1
'(.z-J)2"^-^)2""'"" (2- m)2 ~ (« - m^2 ~ '" + (« - n)2 + (2 - Wj)2
\ ^ — ^ -s — ^ z — m ' z — nii z — n z — n^ '" ) '
745] AND THE POLYHEDRAL FUNCTIONS. 161
where it is to be observed that
£ (p - 1) + 2 (q - 1) - S (r + 1), = 8 - o-j + S - <72 - (8 + o-3) = 8 - o-j - o-2 - o-:5 = - 2 ;
consequently the function is diaphoric, and therefore curtate.
It is to be remarked that the function, although presenting itself in a form
unsymmetric in regard to the factors of P and Q, and of R, is really symmetric
as regards the three sets of factors ; this is obvious a priori, and it will be presently
verified.
23. For the calculation of the second term
/ Y 1 1 1 Y
x'- ( a, b, c .'. 0 — — , ^ , ,
\ AX — ax — ox — cj
we have
/(a? -a), ff(sn-b), h(a;-c)=flP, SIQ, flR,
where H is a determinate function of z\ hence
x' x1 xr _p ir <? rr R n'
— & "* r\ i f\ ' n > r> ' r» •
a,_0' x-b' x-c~ P IT Q ' I
'J'hen substituting these values, by reason that the function is diaphoric, the terms
in ^r disappear, and we have
>••( v, X l l l V
c-(a, b, c • •()-_—» - _i , -^-)
F
= a, b,
which is
r
-a b
— a., u,
z — m z — n
We have S/) = 2g = 2?', = 8 : and hence by what precedes, this function, considered as
a function of the inverts — ;, &c., is diaphoric, and therefore curtate.
z — I
24. We have therefore
/ Y 1 1 1 \"
\x, z\ + x'- a, b, c /. 0 , f , - =
V AX -a x-b x- c]
-2
^ — I z — m z — n
, b, c .-.()2-7)
'*' V >v /
z — I' z — m' z — n
where the whole function on the right-hand side is curtate.
c. XL 21
162 ON THE SCHWARZIAN DERIVATIVE [745
25. We have to bring the function on the right-hand side into the reduced form
a A
(z — a)2 z — a
for the purpose of getting rid of the non-essential inverts (if any).
We write
^ p-1 __ £-1^1-1
viz. z — I here denotes any particular factor, and z—l-^ represents any other factor of
the same set; and so in other like cases.
26. The whole coefficient of — . is
- I)2 + a.ps, = i
an expression which, regarded as a function of a and p, is represented by (ap): the
parentheses are used only to avoid ambiguity, and are omitted when p is a number,
thus al = a, a2 = — f + 4a, and so in other cases.
27. The whole term in , comes from
z — l
_
z — I \ z — l-i z — m z — n
-- - - - -
z—l[ z — l z —m z—n
viz. each term such as - -, - r is to be replaced by — ,- [ — , — ' , 1 , givi
z-l.z-l, J l-li\z-l z-lj
vng
rise to the term , — T -- , , or contributing the term 7 — .- to the coefficient of
l — ^ z — l l — l^
The whole coefficient thus is
— m — n
>-- - - ,--.
I — m I — n
28. Suppose first that z — l is a multiple factor of P, viz. a factor with an index p
ry n'
greater than 1: then, for z = I, we have Q+R = 0, Q' + R' = 0, and thence ^ = -7^,
y Vt
Q T
that is, 2 , = S , . We have therefore
I — m l — n
p (- a - b + c) 2 -3- .+p (_ a + b - c) 2
r
i
I — n
I — in I — nj '
745] AND THE POLYHEDRAL FUNCTIONS. 163
*-< Q 1*
moreover, in the top line, the terms 2, r~- - and — S7 — destroy each other. The
I — m l — n
whole coefficient of ,, when z— I is a multiple factor of P, thus is
z ~~ I
•2— l -2 -1
I — m l — n
— n
a form which is now symmetrical in regard to the inverts , and , -- .
I - m l-n
29. The value just obtained must be equal to
(1 -? + &*•) 12 *« - -1 + 2 if - 1 - 2' -U ;
V i — W 4—71 l — lj
viz. comparing the two forms and reducing, they will be identical if only
(1 -p + 2ap) J2't+f' _ 2 *(1+^>g-^ - 2 tlL^TP]. = 0,
( « — tx t — m t — n }
and it can be shown that the function inside the { } is in fact = 0.
O 1*
30. We have, as before, S7 - = 2 / --- 5 or writing each of these quantities
I — m l — n
the equation to be verified is
I — h 1 l — m ^ l—n
We have
that is,
The first derived function of the numerator is X^ (z — l)+ X1 — pX/, which for
z=l is X1 — pX', which is =0; and, for the denominator, it is X'(z — l) + X, which
is also =0. Passing to the second derived functions, we find
„ Pi ^XJ
z- 1, 2X' X'
From the equation
= - -
X z — I z — I
21—2
164 ON THE SCHWARZIAN DERIVATIVE [745
we find in like manner
" I - I, X'
and we thence obtain (z being always = I)
** i — "&> '
z — LI X
so that the equation to be verified becomes
— m l — n
31. But from the equation @, = PQ' - P'Q, = KP0Q0R0, we find XY1-X1Y =
and then, differentiating, XY-! + X'Yl — X^Y — X1 Y' = KR0': writing in these equations
z = I, they become
X'Y.-X.'Y-X.Y^KR,,
so that, dividing the second by the first,
_Z'F1 X' T _R^
X, Y "*" X, + F ~ M0 '
Y 0'
or, recollecting that X1=pXf and = » we nave
,-Bn
that is,
= (w + !)<!> — p"Z^ — — »2 , — ,
I — m L — n
the required relation.
32. The result is that, z—l being a multiple factor of P, the coefficient of the
1
term , is
z—l
33. In the case where z — l is a simple factor of P we have p=I, and the
coefficient is
— 22' -S.-- -2--(b-c)(2. - -2,
t — tj fc — m l — n) \ l — m l —
745] AND THE POLYHEDRAL FUNCTIONS. 165
34. Of course the formulae for the coefficients of , -- /v, and - — , give at once,
\Z — li) Z ~— (j
by a mere change of letters, those for the coefficients of — rr, - , and
» ( Z ~~ tivj" & iff/
__ , --- ; and the function in question,
(z -rif z — n
{a?, jrl+a^fa, b, c .'.Q -- - - i, --- ) ,
V /js — ax — o x — cJ
is now obtained in the required form
Jb?) M ^L _* -£_
(z-l)2" (z-mY" (z-rif" z-l" z-in" z-n"
where (ap) denotes £ (1 - p0-) + ap2, and the like for (bg) and (c?'); and where, z-l
being a multiple factor of P, the coefficient A contains the factor (ap); and similarly
for B and C.
35. Suppose that the coefficients a, b, c are no one of them =0; we have
al, =a, which does not vanish; that is, z — I being a simple factor of P, the
expression contains -- 7V> , or the invert - — , is essential : and similarly, z — m being
\Z ~~ i)~ Z ~~ I
a simple factor of 0, or z — n a simple factor of R, the inverts and
z — m z — n
are essential. But for z — I a multiple factor of P, the coefficient (ap) of the term
(Z—L)
may vanish, viz. this will be the case if a = | ( 1 -- J ; and, when this is so, the
coefficient A of the corresponding term — ; also vanishes; that is, — , is a non-
% ^~ v & v
essential invert. And similarly for any multiple factor z-m of Q or z — n of R, the
invert ---- — or - may be non-essential.
z — m z—n
36. If P, Q, R contain each of them only multiple factors of the same index,
say of the indices p, q, r for the three functions respectively, viz. if the functions
are F(U(z-l))*, G (U (z - m})* , H(U(z-n))r} the result contains only the six terms
written down : and then, if a, b, c are = £ (l - - ] , | f 1 - -J , if1" ^) respectively
the result is = 0 : viz. we then have
\x, z\ + x- ( a, b, c .'. I) , — — r i - 1 = 0,
V X«— ax— b as — c)
or we in fact have, for the values in question of a, b, c, a solution
f(x-a) : g(x-b} : h(x-c) = P : Q : R
of this differential equation of the third order.
166 ON THE SCHWARZIAN DERIVATIVE [745
37. The reasoning applies directly to lines 2, 3, 4, 5 of the PQR-Tab\e : and
with a slight variation to line 1 ; viz. here the factors of R (= — 1 + zn) are all simple
factors, but in virtue of c = 0 and a = b, the corresponding inverts disappear, and, the
other inverts also disappearing, the value of the function is = 0. Hence lines 1, 2,
3, 4, 5 of the PQ-R-Table giye eacn °f them a result = 0, for the values of (a, b, c)
appearing by the table itself, and shown explicitly in the corresponding line of the
Annex.
Thus line 3 shows that the function x, determined by
f(x-a) :g(x-b) : h(x-c) = (z* + 2 V- Sz- + I)3 : - 12 V-3 (z5 - z)- : - (z4 - 2 \/-Sz- + 1)»,
satisfies
J ) ,
,9' 8' 9"X#-a' x-b' x-c;
and so for any other of the five lines.
38. The indices of the factors of P, Q, R may be such that, for proper values
of the coefficients a, b, c, there are in all only three essential inverts, say •
7^7^' z_ ' belonging to the three functions P, Q, R respectively, or it may be
two, or three, of them to the same function. When this is so, the function of these
inverts is, by what precedes, a curtate function, and it is consequently a function
1 1 1
where alf bl5 cx are the values of the three which do not vanish in the series of
expressions (ap), (b#), (cr).
The remaining lines (III, V, VII, VIII) and IX to XV of the P&R-Table give
such values of P, Q, R, the values of (a, b, c); and the calculation of the values of
(&l} b1} Cj) is shown by the corresponding lines of the Annex. And we have thus
values of x determined by the equations
f(x-a) : g(x-b) : h(x-c} = P : Q : R,
and giving
1 *
*a> b, , — ,,
V Ax -a x-b' x-c -z-a: z- z - c
39. For instance, from line IX we have
f(x-a) : g(x-b) : h (x - c) = (z - 4)3 : - (z - \}(z + 8? : 27 z- ( 1 - — ) ,
\ oo /
4 3 12
the values of (a, b, c) are -, -, ^ ; and since P, Q, R contain factors with the
exponents 3; 1, 2 ; and 1, 2 respectively, the coefficients which present themselves
on the right-hand side are
a3; bl, b2; cl, c2,
745] AND THE POLYHEDRAL FUNCTIONS. 167
which are
3 12 21
= 0; g, 0; 25, ^ respectively.
3 12 21 11
Hence writing a2, b1; c1 = -^, ^, , the corresponding inverts are — -- , ~»
-: and the result is
z
f l ">/* ? i? X -1 1 _J_V_/'3 1? 21 X ! J !Y
^' ^ \9' 8' ZS^JU-a1 x-V »-c/ U%> 25' SO^JU^T' 0-x' ^J1
40. It is hardly necessary to remark that an expression
a-i z —
in fact denotes
ai P! -
(z - a,) (z - b^ '
The particular form of the z inverts is immaterial ; we could by a general linear
transformation upon the z make them to be -- , — j- , with the (alt 61; d)
& ~~~ 6^1 Z ^"~ C/j Z ~^ Cj
arbitrary; or we can give to the a1} bly Cj any particular values we please: there
would be a propriety in making the inverts to be in every case (as in the foregoing
example) - , - , - - ; but the numerical work would be troublesome, and it is
Z Z— 00 Z— 1
not worth while to effect it.
41. The conclusion is that lines (III, V, VII, VIII) and IX to XV of the
PQR-Table, give, for determinate values of (a, b, c) and (a1( bls Cj), solutions
f(x-a) : g(x-l)} : h(x-c) = P : Q : R
of the equation
»/ i. X ! ! ] V / v X ! l l V
{x, *}+aP[&, b, c .'.O— -, -T, - = a:, bj, cx .-. 0 -- , —7-,
-
where a, b, c, a1; 61} d are or can be made arbitrary, but without any real gain of
generality herein. This is the Differential Equation \x, z}.
42. Recurring to the results from the Arabic lines of the PQTZ-Table, but for
convenience writing s instead of z, we have
f(x-o} : g(x-b) : h(x-c} = P : Q : R,
where P, Q, R are now functions of s, a solution of
( , (faV f i, X l l l V A
«, s } + (-J--) a, b, c .'. 0— , —r, ---- =0.
VO*/ V A* — a x — o x — cj
But we have
168 ON THE SCHWARZIAN DERIVATIVE [745
and the foregoing is therefore a solution of
r i. x l l l v
[8, x\ = a, b, c .'. H --- , - — T , — I ,
V AX — a # — 6 x- c)
a differential equation of the third order. This is the Differential Equation [s, x}.
43. From the Roman lines, if we assume
f(x-a) : g(x-b) : 7i(a?-c) = $ : d : ?R,
where S4>, d, 9t are functions of z, not the same functions that P, Q, R are of s,
since they belong to a different line of the Table : we have, as before,
(dx\"i , Y 1 1 1 V / , Y 1 1 IV
\x, z + j- a, b, c .'. 0— , , , - = a^ b,, c, /. ft - - , - r , - 1 .
\dzj \ Ax—ax — b x — c) \ AZ — a* z — b^z — cj
44. We may combine any such result with a properly selected result of the
preceding system, the two results being such that (a, b, c) have the same values in
each of them. (See as to this the foot-note referring to the Annex to the PQR-
Table.) The last equation then becomes
dx / ,
' C
or since
this is
1 1 IV-
a^ rrv *- J '
/dr\~
f -v / \AI**J \ i •> / •*
fa Zl + (dz) ** X> = fa *'*
, , / , Y 1 1 IV
[s, z\ = (al, bj, Ci /.(J - — , — T-, — I ,
1 V A.z-0^ z — h z-cj
the corresponding relation between s, z being of course obtained by the elimination
of x from the two sets of equations
f(a; -a) : g (x - b) : h(x-c} = P : Q : R, and f(x - a) : g (x - b) : h (x - c) = $ : Cl : SR;
viz. the required relation is
F : Q : £ - $ : 0 : 91,
where P, Q, R are functions of s ; *$, d, 3J functions of z ; and, in virtue of
the relations are equivalent to a single equation between z and s. And writing
finally x in place of £, that is, now considering ty, d, $ as functions of a, we have
$ : d : «R=P : Q : 12
as a solution of
/ Y 1 1 \2
fi/v A-1- \
[«, «J = [a,, bn cx /.I)— , 7 , - — 1 ,
V X* — o» x — b^ x — cj
a differential equation of the third order of the foregoing form, {s, x} = given function
of x, but with different values of the coefficients, (a^ bj, Cj) instead of (a, b, c).
745]
AND THE POLYHEDRAL FUNCTIONS.
169
45. It thus appears that there are in all 16 sets of values of (a, b, c), for
which the equation is solved, viz. the 16 sets of values are shown in the right-
hand column of the Annex. For greater clearness I exhibit the integral equations
as follows:
Functions of x.
Functions of s.
1
f t rp ft\ . ft 1 ry* A\ • A / ~» ft\
J \t*j — Uj J . y \*AJ — Uj . 11 \JLt — 0 }
= P : Q : R (1)
Polygon
I
5)
(2)
Double Pyramid
II
,,
(3)
Tetrahedron
III
Ix : -(x+iy : (x-l)2
(3)
»
IV
/» / \ / j_\ Z. / \
j (x Ojj g (x o ) \ fi (x — c)
(4)
Cube and Octahedron
V
(x-\r -(x+iy : 4*
" ( /
»
VI
f (x — a) g (x — b) : h (x — c)
(5)
Dodecahedron and Icosahedi-on
VII
A.'y* (nf* _]_ 1 )•* • \ *¥* 1 i^
(5)
„
VIII
(x-1)2 -(x+iy- : ±x
(5)
,,
IX
P Q : R (IX)
(5)
,,
X
(X)
(5)
„
XI
(XI)
(5)
„
XII
(XII)
(5)
5?
XIII
(XIII)
(5)
II
XIV
(XIV)
(5)
„
XV
(XV)
(5)
J)
The values of the P, Q, R as functions of x, or of s, are taken out of the
PQR-Table: only in the lines III, V, VII, VIII, where P, Q, R are given as
= 4,2, -O + l)2, O-l)2,
and where, as regards V and VIII, there is a transposition of P and R, I have
inserted the actual values of the ay-functions. (See as to this the foot-note referring
to the Annex.)
The Schwarzian Theory. Art. Nos. 46 to 62.
46. Considering the foregoing equation
= a
, l .-- , __ __
as a particular case of the equation {s, x} = Rational function of x, =R (x) suppose,
then we have in 1, I, II, IV, VI solutions of the form x = Rational function of s.
G. xi. 22
170 ON THE SCHWARZIAN DERIVATIVE [745
Consider, in general, a solution of this form, x = F (s) a rational function of s : then
s is an irrational function of x, and if si} s2 are any two of its values, {sl) x} = R(x),
{s2) x}=R(x); that is, {s2, x} = {slt as], and therefore (ante, No. 7) s.2 = — — -, . And
CSi ~r Ct
then x = F(s2)=F( — - — -»), = F(sl): viz. F(s) is a rational function of s, transform -
\csi T a/
able into itself by the transformation s into 7 : and it is moreover clear that
cs + a
between any two roots s whatever of the equation x = F (s) there exists a homographic
relation of the form in question. Further, it is clear that these homographic trans
formations form a group; and consequently that F (s) is a rational function of s,
transformable into itself by the several homographic transformations of a group of
such transformations : viz. taking x to be a rational function of s, it is only in the
case x = F(s), a function of the form in question, that {s, x] can be equal to a
rational function of x.
47. We may, in any equation between x and s, consider these as imaginary
variables p + qi and u + vi respectively ; considering then (p, q) and (u, v) as rect
angular coordinates of points in different planes, we have a first plane the locus of
the points x, and a second plane the locus of the points s: there is between the
two planes a correspondence which is in fact the correspondence of conformable
figures : to the infinitesimal element dx drawn from a point x of the first figure
corresponds an infinitesimal element ds drawn from the corresponding point s of the
second figure, these elements being in general connected by an equation of the form
ds = (a + bi) dx, where a and 6 are functions of x or s ; and this signifies that, to obtain
the pencil of infinitesimal elements or radii ds proceeding in different directions from
the point s, we alter in a determinate ratio the absolute lengths of the infinitesimal
elements or radii proceeding from the corresponding point x, and rotate the pencil
through a determinate angle : this ratio and angle of rotation, or say, the Auxesis
and the Streblosis, being of course variable from point to point. Or, what comes to the
same thing, if dx and d-^x be consecutive elements of the path of the point x, and
ds, d^s the corresponding consecutive elements of the path of the point s, then the
ratio of the lengths of the elements dx, d^x is equal to that of the lengths of the
elements ds, d^s; and the mutual inclination of the first pair of elements is equal
to that of the second pair of elements. In particular, if at any point the path of x
is a curved line without abrupt change of direction, then at the corresponding point
the path of s is a curved line without abrupt change of direction. In what precedes,
we have the relation at ordinary points; but there may be critical corresponding
points (x, s), the relation at a critical point between the corresponding elements dx,
ds being of the form ds = (a + bi) (dx)*, (X a positive integer or fraction) : here the
angle between two elements ds is = X times that between the two elements dx ; or,
if the path of the point x through the critical point is without abrupt change of
direction, say if the angle between the two consecutive elements is the flat angle TT,
then the angle between the two consecutive elements ds is = A/TT : viz. there may be
in the path of the point s an abrupt change of direction.
745]
AND THE POLYHEDRAL FUNCTIONS.
171
48. I consider the foregoing equation {s, x} = R (x), where R (x) is a rational
function, and is now taken to be a real function of x : we may assume s' = ip'6'eie,
where the accents denote differentiation in regard to x, and where p, 6, and there
fore also 6', are real functions of x. We have
and thence
6"
&" f&'\*
"&
and thence
.
pV p
Putting this = R (x), and assuming that as is real, we have
{p,
The last equation gives p"& = 0, that is, ff = 0, which gives s = 0, and may be
disregarded ; or else p" = 0, therefore p', a real constant, = 7 suppose, and {p, x] = 0 :
hence for the solution of the equation {s, x} = R (x), we have s' = iy0'ei9, 0 a real
quantity determined by {0, x} +£0'2 = R(x): and then, integrating the equation for s',
we have s = a + fii + ye™, a, 0, y real constants.
49. The conclusion is that, if {s, x} = R (x), a real function of x, and if x be
real, that is, if the point x move along a right line (say the #-line), then s - a + @i + <yeie
(6, and the constants a, /3, 7, being real), that is, the point s moves in a circle,
coordinates of the centre a, /3, and radius =7.
50. Suppose a, b, c are any real values of x representing points a, b, c on the
; and A, B, C any given imaginary values of s representing points A, B, C
22—2
172 ON THE SCHWARZIAN DERIVATIVE [745
in the s-plane : since {s, x}=R (x) is a differential equation of the third order, the
integral contains three arbitrary constants, and we may imagine these so determined
that to the values x=a, b, c shall correspond the values s = A, B, C respectively.
If there is not on the #-line any critical point, as the point x moves continu
ously along this line the point s will move continuously along a circle, which (in
asmuch as a, b, c and A, B, C are corresponding points) must be the circle through
the three points A, B, C*.
51. If however the points a, b, c are critical points, such that the element ds
at the corresponding points A, B, C are equal to multiples of (cfo)x, (doty* (dx)v re
spectively, then to the flat angles TT at a, b, c correspond in the path of s the
angles XTT, /XTT, VTT at the points At B, C respectively: and, assuming that a, b, c
are the only critical points on the #-line, the path of s is made up of the three
circular arcs CA, AB, BG meeting at angles XTT, /JLTT, vir respectively. The arcs are
completely determined by these conditions ; for supposing the arc BC to make with
the chord BG, at the points B and C, the angles /, /, and similarly the arcs CA
and AB to make with the corresponding chords the angles g, g and h, h, then the
conditions give XTT, fiir, V7T = Z.A + g + h, ^B + h +/, ^ C +f+g, where the angles
referred to are those of the rectilinear triangle ABC: we have thus the values of
/, g, h; and the arc BC is the arc on the chord BC meeting it at angles f, f:
and the like as regards the arcs CA and AB respectively.
52. The foregoing equation
r i f v X ! ! * V
{s, x] = a, b, c /. fl - -- , -- T , - ,
V AX — a x — b x — cj
where a, b, c have the values \ (1 - X2), \ (1 - /A2), \ (1 - v2), and X, p, v are real and
positive, has x = a, b, c for critical points of the kind in question. In fact, writing
x — a = h, the equation is of the form
- X2) a0
, -j£—' + -j? + a1 + aJi+ ...,
which is satisfied by
d , ds 1 + X , , 7 , ,
a^a— -r-^+M+MF+-J
we thence obtain an integral of the form
s = M~x (1 + kji + k2h* +...), =k<}> for shortness.
This is a particular integral, but we have' from it the general integral
* Since there is no critical point on the x-line there can be no abrupt change of direction in the path
of s, that is, the path of s cannot consist of circular arcs meeting at an angle: but it is in the text
further assumed that the path of s cannot consist of different arcs of circle, the one continuing the other
without any abrupt change of direction.
745] AND THE POLYHEDRAL FUNCTIONS. 173
If A be the value of s corresponding to h = 0, then /3 = BA , and we find
l4__ -A+^,
"l'S'7J.M'^^/-t] ' — -*1 T 5.7 " ~ "7 T • •• 5
__
— i £7 A. ' "'S'7J.'^/-t ' — -* T 5.7 " ~ "7
7 + ofc<p V bk<pj \ OK<p/ OK 9
viz. reducing - to its principal term AA, and then writing ds, dx for s — A, and h(=x — a}
respectively, we have ds = K (dx)K, or x = a is a critical point with the exponent A, ;
and similarly x = b and x = c are critical points with the exponents yu, and v respectively.
53. Hence in the equation
/ Y 1 1 1 \2
fi/i A-1- * * i
[8, X \= a, D, C .'.(I — -- , -- r, -I,
V Ax — ax — bx — c]
as the point x, passing successively through a, b, c, describes the #-line, the point s,
passing successively through A, B, C, describes the sides AB, BC, CA of the curvilinear
triangle ABC. To points x indefinitely near the «-line correspond points s indefinitely
near the boundary AB, BC, CA of the triangle, viz. to points x indefinitely near to
and on one side, suppose the upper side, of the #-line, correspond the points s
indefinitely near to and within the boundary of the triangle : and in like manner to
whole series of the points x on the same upper side of the #-line, correspond the
whole series of points s inside the triangle.
54. We have attended so far only to one of the points s which correspond to
a given point x, but considering the set of points s which correspond to the same
point x, we have in the s-plane entire circles forming by their intersections curvilinear
triangles ABC, ABC', &c. ; we have thus two systems, say ABC, &c., and ABC', &c.,
of triangles, such that to a point x on the upper side of the #-line correspond
points s, one of them within each of the triangles ABC, &c., and to a point x on
the lower side of the #-line correspond points s, one of them within each of the
triangles ABC', &c. ; and so consequently that, to the two half-planes on opposite sides
of the #-line, correspond the two sets of triangles ABC, &c., and ABC', &c., respectively.
55. In order that the relation s and x may be an algebraical one, it is necessary
that the two sets of triangles should completely cover, once or a finite number of
times, the whole of the s-plane : and this implies that the angles XTT, JJLTT, VTT have
certain determinate values; and, in fact, that dividing the surface of a sphere into
triangles, each with these angles, the curvilinear triangles ABC, ABC', &c., are the
stereographic projections of these triangles. It was by such considerations as these
that Schwarz, in the Memoir of 1873, p. 323, obtained the series of values I to XV
of X, /*, v, giving for a, b, c, = £(1-V), ^ (I - ^), % (1 - v°-), the series of values
mentioned in the Annex of the PQR-T&lale : and thus showed a priori that the equation
{*, «}=»(», b, c .*.Q , =-, — — - }
\ \x — ax — bx — c)
is algebraically integrable for these values of a, b, c; and only for these values, or
for values reducible to them.
174 ON THE SCHWARZIAN DERIVATIVE [745
56. As an instance, take the double pyramid form: the integral equation is
f(x-a) : g(x-l) : h(x-c) = 4,sn : -(s'l-l)2 : (sn + l)z,
or say
(c - a) (x - b) _ Qn-l)2.
(a-b)(x-c)~ (sw+l)2'
sn _ ]\2
or if, for greater simplicity, we assume a, b, c = 1, 0, oo , this is #= n or say
1 -f V ic
_(sw — 1) = V#(sn + l), that is, 5n=— — -f-, a solution of the differential equation
1 + V x
I, «.i - f _ in -«
l«, « -(^, 2(
In particular, if n = 3, we have x =
3 4 3
or s3 =
, a solution of
1 \2
a;_
57. We have here the spherical surface divided by the equator and three meridians
into twelve triangles, each with the angles \ir, \TT, ^TT : and then, projecting from the
South pole on the plane of the equator, we have the annexed figure of the s-plane,
JB\
B,
divided into 12 curvilinear triangles, each with these same angles 90°, 90°, 60° ; the
plane is divided by the shading into two systems, each of 6 triangles. The figure
of the #-plane is by the #-line divided into two half-planes, one shaded, the other
unshaded ; and we have on the line the point c at oo , a at the origin, and b at
the distance unity.
745] AND THE POLYHEDRAL FUNCTIONS. 175
58. Take x real ; then, if x is positive and less than 1, s3 is real and positive,
and we have for s the infinite half-lines at the inclinations 0°, 120°, 240°, while if
x is positive and greater than 1, s3 is real and negative, and we have the infinite
half-lines at the inclinations 60°, 180°, 300°. If x is real and negative, then s3 is of
1 — ki
the form ,- . , = cos 0 + i sin 6 ; whence s is of the same form, or the locus of the
1 + ki
~^ Y y>
point s is a circle radius unity. Writing s3 = — — -._ , and supposing that the point x
1 + V#
moves along the #-line from b through a to c at — oo , and then from c at + oo to 6,
the point s describes the sides BA, AC, CB of the shaded triangle marked K.
59. Suppose that the point x is at k, in the shaded half-plane at an indefinitely
small distance from a ; say we have x = — %K2i, (K small), then taking for *Jx the value
1 — K. ( 1 — i)
*(l — t), we have s3 = =— — -^ /, =1 — 2«(1 — i) nearly, and hence a value of s is
JL "T" fC ^ -L "~~ %J
= 1 — f/c + f/tt, which belongs to a point K near A, and within the shaded triangle:
we have thus, in respect of this value of s, the shaded half of the #-plane corre
sponding to this shaded triangle. To the same value x = — 2fc2i, correspond in all six
values of s, giving six points K each lying near a point A within one of the shaded
triangles; and hence the shaded half-plane corresponds to the six shaded triangles, and
the unshaded half-plane corresponds to the six unshaded triangles.
60. Suppose the equation is
i i / u X * 1 1 \2
{s, as = a, b, c /.Q— -, — •=• , -) ,
V \x — ax — bx — c)
that is,
_ - (b — c) (c - g.) (a — b) / a b c \
x — a.x—b.x — c \b-c.x — a c — a.x—b a — b.x — c) '
where a, b, c are real, but a, b, c are imaginary. It is to be shown that, if the path
of x is the circle passing through the points a, b, c, then the path of s is a circle
passing through the corresponding three points.
61. We may find a, /3, 7, 00, 0lf 0,, such that a, b, c are =a + @i + fy&i, a + fti + ye9^,
a + £*+7e«»< (this is, in fact, finding a and ft the coordinates of the centre, and 7 the
radius of the circle through the three points a, b, c) : we then have x — a. + @i + yeei,
6 a variable parameter, the equation which expresses that the point x is situate on
the circle in question.
We have x - a = 7 (eei - er8*), = ^ <e+0<>> {e± <«-«•>' - e~* {e~e^} ; the second factor is
isin£(0— 00), =iP suppose, or the equation is x — a = iPye*(0+e°}i, say
x — a = iPy expi £ (0 + 00).
Similarly x - I . = iQy expi 1 (6 + 6,), and as- c = iRy expi £ (6 + 02); where P, Q, R denote
sin |(0 - #0), sin £ (6 - 0J, sin £ (0 - 02) respectively. In like manner, we have b - c, c-a,
a - b, = iFry expi | (0! + 02), iGV expi £ (03 + 00), ^7 expi £ (00 + 0,), where JP, G, 5" denote
0-^) respectively.
176 ON THE SCHWARZIAN DERIVATIVE [745
We have
b — c . c — a .a — b — FGH • \ /A t A . » om
- «-„.«-»..- o = -PQR expl * (e" + * + * - 3">'
with the like values for - r and - , - . Hence the right-hand side of
c — a.x — b a — b.sc—c
the equation is
FGH /a b
62. Considering now the left-hand side of the equation, we have
substituting for x its value = a + fii + <yeei, this becomes
that is,
1
72
Assume s = L + Mi + Ne&i, L, M, and N constants ; then using the accent to denote
differentiation in regard to 0, we find without difficulty {s, 6} = [®, #}+|-®'2, and the
value of {s, x} becomes
= - -2 ({0, 6} + $&* - i) expi (- 20).
Hence, substituting the values of the two sides of the equation, the imaginary
factor expi (— 20) divides out, and the equation becomes
AI irvs i _ a b c
~
an equation, in which everything is real and which thus determines S as a real
function of d : and we have therefore the theorem in question.
Connexion with the differential equation for the hypergeometric series. Art. Nos. 63 to 68.
63. Take p, q given functions of x, and y a function of x determined by the
equation
745] AND THE POLYHEDRAL FUNCTIONS. 177
again P, Q given functions of z, and v a function of z determined by the equation
d?v
j-.>
dzi
and assume
y = wv.
Substituting this value of y in the first equation, we obtain for v an equation
of the second order (the coefficients of which contain w), and we may make this
identical with the second equation ; viz. comparing the coefficients of the two equations,
we thus have two equations each containing w ; and by eliminating w we obtain a
differential equation of the third order between z and x. This is, in fact, the basis
of Kummer's theory for the transformation of a hypergeometric series : the equation
between z, x will be found presently in a different manner.
64. But if with Schwarz, instead of making the equation obtained for v as above
identical with the given equation for v, we merely assume that the two equations are
consistent, then there is nothing to determine the value of z, which may be regarded
as an arbitrary function of x] y and v are then functions of x, and w denotes the
quotient y-r-v of these two functions, and as such satisfies an equation the form of
which will depend on the assumed relation between z and x. In particular, if P and
Q denote the same functions of z that p and q are of as ; and if we assume z = x,
P, Q will become =p, q respectively : the given equation in v will be
d2v dv
and w will thus denote the quotient of any two solutions of the equation
d-y
viz. writing X = p- + 2 -^- — 4<q, then, by what precedes, the equation for w will be
doc
{w, x}=-
65. Returning now to Kummer's problem, and considering y, v as solutions of
the two differential equations respectively, w is a function independent of the particular
solutions denoted by these letters : we have y = ivv, and taking any other two solutions
77 ?J
we have y1 = wv1, so that — = -; calling each of these equal quantities s, we have s
y\ vi
denoting the quotient of two solutions of the equation in y, and also the quotient
dp
dx
of two solutions of the equation in v ; whence, writing as before X =p2 + 2 ~- — 4>q,
dP
and similarly Z = P2 + 2 -^ -- 4Q, we have
and since in general
^ ^ = (£) ^ z^ + & *'*'
C. XL 23
178 ON THE SCHWARZIAN DERIVATIVE [745
we obtain
-I
as the required equation for the determination of z as a function of x. The process
does not give the value of w, but this can be found without difficulty, viz.
-.
ax
If z, x are regarded each of them as a function of the new independent variable
9, then the equation is
66. Jacobi's differential equation of the third order for the transformed modulus X,
Fund. Nova, p. 78, [Ges. Werke, t. i, p. 132], is
where the accents denote differentiations in regard to an independent variable 6 : viz.
dividing by 2fc'2X'2, this becomes
which is thus a particular case of Kummer's equation, k, X corresponding to x, z
respectively, and the values of X, Z being
67. In the case of the hypergeometric series, the two differential equations of the
second order are
d?y 7-(a + ff + 1) x dy __ aff.y _
Hence
__
da? x .I—a; dx x.l — x
drv y- (of +^ + 1)2 dv _ _affv_ = Q
dz- z . 1 — z dz 2.1— z
^'))-(« + /3 + l)A' = 7 7-g-^-l = _- ayg
« . 1 — X ~ X I — X * X.I — X'
;ind hence
1.9 _ , .
^ *q~
x.l-x
viz. writing
X2 = (1 - 7)2, a = i (1 - X2),
745] AND THE POLYHEDRAL FUNCTIONS.
and putting in the formula x—\, = — (!—#), we have
-a + b — c
.x' x — oo ' x — I/
with a like formula for £ ( P2 + 2 -, 4Q ] . We then have
V az J
y = wv,
~dz '
and the differential equation of the third order for the determination of z is
/ , Yl 1 1 \- fdz\9 / , Yl 1 1 V
{z, «} + (%, b1} cx .*.(}-, -, -^r-J j- - (a, b, c .-.«-, -, - -T =°.
V )(z z— en z— 17 Vc?^/ V X« x — oo a; — I/
where au bls Cj are the same functions of a, ft', y' which a, b, c are of a, ft, yr
This is, in effect, Rummer's equation for the transformation of the hypergeometric series.
68. And in like manner the Schwarzian equation for the determination of s, the
quotient of two solutions, is
r i / u X1
[s, x = a, b, c .•.()-,
V X«
-, = .
— oo a; — IJ
PART II. THE POLYHEDRAL FUNCTIONS.
Origin and Properties. Art. Nos. 69 to 80.
69. The functions in lines 1, ...,5 of the PQJ?-Table are connected with the
geometrical forms :
'1. Polygon or
^2. Double Pyramid *,
3. Tetrahedron,
4. Octahedron and Cube,
5. Dodecahedron and Icosahedron,
(these figures being regarded as situate on a. spherical surface), and with the stereo-
graphic projections of these figures.
Prof. Klein regards 1 as belonging to the polygon and '2 to the double pyramid : it seems to me
that the fundamental figure, to which 1 and 2 each of them belong, is the polygon.
23—2
180 ON THE SCHWAEZIAN DERIVATIVE [745
Consider a spherical surface and upon it any number of points: take at pleasure
any point as South Pole, this determines the plane of the equator ; and the stereo-
graphic projection of any point is the intersection with the plane of the equator of
the line joining the point with the South Pole.
To fix the ideas take the radius of the sphere as unity: let the axes of x and y
be drawn in the plane of the equator in longitudes 0° and 90° respectively, and the
axis of z upwards through the North Pole : the position of a point on the sphere
is determined by means of its N.P.D. 6 and longitude /: moreover we take X, Y, Z
for the coordinates of the point on the surface, and x, y for those of its projection ;
and we then have
X, Y, Z=sinO cos/, sin 6 sin/, cos 6 ;
X
tf =
and conversely,
X, Y, Z=2x, 2y, \-a?-y\ - (1 +x'~+ y-}.
We represent the point (X, Y, Z) on the spherical surface by means of the
magnitude x + iy, = tan £0 (cos/+ i sin/), or say by the linear factor, s — (x + iy): and
similarly any system of points on the surface by means of the system of magnitudes
x + iy, or say by the function IT [s — (x + iy)}, denoting in this manner the product of
the linear factors which correspond to the different points respectively.
70. It will presently appear that, if (considering a different stenographic pro
jection, that is, a different position of the South Pole) we take x, y' as the coordinates
of the new projection of the point, then x' + iy' is a homographic function
a (x + iy) + b -i- [c (x + iy) + d}
of x + iy: and consequently that the functions of s, which belong to different pro
jections, are linear transformations one of the other: but at present we consider a
single projection.
It may be proper to remark that the figures in question are spherical figures
having summits which are points on the spherical surface, edges (or sides) which
are arcs of great circle joining two summits, and faces which are portions of the
spherical surface : the mid-points of the sides, and the centres of the faces are of
course points on the spherical surface.
71. (1), (2). Considering a regular polygon formed by n summits on the equator,
the longitude of one of them being 0°, then the stereographic projections correspond
with the points themselves, and the values of x + iy are
27T . . 2-7T (n-l)2w . .
1, cos M sin — , .... cos h i sin
n n n
The corresponding function of s is sn— 1.
AND THE POLYHEDRAL FUNCTIONS.
181
The values of x + iy for the mid-points of the sides are
TT . TT 3?r . . STT ('2n-l}-rr . (2/1-1) TT
cos — h i sin - , cos --- h i sm — . .... cos - - - -- (- 1 sm 2 - - — .
n n n n n n
The corresponding function of s is sn+l.
The North and South Poles, which form with the n points a double pyramid of
n+2 summits, correspond to the values s = 0 and 5=00. We have thus
as the function corresponding to the double pyramid.
72. (3). Considering for a moment the tetrahedron as a figure with rectilinear
edges, this is so placed that two opposite edges are horizontal, and that the vertical
planes passing through the centre and these two edges respectively are inclined at
angles ±45° to the meridian: viz. the upper edge has the longitudes 135°, 315°,
and the lower edge the longitudes 45°, 225°. We thus explain the position of the
spherical figure.
Corresponding to the summits we have the function s* — 2i V3 s2 + 1.
In fact, the equation s* — 2i V3 s2 + 1 = 0 gives s2 = i (V3 + 2), and hence the values
of s are the four values of x + iy shown in the annexed table for the values of
X, Y, Z, and x + iy for the summits of the tetrahedron,
long. X
Y Z
x + iy
1
I I
l+i
V3
135° -
225° -
21 5° -u
V3 V3
V3-1
V3 + 1
-1 -t
V3-1
l + i
Corresponding to the centres of the faces, or summits of the opposite tetrahedron,
we have the function s*+ 2tV3s2 + l.
Corresponding to the mid-points of the sides, we have the function
viz. the points in question are the North Pole s = 0, the South Pole s = oo , and
the four points s=±l, s=±i on the equator at longitudes 0°, 90°, 180°, 270°
respectively.
182 ON THE SCHWARZIAN DERIVATIVE [745
78. (4). The octahedron is placed with two of its summits as poles, and the
other four summits in the equator at longitudes 0°, 90°, 180°, 270° respectively:
the values of s are, as in the last case, 0, oo , +1, ±i, and the function is
The function for the centres of the faces, or summits of the cube, is s8 + 14s4 + 1.
The function for the mid-points of the sides of the octahedron or of the cube is
s12 - 33s8 - 33s4 + 1.
74. (5). The Icosahedron is placed with two of its summits for poles ; five summits
lying in a small circle above the plane of the equator at longitudes 0°, 72°, 144°, 288°,
and the remaining five summits in the corresponding small circle below the equator at
longitudes 36°, 108°, 180°, 252° and 324°.
The function for the summits of the Icosahedron is
»(l--
The function for the centres of the faces of the Icosahedron, or summits of the
Dodecahedron, is s20 - 228s15 + 494s10 + 228s3 - 1.
The function for the mid-points of the sides of the Icosahedron or the Dodecahedron
is
s30 - 522s25 + 10005s20 + Os15 - 10005s10 + 522s5 + 1.
I give for the present these results without demonstration.
tJT
75. Writing - for s so as to obtain homogeneous functions (*]£#, y)n, — it will be
(7
recollected that the x, y of these functions have nothing to do with the x, y of
the foregoing values x + iy — the forms which have thus presented themselves may be
denoted as follows :
(3): /3 = (1, -2»V3, 1£*2, tf-r,
A3 = (l, +2tV3, l&r2, y-}-,
£3 = xy O4 - y4),
(4): f*
/i4 = (l, 14, l^ar4, ?/
«4 = (1, -33, -33,
(5): /5 = ay(l, 11, -
k5 = (I, - 228, + 494, + 228, - 1$V, y5)4,
*5 = (1, -522, 10005, 0, -10005, 522, 1$V, y5)6,
where observe that /4 is the same function as £3. In each set of functions /, h, t,
we have h and t covariants of f, viz. disregarding numerical factors,
h is the Hessian, or derivative (/, /)2, and t is the derivative (f, h).
745] AND THE POLYHEDRAL FUNCTIONS.
76. Since /4 is the same function as £3, we have of course /4, A4 and t4>
themselves covariants of f 3 : but it is convenient to separate the two systems.
77. It is to be observed that f'3 is a quartic function having its quadrinvariant
(/) = 0 ; but independently of this, that is, qua quartic function, it has only the
covariants A3 and £3 (the Hessian and the cubicovariant respectively), viz. every other
covariant is a rational and integral function of f3, A3 and £3. In particular, A4 and
t± are rational and integral functions of /3, A3 and £3 ; but inasmuch as f'3 and
A3 are not covariants of /4, this is not a property of A4 and £4 considered as
covariants of f4>, and the relation in question need not be attended to.
78. It has just been stated that /3 qua quartic function has (in the sense
explained) only the covariants A3 and £3 : f4> qua special sextic function and fo qua
special dodecadic function have the like property, viz. /4 has only the covariants A4
and £4 ; f5 only the covariants A5 and to. Hence fS, f4>, fo are " Prime-forms " in
the sense defined in the paper by Fuchs, of 1875, viz. a Prime-form has no covariant
of a lower order than itself, and also no covariant of a higher order which is a power
of a form of a lower order.
79. The same functions have also the property that they are functions trans
formable into themselves by means of a group of linear transformations, and in this
point of view they were considered in the nearly contemporaneous paper by Klein, of
1875; it is in this paper shown that the functions so transformable into themselves must
be Polyhedral functions as above, the linear transformations in fact corresponding to
the rotations whereby the spherical polyhedron can be brought into coincidence with
its own original position. This theory will be presently given.
80. It is to be observed that, if U, V are functions (*$#, y)n of the same
order n, then using the accent to denote differentiation in regard to x, UV - U'V
and (U, V) differ only by a numerical factor: and further that, writing as before
cc
*=-, and in the expression UV - U'V regarding U, V as functions (*}[s, l)n, and
t/
the accent as denoting differentiation in regard to s, we have UV — U'V and (U, V)
differing by a numerical factor only. We have in the PQP-Table, lines 3, 4, 5,
P, Q, R equal to given numerical multiples of W, F, fa, the indices a, /3, 7 being
such as to make these to be functions of the same degree: hence, neglecting
numerical multipliers, PQ' - P'Q is equal to a function (/*?, F), which is = A""1^-1 (A, t) :
and the theorem that PQ' - P'Q, = QR - Q'R, = RP' - R'P, contains only factors of
P, Q, R is in fact the theorem that (A, t\ (A, /), and (t, /) are each of them equal
to a term or product of /, A, t : which is a result included in the theorem that /
has only the covariants A and t. And by this last theorem we know already how
from R, assumed to be known, we can derive P and Q : viz. R is a power of /;
and we thence have A = (/, /)2 and t = (h, /), equations giving the functions A and t,
upon which P and Q depend.
184 ON THE SCHWARZIAN DERIVATIVE [745
Covariantive Formulce. Art. Nos. 81 to 84.
81. The various covariantive formulae will be given with their proper numerical
coefficients.
Tetrahedron function. /, h, t stand for the before-mentioned values,
/3, A3, $ (P, Q, R = h3, -12iV3.£2, -f3).
For /3.
^_ o
(a, b, c, d, e) = l, 0, -- , 0, 1.
(/, A) = 32i V3 . t, (/, /)4 = 5767= 0, (/ A)4 = 1152/ = 1152 .
--,
&»-/»- 12* V'S^'-O,
/A = (l, 14, l$a-, 2/4)2
It is convenient to remark that t2, f3, h3 being of the same order we have
t- (f3, h3) +f3 (k3, Z2) + h3 (t\ f3} = 0,
that is,
t2 . 3 . 3/2A2 (/ h) +f3 . 3 . 2hH (h, t) + h3.2. 3tf2 (t, /) = 0,
an equation which, substituting for (/, h), (h, t), (t, /) their values, reduces itself to
the before-mentioned relation h3 — fs — I2i»J'3P= 0 ; and we have thus a verification oi
the values of (/, K), (h, t) and (t, /). The like remark applies to the other two
cases, which follow.
82. Hexahedron function. /, h, t stand for the before-mentioned values
/4, A4, *4 (P, Q, R = h3, -t\ -
For /4.
(a, 6, c, d, e,f, #)=(0, i 0, 0, 0, -£, 0).
y = 0, I (/ /)« = (720)2 . f,
(/, 0 = - 12A2, | («, O2 = 2* . 3MP .
(A, t) = - 1728/3,
7i3-«2-108/4=0.
745] AND THE POLYHEDRAL FUNCTIONS. 185
83. Dodecahedron function. /, h, t stand for the before-mentioned values
/5, h5, to (P, Q, R=hs, -t\ -1728/5).
For /5.
(a, b, c, d, e,f, g, h, i, j, k, I, m) = (0, &, 0, 0, 0, 0, £, 0, 0, 0, 0, -&, 0).
$ (/ /)« = - 121/>, \ (f, fY = 0, \ (/, /)« = \ (924)* (720)" . if/*,
I (/, /)8 =0, i (/, f)w = 0, J (/, fr = i (924)* (720)< . ff*,
(/ A) = - 20*, i (A, A)2 = 173280/3,
(/*) = - 30#, | (*, *)2 = 9082800/3/*,
(A, $) = - 86400/5,
84. We have
t = (xw + yw) (1, 522, -10006, -522,
Write
^ = (^ + 2/2).(l, 2, 6, -2, l$a;, y)4,
then
« = f(l, -10, 45$f,/).
Or putting
(^ + y2)(i> 2. 6, -2, i$a?f
V/ ' V«y («10 +
that is, ^ = jjV/. then
- 10p3 + 45^ = -y^j . (Klein.)
Investigation of the forms fb and ho. Art. Nos. 85 and 86.
jfy _ ^
85. Writing for shortnessf A: = tana = ^ — > and ff = cos 36° + * sin 36°' then tne
values of x + iy corresponding to the summits of the Icosahedron are
0,
k, kg*, kg*, kg6, kg*,
k->g, &-y, k-y, k-y, k-y,
oo;
and the function /5 is thus
* The numerical coefficients -¥\ and ff are Klein's B and A: the latter of them is the ordinary
quadrinvariant of a dodecadic function; the former is an invariant linear as regards the coefficients of /,
and existing only for the special form / in question : viz. writing for a moment
then (/, /)6 contains the factor X-, and (/ containing the factor X) the form is
4(/./)6
which is linear as regards X. We have also
say 4 = !fX2, U=-¥\X; or 8iB*=A. Of course in the case of a general dodecadic function /, we have
(/, /)ti, an irreducible covariant, not breaking up into factors.
t a is the a, 7 is the 7, and •/ the a-/3 of the Table, No. 99.
c. XL 24
186 ON THE SCHWARZIAN DERIVATIVE [745
where the product of the last two factors is sw + (/c~5-k5) s5 -I. We have
kr* = ^ (80 V5 + 170), = I (5 V5 + 11),
fr = ^(80^5-176), =|(5V5-11),
and consequently k~5-k5 = Il; or the function is
86. Similarly, writing for shortness* I — tan £7, £'=tan|7', where
5 + 2V5 . , 10-2^5 cos 7 3 + V5
cos2 7 = — - — , sm2 7 = -- --— - ; and therefore -. — * = /- ;
15 15 sm7 4
, =
15 15 sin 7' 4
and # = cos 36° + i sin 36° as before, then the values of x + iy for the summits of the
dodecahedron are
l~\ l-y, l-y, i-y, i-y.
The function ho is therefore
= 510 + S5 (Is - I'*) + 1 . s10 4- S5 (l's - l'~5) - 1.
We have
7_5 75 (1 + cos 7)5 - (1 - cos 7)5 2 cos 7 . „
• ~ * = — ^5 ~^ = • 5 - (5 + 10 cos2 7 + cos4 7)
sin8 7 sin5 7 f '
_ 2 cos 7 384 + 64 V5 128 cos 7
- sin*7 ' ~ ~4T =T5 sin^7(6 + Vo) = 114 + °° V° ;
viz. this last identity depends on
ff (3 + V5) (6 + V5) = (114 + 50 v/5) sin4 7)
that is,
160 (3 + V5) (6 + V5) = (114 + 50 V5U120 - 40 Vo),
or
2 (3 + V5) (6 + V5) = (57 + 25 V5) (3 - V5),
or finally
(7 + 3V5)(6 + V5)= 57 + 25V-5,
which is right.
Similarly
l'-*-r* = 114-50^5,
and observing that the sum and product of 114 + 50^-5, 114 - 50 x/5 are =228 and
496 respectively, the required function of s is
(Sio -i)2_228 (s15 -s5) + 496s10,
= s20 - 228s15 + 494s10 + 228s5 + 1,
which is the required value of /to.
* a is the a, 7 is the 7, and 7' the a-/3 of the Table, No. 99.
745] AND THE POLYHEDRAL FUNCTIONS. 187
Invariantive property of the Stereographic Projection. Art. Nos. 87 to 93.
87. The before-mentioned theorem that the functions derived from two different
stereographic projections of the same point are linear transformations one of the other,
may be thus stated :
Considering on the surface of a sphere, two fixed points A and B; and determining
the position of a point C, first in regard to A by its distance 0 and azimuth / and
x
next in regard to B by its distance & and azimuth /', the azimuths from the great
circle ABx which joins the two points A and B, then we have
tan \Q (cos / + i sin / ), and tan \& (cos /' + i sin /'),
homographic functions one of the other : calling them s, s', and putting the distance
AB=c, the relation between them in fact is
. s — tan ic
o — - _ *L
1 +stan |c '
or, what is the same thing,
tan£c(l + ss) = s — s' ;
or, observing that
ss' = tan \Q tan \& (cos (/+/') + i sin (/+/')},
we have the two equations
tan |- c [1 + tan |0 tan %0' cos (/+/')} = tan \Q cos/ - tan £#' cos/',
tan £c { tan \Q tan \& sin (/+/')} = tan \6 sin/— tan \& sin/'.
88. If we denote the angles of the spherical triangle by C, A, B, and the
opposite sides by c (as before), a, b, then 6, 0' = b, a; /, f' = A, ir — B, whence
s, s' = tari | b (cos A + i sin A), — tan |a (cos B — i sin B) :
or we have between the sides a, b, c and angles A, B of a spherical triangle the
relations
tan £c {1 — tan |a tan |6 cos (A — B)} = tan ^b cos A + tan \a cos B,
tan \c { — tan \a tan ^b sin (A — B}\ = tan ^b sin A — tan |a sin B;
24—2
188
ON THE SCHWARZIAN DERIVATIVE
[745
equations which may be verified by means of the ordinary formulae of Spherical
Trigonometry.
89. But it is interesting to give the proof with rectangular coordinates.
Taking (X, Y, Z), (Xlt Y1} ZJ for the coordinates, referred to two different sets
of axes, of a point on the spherical surface: also x, y, xl, yl for the coordinates of
the corresponding stereographic projections, we have
(xl} rlv4)-(«, £, 7
', 13', y
X : Y : Z : 1 = 2aj : 2y : 1 - a? - f : 1+ a? + f ,
X, : Y, : Z, : 1 = 2^ : 2yx : l-a;*-y* : l+tf + yf,
and thence
a?j : yl : 1 = 2or + 2/3y +7 (1 — a? — y2)
: 2afa; + 2j3'y + 7' (l-x2- y2)
: 1 + x2 + f- + 2a"x + 2@"y + 7" (1 - a? - f).
/y» a I f** ni
90. Introducing z, z^ for homogeneity, or writing -, - and — , — in place of
Z Z> — j Z \
x, y and xlt ylf respectively, we have
xl = 2ax + 2fiy +7 (z- - y? - y2), = ( - 7 , - 7 , 7 , £ , a , 0$«, y, 0)-,
/(^-^-.V2), =( ~7> -7, y,ff,*,01i „ )2,
- y
+ 7",
and thence without difficulty
* = --77 {(1 + 7")
7 + 7"
{(1 + 7")
K! - 7")
iy}},
ty)J,
^ _ iy, = _ /2 {(1 - 7") ^ - (a" + iff') (x - iy)} {(1 + 7") * + < a" - iff') (x + iy)},
viz. the form is zl : xl + iyt : x1 — iy1 = MN : NL : LM (L, M, N linear functions of
z, x 4- iy, x — iy) : showing that the relation between two stereographic projections of
the same spherical figure is in fact that of a quadric transformation, the fundamental
points in each figure being an arbitrary point and the two circular points at infinity:
or, what is the same thing, to any line in the one figure there corresponds a circle
in the other figure, which is the " circular relation " of Mobius.
91. The actual values are
xi + iyi = 1 + j" (1 - y') z - (a" - iff') (x + iy)
z, 7 + 7 » ' (1 + 7") * + («" - »£") (x + *y) '
xi - iyi = !+ 7" (1 - y") z - (a" + iff') (x - iy)
Zi 7 - 7ft ' (1 + 7") z + («" + iff) (x - iy) '
74 5 J AND THE POLYHEDRAL FUNCTIONS. 189
np \ *i 7/
viz. attending only to the former of these, we have - — — a homographic function of
z\
-— , which is the before-mentioned theorem.
z
92. Supposing that the transformation from (X, Y, Z) to (Xlt Yl} Z^) is made by
a rotation, the coordinates of which are \, ji, v: that is, if /, g, h are the inclinations
of the resultant axis to the axes of x, y, z respectively, and 6 the angle of rotation,
putting X, /j,, v = t3ui^6cosf, tan|0cos#, tan |# cos h: then the coefficients of trans
formation are
«', P, 7
«", 0", 7"
2 (i/X + //,) , 2 (/«/ - X) , 1 - X2 - p? + v-
Substituting these values, the formulae become, after an easy reduction,
i] (x + iy} + (X, + ip) z
Zi (X - t» (a; + iy) + (v-i)z '
®\ — ij/i _ — (v— i) (x — iy) + (X — ip) z
z^ (\ + i/j,) (x - iy) +(v + i)z '
attending to the former of these, and writing for greater simplicity
xl + iy-, x + iy
'
respectively, we have
or writing this
_ — (v + i) s + (\ + if
then A : B : C : D = — v — i
v — .
93. I call to mind that the condition, in order that the homographic transformation
s1 = (As + B) -f- (Cs + D) may be periodic of the order n, is
(A + D)- - 4 (AD - BC) cos2 — = 0,
in being an integer different from zero and prime to n. In particular, when n = 2, it
is ^1+D = 0: w = 3, it is A*+ AD + D- + BC = 0 : n = 4, it is A2 + D2 + 2BG= 0 : and
n = 5, it is (A + D)* - i (3 ± V5) (AD - BC) = 0.
Groups of homographic transformations. Art. Nos. 94 and 95.
94. The formulae just obtained serve to connect the theory of the rotations of
a polyhedron with that of the homographic transformations s into (As + B) + (Cs + D) :
and, corresponding to the rotations which leave the polyhedron unaltered, we have
groups of homographic transformations. We have thus, corresponding to the cases of
the tetrahedron, the cube and the octahedron, and the dodecahedron and icosahedron
respectively, groups of 12, of 24, and of 60 homographic transformations s into
190 ON THE SCHWARZIAN DERIVATIVE [745
(As + B) + (Cs + D). The group of 60 and the group of 24 include each of them as
part of itself the group of 12 : it is further to be remarked that the group of 12
may be regarded as that of the positive substitutions upon four letters abed, the
group of 24 as that of all the substitutions upon the four letters, and the group of
60 as that of the positive substitutions upon five letters abode.
95. I call to mind that a group of functional symbols 1, a, /3, ... can always
be expressed in the equivalent form 1, ^aS-"1, ^/S^"1, ... where ^ is any functional
symbol whatever : clearly, a, /3, . . . being homographic transformations, then, S- being
any homographic transformation whatever, the new symbols ^a^-"1, S-/3S-"1, ... will also
be homographic transformations ; and thus the group of homographic transformations
can be expressed in various equivalent forms : these correspond to the different
positions of the polyhedron in regard to the axes of coordinates: and there are in
fact three cases which it is proper to consider, viz. attending for the moment to the
dodecahedron, we may have the axis of z passing through the midpoint of a side,
through the centre of a face, or through a summit; that is, in the language
presently explained, the cases are 1°, Pole at a point © ; 2°, Pole at a point A ;
3°, Pole at a point B.
The regular Polyhedra. Art. Nos. 96 to 103.
96. We require a theory of the regular Polyhedra considered as systems of points
on a sphere. I refer to my two papers [375] and [679]. In the latter paper, I
remark that, considering the five regular figures drawn in proper relation to each
other on the same spherical surface, the only points which have to be considered are
12 points A, 20 points B, 30 points @, and 60 points 3>. Describing these by
reference to the dodecahedron, the points A are the centres of the faces, the points
B are the summits, the points © are the midpoints of the sides, and the points 3>
are the midpoints of the diagonals of the faces. Or describing them by reference to
the icosahedron, the points A are the summits, the points B are the centres of the
faces, the points © are the midpoints of the sides: viz. each point © is the common
midpoint of a side of the dodecahedron and a side of the icosahedron, which there
intersect at right angles: and the points 4> are points lying by threes on the faces
of the icosahedron, each point <& of the face being given as the intersection of a
perpendicular AS of the face by a line BB joining the centres of two adjacent
faces and which intersects A® at right angles.
97. The points 3> are comparatively unimportant, and it is proper in the first
instance to attend only to the 12 points A, the 20 points B, and the 30 points 0:
these form 6 pairs of opposite points A, 10 pairs of opposite points B, and 15 pairs
of opposite points ©. Considering the diameters through each pair of opposite points
©, we have thus a system of 15 axes, which in fact form 5 sets each of 3 rect
angular axes: attending to any one of such sets, the diametral plane at right angles
to one of the three axes contains of course the other two axes : it contains also
two axes each through a pair of opposite points A, and two axes each through a
pair of opposite points B. If instead of the plane we consider its intersection with
the sphere, we have thus on the sphere 15 circles each containing 4 points ©,
745]
AND THE POLYHEDRAL FUNCTIONS.
191
4 points A and 4 points B. The fifteen circles intersect by fives in the pairs of
opposite points A, by threes in the pairs of opposite points B, and by twos in the
pairs of opposite points © ; the mutual inclinations of successive circles at the points
A, B, © being =36°, 60° and 90° respectively. The whole number 15.14, =210, of
the intersections of the circles two and two together is thus made up of the 12
points A each counting 10 times, the 20 points B each counting 3 times, and the
30 points 0 each counting once ; 210 = 120 + 60 + 30.
98. The angular magnitudes which present themselves are all obtained from
the dodecahedral pentagon, as shown in the annexed figure, in which the angle
subtended by a side at the centre is = 72°, and the angle between two adjacent
sides is =120°.
We write A® = a, B®=@, AB = j, B,B^ = x, ^B1B4B = 0, ®B4=g, Z
From the triangle A®B, the angles of which are 36°, 90°, 60° and the opposite
sides ft, 7, a, we find the values of a., ft, 7, and these are such that a + ft + 7 = £TT.
From the triangle B4BB^, where the sides B4B, BBL, and the included angle are
2/3, 2/3, 120°, we have the opposite side x, and the other two angles each = 0.
From the triangle B4BS, where the sides B4B, B®, and the included angle are
2/3, ft, 120°, we find the opposite side g, the angle BB4®, = <£, and the angle
B4%B, =45°.
Hence each of the angles B4SB, Ba®Blt being =45°, the angle Bt@Ba is =90°:
in this triangle the hypothenuse B2B4 is =x, and each of the other two sides is
= g: whence we have cos x = cos2 g, as is in fact the case, and moreover the values
give x + 2^ = 180°. Also each of the other angles is found to be =60°; that is, we
have Z B2BJ& = 60°, or the whole angle at B4 being =120°, the sum of the remaining
angles B3B4Bn_ and BB4® is =60C: that is, 0 + 0 = 60°.
From the triangle ©J^©' where the two sides and the included angle are
/3, ft, 120°, we find ©©' = 36°.
192
ON THE SCHWARZIAN DERIVATIVE
[745
And from the triangle ©54@", where the two sides and the included angle are
g, g and (120° - 2<£=)20, we find 08" = 60°.
99. We thus arrive at the following Table:
where as above
« + £ + 7 = 90°,
x + 2g = 180°,
6+<f> =60°.
cos
01 o JO'
/5-V5
/5 H- ^5
A®
a
0
ol 4-o
OA° KK'
V 10
,/»-!
V 10
V5+1
_ofe)
P
2V3
2^/3
A n
owo 9.-y
/ 10 -2^/5
/5 + 2V5
(BB)
(£&)
D B D
y
X
9
70* 32'
54 44
°.7° <lfi'
V 15
2^/2
3
5/2
V3
x/3
V 15
1
3
1
T3
J*
JiJtJb
T)(3i T)
V
92 14
2^2
-s/3 (V5 - 1)
1J2
V5 + 3
9
2a
2/8
9,/
63 26
41 50
74 44
4^/2
2
%/»
2
3
2(^/5 + 1)
4x/2
1
N/5
^5
3
4-^/5
Jy
3^5
3^/5
n R
/5-2V5
/10 + 2^/S
a p
V 15
V 15
18°
V5-1
/5 + N/5
4
V 8
0®
36°
/5-V5
^5 + 1
V 8
4
100. We now construct three figures of the points A, B, © ; viz. these are
stereographic projections, each showing the Northern hemisphere projected on the plane
of the equator by lines drawn to the South Pole: hence, for any pair of opposite
points not on the equator, only the point in the Northern hemisphere is shown:
but for a pair of opposite points on the equator the two points are each of them
shown. In fig. 1 the North Pole is taken to be a point © ; in fig. 2 it is a point
A ; and in fig. 3 it is a point B. The position of any point on the sphere is
determined by its N.P.D. and its longitude, measured from an arbitrary origin,
say from the point E of the centre left-handedly : then, in the three figures, the
positions are as follows.
194 ON THE SCHWARZIAN DERIVATIVE
102. Fig. 2. Pole at A.
e
N.P.D.'s
Longitudes.
A
0
—
5A
2a = 63° 26'
0° 72° 144° 216° 288°
5A
180° - 2a = 116 34
36 108 180 252 324
A
180
—
5B
y= 37 22
36 108 180 252 324
5£
90° - a + ft = 79 12
36 108 180 252 324
5B
90 + a - /? = 100 48
0 72 144 216 288
bB
180 - y = 142 38
0 72 144 216 288
5®
o= 31 43
0 72 144 216 288
5®
90° - a = 58 17
36 108 180 252 324
10©
90
(36 108 180 252 324) + 18°
5®
90 + a = 121 43
0 72 144 216 288
5©
180 -a= 144 17
36 108 180 252 324
745]
AND THE POLYHEDRAL FUNCTIONS.
195
103. Fig. 3. Pole at B.
KP.D.'s
Longitudes.
3A
y= 37° 22'
30° 150° 270°
3A
90°- a + £= 79 12
90 210 330
3A
90 + a - /3 = 100 48
30 150 270
3A
180 - y = 142 38
90 210 330
B
0
__
3B
2/3- 41 50
90 210 330
6B
x= 70 32
(30 150 270) + »
= 37° 46'
QB
180°- x= 109 28
(90 210 330) + &
= 37 46
3B
180 -2)8= 138 10
30 150 270
B
180
—
3®
/8 = 20 55
90 210 330
6®
g= 54 44
(90 210 330) + <^>
- 22° 14'
3®
90°-£ = 69 5
30 150 270
6®
90
0 60 120 180°
240° 300°
3®
90 + /?= 110 55
90 210 330
6®
180 -g = 125 16
(30 150 270) + <f>
= 22° 14'
3®
180 -0= 159 5
30 150 270
25—2
196 ON THE SCHWARZIAN DERIVATIVE [745
The groups of homographic transformations, resumed. Art. Nos. 104 to 117.
104. The axes of rotation for the dodecahedron and the icosahedron are 15 axes
each through a pair of opposite points ©, 6 axes each through a pair of opposite
points A, and 10 axes each through a pair of opposite points B\ or say 15 ©-axes,
10 jB-axes and 6 .4-axes : the corresponding angles of rotation are 180°, 72° and 120° ;
so that (excluding in each case the original position or that of a rotation 0) we have
in respect of each ©-axis 1 position, in respect of each A-axis 4 positions, and in
respect of each 5-axis 2 positions; in all, including the original position,
1 + 15 + (6 x 4) + (10 x 2), = 60 positions,
that is, a group of 60 rotations.
To find, in any one of the three forms, the group of homographic transformations,
we can in each case obtain from the foregoing tables the values cos/, cosg, cosh of
the cosine-inclination of an axis of rotation to the axes of coordinates, and thence
calculate the values of
X, fj,, v = tan ^S-cos/, tan |S- cos g, tan |S- cos h,
and thence the values of
A, B, C, D = — v — i,
viz. in the case of a ©-axis, S- is = 180°, (so that here tan £S- = oo , or the values of
A, B, C, D are = — v, \+ifji, \-ifju, v, that is, —cos h, cosf+icosg, cosf—icosg, cos A);
in the case of a JB-axis, the values are ^ = 120°, 240°, and therefore tan^= + \/3;
and in the case of an J.-axis, they are S- = 72°, 144°, 216°, 288°, and therefore
V5 - 1 V° + 1
105. The ©-form was first given in my paper of 1879, but in obtaining it I
used results given in the paper of 1877. As regards the identification with the
substitution-symbols, since there is nothing to distinguish inter se the letters a, b, c, d, e,
any transformation A, B, C, D of the fifth order might have been taken for abode,
but No. 37 of the group having been taken for this substitution abcde, I do not
recall in what manner I found that, consistently herewith, the transformation No. 2
(— 1, 0, 0, 1, that is, s into -s) of the second order could be taken for ab.cd. But
there is no sub-group of an order divisible by 5 ; and hence, these two transformations
being identified with the two substitutions, the other transformations correspond each
of them to a determinate substitution.
745]
AND THE POLYHEDRAL FUNCTIONS.
197
106. Homographic Transformations. The group of 60. Pole at ®.
(Ax + B) •*• (Cx +D)
2
3
4
5
6
7
8
9
10
11
12
18
14 !
15 i
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
-1
0
0
2
2
2
2
2
2
2
2
-i
-1
1
i
1
-1
3 + v/S)
2
2
2
2
2
2
2
2
-2
afc . cd
ac . bd
ad . be
be . de
ae . be
ad . ce
ad . be
ae . cd
ab . de
be . cd
ab . ce
ac . be
bd .ce
ae . bd
ac . de
abc
acb
adc
acd
adb
abd
bed
bdc
aec
ace.
bed
bde
bee
bee
aed
ode
cde
ced
aeb
abe
abcde
acebd
adbec
aedcb
adceb
acbde
aedbc
abecd
acbed
198
ON THE SCHWARZIAN DERIVATIVE
[745
46
47
48
49
50
51
52
n
54
M
M
57
68
59
00
l-x/5)
l-x/5)
5)
3-^/5 + ^ l-x/5)
-1
2
2
2
2
2
2
2
2
abdce
aecdb
adebc
aecbd
acdeb
abedc
adbce
aebdc
abced
adecb
acdbe
abdec
adcbe
aebcd
acedb
107. Taking out of the foregoing group of 60 a group of 12 contained in it,
viz. that corresponding to the positive substitutions of the four letters abed, it is
easy to see, that there is a transformation (i, 0, 0, 1), that is, s into is, which can
be taken for the substitution adbc, and also to complete thence the group of 24. And
we have thus the following Table.
Groups of 12 and 24. Pole at @.
(Ax
1
1
0
0
1
2
— 1
0
0
1
3
0
1
1
0
4
0
1
-1
0
5
-i
i
1
1
6
- 1
i
1
i
7
-i
1
i
8
— i
-i
1
-1
9
i
i
1
-1
10
1
i
1
—i
11
-1
-i
1
-i
12
i
-i
1
1
13
i
0
0
1
14
— i
0
0
1
15
0
i
1
0
16
0
i
-1
0
17
1
-1
1
1
18
— /
_ i
1
i
19
?'
1
1
i
20
1
1
1
— 1
21
_ j
-1
1
-1
22
i
-1
1
-i
23
-i
1
1
-i
24
-1
1
1
1
1
ab . cd
ac . bd
ad . be
adbc
acbd
cd
ab
acdb
bd
abed
be
abdc
ac
a deb
ad
745] AND THE POLYHEDRAL FUNCTIONS. 199
108. The group of 60 was obtained in the J.-forrn by Gordan in his paper.
The passage from the ©-form to the J.-form is made as follows: let X, Y, Z be
the coordinates of a point when the axes are as in the ©-form, Xl} Y1} Z, the
coordinates of the same point when the axes are as in the A-form : we may write
X, Y, Z^bX.-aZ, : Y, : a
where
/
' V
5-V5 5 + V5
then, if the equations of an axis of rotation referred to the first set of coordinates
are X : Y : Z = L : M : N, those of the same axis referred to the second set of
coordinates are
bX. + aZ, : Y, : -aX^bZ^L : M : N;
or taking these to be
X, : 7, : Z, = Lt : M, : N,,
we may write
Llt Mlt N^bL + siN, M, -aZ
these values are such that
I,2 + Mf + NI- = L2 + M2 + N2,
and hence, \, /A, v and \1} fj,l, ^ being the rotations, we may write
L, M, N=$\, v, *v, LI, Mlt ^=^\15 Vi, X;
where S- has the same value in each set of equations. From the equations
A : B : C : D = — v — i : \+i/j, : \ — i/j, : v — i,
we have
B + C : B-C : D-A : D + A =\ : ip : v : - i
= L : iM : N : - fb
and similarly
B. + C, : B.-C, : D,- A, : D. + A^L, : iM, : N, : -
Hence we may write
#+<?!= \>(B + C) + a,(D-A),
B1-Cl= B- C,
or say,
A , =
which are the values for a transformation (A,, B1} Clt A) in the ^-form: of course,
as only the ratios are material, the values may be multiplied by any common factor.
200
ON THE SCHWARZIAN DERIVATIVE
[745
109. The results are exhibited in terms of e, an imaginary fifth root of unity :
taking e = cos 72° + i sin 72°, we have
5 + V5
where the upper signs belong to e, e° and the lower to e4, e3. It may be remarked
that
1_ /5 + V5 1_ /5 - V5 b V5 + 1 a V5 - 1
a~V~ 2 ' b~V~ 2 ' a~ 2 ' b~ 2
For instance, we have in the ©-group (A, B, G, D) = (— 1, 0, 0, 1); ab.cd: and thence
in the A -group Ait B1} C1} D1 = (-2b, 2a, 2a, 2b); ab.cd: or say this is
-1, |, £, l), =(-1, e + e4, e + e4, 1);
which in the Table is given as (— e3, e2 + e4, e2 + e4, e3) ; ab . cd.
By effecting the passage to the A -group in this manner, we of course obtain the
proper substitution corresponding to each transformation : but I found it easier starting
from two transformations and the corresponding substitutions, to obtain thence by
successive compositions the entire group.
110. Homographic Transformations. The group of 60. Pole at A.
0No. (As +B) ~(Cs +D)
1
1
1
1
1
2
4
0
-1
1
0
ad . be
3
13
0
-e4
1
0
ac . be
4
9
0
-e3
1
0
ae . cd
5
10
0
-e2
1
0
ab . de
6
14
0
-e
1
0
bd . ce
7
6
e + e2
e4
1
-(e+e3)
ae . be
8
5
e+e3
1
e4
-(e+e3)
be . de
9
16
e + e3
e
e3
-(e + e3)
ac . de
10
3
e + e3
e2
e2
-(e + e3)
ac . bd
11
15
e + e:s
r5
e
-(e + e3)
ae . bd
12
12
-1
e + e3
e2 + e4
1
ab . ce
13
11
— e
e3+l
e2 + e4
e
be . cd
14
7
-£2
1+e2
e2 + e4
e2
ad . ce
15
2
-e3
e2+e4
e2 + e4
e3
ab . cd
16
8
-e4
e4+e
e^ + e4
e4
ad . be
745]
AND THE POLYHEDEAL FUNCTIONS.
201
17
21
6»+l
e
1
-(e + e3)
adb
18
35
ea+1
e2
e4
-(e + e3)
aeb
19
30
e3 + l
e3
e3
-(e + e3)
bee
20
34
e^+l
e4
e2
-(e + e3)
ced
21
19
€» + !
1
e
-(e + e3)
adc
22
33
e + e4
£2
1
-(e + e3)
cde
23
20
e + e4
e3
e4
-(e + e3)
acd
24
22
e + e4
e4
e3
-(e + e3)
abd
25
36
e + e4
1
e2
-(e + e3)
abe
26
29
e + e4
e
e
-(e + e3)
bee
27
31
— e
e2 + e4
e2 + e4
1
aed
28
17
-e2
e4 + e
e2 + e4
e
abc
29
27
-e3
e+e3
e2 + e4
e2
bed
30
25
-e4
es+l
e2+e4
e3
aec
31
23
-1
1+e2
e2 + e4
e4
bed
32
24
-e4
1 +€2
e2+e4
1
bdc
33
32
-1
e2 + e4
e2 + e4
e
ode
34
18
-e
e4 + e
e2 + e4
e2
acb
35
28
-e2
e+e3
e2 + e4
e3
bde
36
26
-e3
eO+1
e2+e4
e4
ace
37
44
e
0
0
1
abecd
38
43
e2
0
0
1
aedbc
39
42
e3
0
0
1
acbde
40
41
e4
0
0
1
adceb
41
38
e2 + e4
1
1
-(e + e3)
acebd
42
46
e2 + e4
e
e4
-(e + e3)
abdce
43
58
e2 + e4
e2
e3
-(e+e3)
adcbe
44
55
e2 + e4
e3
e2
-(e + e3)
adecb
45
50
e2+e4
e4
e
-(e + e3)
acdeb
46
51
1+e2
e3
1
-(e + e3)
abedc
47
39
1+e2
e4
e4
-(e+e3)
adbec
48
47
1+e2
1
e3
-(e + e3)
aecdb
49
59
1+e2
e
e2
-(e + e3)
aebcd
50
54
1+e2
e2
e
-(e + e3)
abced
51
56
-e2
e3 + l
e2 + e4
1
acdbe
52
49
- e3
1 +e2
e2 + e4
e
aecbd
53
37
-e4
e2 + e4
e2 + e4
e2
abcde
54
45
-1
e4 + e
e2+e4
e3
acbed
55
57
— e
e+e3
e2 + e4
e4
abdec
56
48
-e3
e4 + e
e2+e*
1
adebc
57
60
-e4
e+e3
e2 + e4
e
acedb
58
53
_ j
e3+l
e2 + e4
e2
aebdc
59
52
— e
1+e2
e2 + e4
e3
adbce
60
40
-e2
e2 + e4
e2 + e4
e4
aedcb
C. XI.
26
202
ON THE SCHWARZIAN DERIVATIVE
[745
111. Selecting the transformations which correspond to the positive substitutions
abed, and completing the group of 24 we have
Homographic Transformations. The groups of 12 and 24. Pole at A.
(As +B) -HC* +D)
1
1
0
0
1
1
2
0
-1
1
0
ad . be
|
e + e3
e'-
e2
-(e + e3)
ac . bd
4
-e3
e2 + e4
e2 + e4
e3
ab . cd
5
_e2
e + e*
e2 + e4
e
abc
fi
— 6
e+e4
e2+e4
e2
acb
7
e+e4
r!
e4
-(e + e3)
'/'•li
8
e3 + l
1
e
-(e + e3)
adc
9
e+e4
e4
e3
-(e + e3)
aid
10
l»+l
e
1
-(e + e3)
adb
11
_ i
1 + e2
e2 + e4
e4
bed
12
-e4
1 + e2
e2 + e4
1
bdc
11
1
l + 2e4
l + 2e
-1
ab
14
- e2 + e3
1 + e + Se4
-1-Se-e4
62_e3
cd
15
e2-e4
3 + e + e3
-l-Se-e3
-e2 + e4
ac
16
-1+e2
-l-e2 + 2e4
l + e2-2e3
1-e2
bd
17
2+e3 + 2e4
- 2 - 2e2 - e3
2e + e3 + 2e4
2e + 2e2 + e3
ad
18
2+2e2 + e3
2 + e3 + 2e4
-2e-2e2-e3
2e + e3 + 2e4
be
19
-2 + e+e3
-e + e3
-e + e3
e + e3-2e4
abed
20
1
-1
1
1
abdc
21
1
1
-1
1
acdb
22
l + e + 3e4
e2-e3
e2-e3
l + 3e + e«
acbd
23
l + 2e4
-1
-1
-l-2e
adbc
24
3 + e + e3
- e2 + e4
-e2+e4
l + 3e + e3
adcb
As an example of the calculation we have (A, B, C, D) = (0, i, — 1, 0); ab. Hence
A,, Blt Glt A=
a a
The second and third coefficients are
1 . /
— *V
5 +
which, in virtue of the values of e and e4, are = 1 + 2e4 and 1 + 2e respectively: or
the result is as above (1, 1 + 2e4, 1 + 2e, —1).
745] AND THE POLYHEDRAL FUNCTIONS. 203
112. In like manner for the passage from the ©-form to the .#-form, if X, Y, Z
be the coordinates of a point on the spherical surface in regard to the ©-axes,
X2, Y.,, Z» those of the same point in regard to the .B-axes, we may write
X : Y : Z=X,
where
2V3 ' 2V3 '
Hence X : Y : Z= L : M : N, being the equations of an axis of rotation in the
first set of coordinates, those of the same axis in the second set of coordinates
will be
J\. o : b JL a -|- 8i£/» : — a -i o i D£» == x/ '. J.VI : JM ,
or calling these
X, : Y., : Z, = L2 : M^ : N.,,
we have
L,, M2, N, = L : })M-aN : aj
these values are such that
ia2 + M,2 + N./ = L- + M- + N2,
or X, yu,, v, X2, fa, v-2 being the rotations, we have
L, M, N=^\, V, *v\ L2, M.2, N, = ^\,
where ^- has the same value in the two sets of equations. We have thus
B +C : B -C : D -A : D + A = L : 2M : N : -?$,
B2 + C, : B, - C, : D, - A, : D, + A, = Z2 : 2Jf2 : N, : - ?$,
and hence
B, + C, = B + C,
B,-C,= \)(B-C)-ai(D-A),
D. -A, = - ai (B-C) + b (D- A),
D2 + A,= D + A-
and thence
At= Sii(B-C)-}) (D-A)
B,= b B-C-MD-A
(D - A) + (D + A}.
113. As an example of the transformation, take
(A, B, C, D)= 2, -3 + V5 + i(l-V5), -3 + V5 + i(-l+V5), -2 [bc.de]:
then
B-C, B+C, D-A, D + A=i(\-*Jo\ -3 + V5, -2, 0;
26 — 2
204
and thence
ON THE SCHWARZIAN DERIVATIVE
1
2TV3
1
1
23
v/5) + (-
viz. multiplying by 2 \/3, these are
8" / /^ [ rt / X \ i_ O / O / O i^ / "* \ si ( £\ 9 / P\ \ i O / Q / __
j v \ ^™" O "i ^ ^ O ) j~ *- M O I ~"~ O *i \f OK Z I O ^^ ^ ^ O ) "T* ^ Y ^ V "™"
that is,
or since
2 -f V3 = - 2i&> and - 2 + ^'3 — 2ia)-,
dividing by 4 these are
2, i (3 - <\/5) <y, i (- 3 + V5) <w2, - 2,
as in the table.
114. Homographic Transformations. The group of 60. Pole at B.
-8,
1
1
0
0
1
1
2
0
1
1
0
ac . bd
3
0
u
1
0
ae . bd
4
0
u-
1
0
bd . ce
5
2
i( 3-x/5)
i( -3 + ^5)
-2
ab .cd
6
2
i(-S-^5)
i ( 3 + ^5)
-2
ad . be
7
2
*( 3-^/5)0,
i( -3 + ^/5)0?
-2
be .de
8
2
•t ( -- 3 - N/5) w
i( 3+^5)^
-2
be . cd
9
2
i( 3-N/5)w2
i( _3+x/5)w
-2
ad . be
10
2
i(-3-N/5)w2
i( 3+^/5)0,
-2
ab . de
11
2
(-JB-iJS)*
(-v/S + zVS)^
-2
ab .ce
12
2
-v/3-ix/5
-V3 + W5
-2
ac . be
13
2
(_x/3_,t-x/5)w2
(.^3 + ; ^5)0,
-2
ae . be
14
2
s/8-»V6
x/3 + ^5
-2
ac . de
15
2
( V3-iV5)w
( x/S + 'V^Jw2
-2
ad . ce
16
2
( \/3 — i ^/5) w2
( ^3 + U/5)W
-2
ae .cd
745]
AND THE POLYHEDRAL FUNCTIONS.
205
17
W
0
0
1
ace
18
w2
0
0
1
aec
19
x/3-iV5
2
-2
V3 + W5
bed
20
— ^/3 — i >/5
2
-2
-^3 + 1^/5
bde
21
-x/3-i^S
2w2
-S«
-v/3 + iV5
bdc
22
v/S-i^S
2w2
-2W
/o i ,• /r
V "T * v«
bed
23
-V3-iV5
2w
- 2w2
-V3 + W5
abd
24
^3-iVS
2 co
- 2w2
V3 + W5
adb
25
2w2
-x/3-iV5
-V3 + W5
-2w
abc
26
2w
-V3-W5
- x/3 + 1 v'5
-2w2
acb
27
2o>2
-X/3-W5
(-v/S + 'VS)^
-2
abe
28
2
-v/3-W5
(-v/S + i^w2
-2w2
aeb
29
2w
X/3-W5
\ ** ~T ^ 'V
-2w2
acd
30
2w2
x/3-tVS
X/ *J ~T 1 *V
-2w
adc
31
2w2
X/3-V5
( v/o~f~z ^/o) w
-2
ade
32
2
v/3 - / x/5
( v/3 + jV5)"2
-2w2
aed
33
2
-v/3-i^S
(-v/3 + iV5)w
-2w
bee
34
2w
/D ,' /-
~ V " ~ * V "
(-N/3 + tV5)w
-2
bee
35
SM
V3-V5
( V3 + W5)»
-2
cde
36
2
J3-IJ5
( ^/a+tvsjo,
-2o,
ced
37
2
i( 3 - ^5) w2
i(-3 + v/5)
-2W2
adceb
38
-V3-W5
+ 2w2
-2
(-V3 + W5)w2
acbde
39
V3-W5
2
-8w
( v/3 + W5)w
aedbc
40
2
i( 3-^/5)
i(-3 + x/5)w
-iH
abecd
41
2
i( 3-V5)w
<<-8+^/6)
-2w
aedcb
42
""* /v ^ \>
2w
-2
(-V3 + W5)"
adbec
43
v^o — 1> w &
2
- 2w2
( VS + i^w2
acebd
44
2
i( 3-V5)
i (-3+^5) co2
-2w2
abcde
45
2
* ( 3 - v/5) w2
^(-S + Vo)^2
-9w
adebc
46
<V — Z (^<^
2w2
-2w2
( V3 + »V6)w
aecdb
47
~~ M & — ^ /^ "
2w
-2w
(-v/3 + zV5)w2
abdce
48
2
i ( 3 - ^5) w
i(-3 + x/5)w
— 2 co2
acbed
49
2
i(-B-^«
t( 3 + V5)o;
-2w2
acdeb
50
jy O — 1 ^^
2w
-2w
( >/3 + i ^/o) w2
adbce
51
— v ~ i v
2w2
-2w-
(_V3 + i\/5)w
aecbd
52
2
i ( - 3 - J5) ur
i( 3 + x/5)W-'
-2w
abedc
53
2
i(- 3—^/5) w
M 3 + v/5)
-2w
aebcd
54
-x/3-W5
2w
-2
(-v/S + iVS)"
abdec
55
x/3-^5
2
-2or
( VS + WS)"2
acedb
56
2
t ( - 3 - ^5)
i( 3 + V5)o;2
-2w2
adcbe
57
2
f(-8-Vfi)
i( 3 + ^/5)0;
-2o>
adecb
58
-s/3-i^/S
2
-2w
(-x/S + i^w
aebdc
59
v/S-i^S
2w2
— 2
( ^3 + W5)w2
acdbe
60
2
i ( - 3 - ^5) w2
i( 3 + ^/5)
-y
abeed
206 ON THE SCHWARZIAN DERIVATIVE
115. We hence derive
Homographic Transformations. The groups of 12 and 24. Pole at B.
(An +B) -=-(<?* +D)
[745
l
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
0
2
2w
2
2
1
2
2
1
1
N/3(
3-^/5)
- x/3 - i ,
2w
/3 - i
2w2
2W2
v/3(
(-3-J5)
3-v/5)
3-^/5)
1
i(
-2w2
-2
3 + ^/5)
i
•iJ5
1
2
0
2
2w
-2W2
-2
-2
-1
-2
- 2
1
1
N/3(
v/3(
1
at
'1C
nd
<ibc
tld)
ula
adl
acd
adc
bed
Me
ab
cd
ac
bd
ad
be
ale
ndi
a be
neb
acd
adl
116. I give also the group of 12, (abce), slightly modifying the form: viz. I
write first ^3 + i^/o=2^2k, and therefore V3 - i V5 = 2 V2. ^ : then for a; I write X^r,
K
and divide the A and .8 by X: the A and 5 then contain -, and the G and D
\
X & X
contain p and assuming - = t, we have ^ = - i. For instance, in the transformation
corresponding to abc, the Ax + B and Cx + D,
and - 3+ 2 5 as - 2o>
become first 2&>2# - 2 V2&, and - 2 V2 T « - 2o>, and then (omitting also the factor 2)
Z" "X 7
a>2# - V2 - and - V2 -^ a- - to, viz. when - = i, they are w-a; - i *J2 and i» . i \/2 - &> ; that
is, the values of A, B, C, D are &>-', -z'V2, zV2, - w. The group is
745]
AND THE POLYHEDRAL FUNCTIONS.
Group of 12. Pole at B.
207
1
0
0
1
1
M
0
0
1
ace
or
0
0
1
aec
1
-iuJ2
iuj2
— or
dbc
1
-f«V2
t«V2
— 0)
acb
1
-t-w^/2
/v/2
- w
abe
1
-i*J2
i w2 V2
- ur
aeb
1
-t«V2
z'V2
— or
bee
1
-iV2
JWv/2
- w
bee
1
- iw V2
f«* v/2
-1
ab . ce
1
- tw2 ^/2
?wV2
-1
ae . be
1
-iv/2
W2
-1
ac . be
117. From the Table of the Groups of 12 and 24, ©-form, it appears that the
group of 12 is
X' x' X' x' x + l ' x + l ' x-l '
— i —(x + i) —(x — i)
x—i' ,v + i ' # — t a; + t
and if we proceed to form the product of the twelve factors s — x, s -- , s + x, &c.,
vC
we have first the three products
1 (x-l\- /a?+l\= /a? + A3 /a?-iV
s2 - ^ . s- - - ; s- + - — •= . s- + -- ; s- - -- :}.«•—{ - .
a;2' \« + l/ w — I/ \x-iJ \x + i)
= s4 + as-' + 1 ; s4 -I- /3s2 + 1 ; s4 + ys* + 1 ;
if for shortness
/ 1 \ ar« + 6^ + 1 _ a? - Gtf + 1
* A 7- -•*. 2 ' 2 2 •
The product of the three quartic functions is
= (s4 + 1)3 + (s4 + 1)2 s2 (a + yS + 7) + (s4 + 1) s4 (£7 + 7« + a/3) + s6 . a2/3y ;
and we have
32^ (V + 1)
- (a^2 - 3-S6-8 - 33s4 + 1)
=
r-* I i /'•* I ^A^) o/i ~t \.. » u\r, w
/r»J / /y4 _ _ 1 \ J
*t/ I »X^ J- I
Hence the product is found to be
= (s12 - 33s8 - 33s1 + 1) - s2 (s4 - 1)- .
4 (x1-- 33s8- 33s4 +1)
^,-J f ,w» 1 \a
w ^ wx if
_l_
208 ON THE SCHWARZIAN DERIVATIVE [745
which is
, fs12 - 33s8 - 33s4 + 1 x™ - SSafi - 33^ + 1
— o« I &* •_._ I
1
We thus verify that the twelve transformations x into x, into -, &c., give each of
cc
them a transformation of the function
x12 - 33s8- 3&c* +1
^(o,-4-!)2
into itself.
The system of 15 circles. Art. Nos. 118 to 127.
118. It has been already remarked that we can from the coefficients (A, B, C, D)
of the homographic transformation pass back to the position of the axis of rotation :
viz. we have
A : B : C : D = — v — i : \ + ifj, : \-ifj, : v — i,
and thence
\:H'.v:\= B + G : -i(B-C) : D - A :i(D + A),
that is,
\,l*,v = -i(B+C), - (B-G\ -i(D-A)- +
The equations of the axis thus are
x iy z
B+C~B-GD-A'
and the equations of the central plane at right angles to the axis are
119. In particular, we may find the equations of the 15 planes at right angles
to the ©-axes : these are in fact the before-mentioned 15 planes, intersecting the
sphere in great circles the projections of which are the circles in the three figures
respectively. Taking the equation of the plane to be Lx + My + Nz = 0, it is at once
seen that the equation of the projecting cone (vertex at the South pole) is
N (a? + y* + z> - 1 ) - 2 (z + 1) (Lx + My + Nz) = 0,
and hence, writing z = 0, we find
lY(a;2 + y- - 1) - 2 (Lx
for the equation of the circle in the plane figure. We have thus the equations of
a system of 15 circles related to each other in the manner before referred to.
120. Taking the ®-form, the equations of the 15 planes are at once found: and
we thence obtain the equations of the 15 circles: viz. writing for shortness
745] AND THE POLYHEDRAL FUNCTIONS. 209
the equations are
2 = 0, (ab . cd) fl = 0,
a; = 0, (ac . bd) # = 0,
y = 0, (ad .be) y = 0,
(- 1 - Vo) a? + (- 1 + V5) y + 2^ = 0, (a& . ce) O - [(- 1 - A
- = 0, (ac? . 6e) and similarly for the other circles.
• = 0, (ab . de)
121. Observe that the arrangement is in sets of 3 planes, or circles, intersecting
at right angles. One of the circles is the circle fl, =a? + y> — 1, =0 corresponding to
the equator, and two of them are the right lines x = 0 and y = 0. The equations of
the remaining 12 circles may be written in the somewhat different form
0 + (V5 - 1) [y - | (V5 - 1) x\ = 0,
fl - (V5 - 1) [y - i (V5 + 3) x\ = 0,
O - (V5 + 3) [T/ + i (V5 - 1) *] = 0,
li - (V5 - 1) [y - i (Vo - 1) ^] = 0,
ft + (Vo + 3) [y + \ (Vo - 1) x] = 0,
ft + (V5 - 1) [y + £ (V5 -!)«] = 0,
ft - (V5 - 1) [y + \ (V5 + 3) x-\ = 0,
ft - (V5 + 3) [y - % (Vo - 1) x] = 0,
O - (V5 - 1) [T/ + \ (V5 -!)«] = 0,
O + (V5 - 1) [y + i (^5 + 3) ^ = 0,
0 + (V5 + 3) [y - \ (V5 - 1) x\ = 0.
It hence appears that 4 and 4 circles have with H = 0 the common chords y + 1(\/5 - 1) x = 0,
y- H\/5-l)# = 0 respectively: and that 2 and 2 circles have with H = 0 the common
chords y + 1 (V 5 + 3) x = 0, y - i ( V5 + 3) x = 0 respectively.
c. xi. 27
210 ON THE SCHWARZIAN DERIVATIVE [745
122. The equations of the 12 circles are, in fact,
fl ± (V5 - 1) [y ± i (V5 - 1) x] = 0, 0 ± (V5 + 3) [y ± | (V-5 -!)«] = 0,
0 + (V5 - 1) [y + £ (V5 + 3) a;] = 0 :
hence the radii are = V5 - 1, 2 and V5 + 1 respectively.
The construction of the 12 circles is as follows. Starting with a circle radius 1.
Lay down the diameters y ± |(\/5 - 1)^ = 0 (A A in the figure), and through the
extremities of each describe 2 pairs of circles with the radii J5 - 1, \/5 + 1 respectively.
Lay down the diameters y ± £ (^5 + 3) a? = 0 (55 in the figure), and through the
extremities of each describe a pair of circles with the radius 2.
123. For the .4 -form, the equations of the fifteen planes are at once found to be
y = o,
ad
.be
— a;
+ (e + e4) z = 0,
ac
.Id
(e +
e4)*
+ * = 0,
ab
.cd
(e2-
e3)*
- i (e2 + e3) y =0,
ac
.be
-(e2 +
e3)^
+ i (e2 - e3) y + 2 (e + e4) z = 0,
ae
.be
-• +
i (e2 + e4 - e - e:!) y+ 2z = 0,
ab
.ce
(€ ~
e4)*
- i (e -t- e4) y =0,
ab
.de
-(e +
eV
+ i (e - e4) y + 2 (e + e4) 2 = 0,
ae
.bd
+ (e2 + e3 +
2) a;
— i (e- — e3) y + 2z = 0,
ad
.be
(e ~
e4)*
+ i (e + e4) y =0,
ae
.cd
-(e +
e4)*
- i (e - e4) y + 2 (e + e4) z = Q,
ac
.de
(e2 + €3 +
2)0
+ i(e*-e>)y+ 2^ = 0,
ad
.ce
(e2-
e3) x
+ i (e- + e3)y = 0,
bd
.ce
-(e2 +
e3)*
- i (e- - e3) y.+ 2 (e + e4) z = 0,
be
.de
— a; —
i (e2 + e4 - e - e3) y + 2z = 0,
be
. cd,
where, as before, the three planes of each set intersect at right angles.
124. Passing to the circles, the first plane of each set gives a right line, and
we have thus five of the circles reducing themselves to right lines inclined to the
axis of os at angles 0°, 36°, 72°, 108° and 144° respectively.
The remaining 10 circles form 5 pairs, the circles of a pair having different
radii, but the two radii being the same for each pair, and so that for the several
pairs the common chords with the circle fl = 0, are the diameters inclined to the
axis of x at the angles 18°, 54°, 90°, 126° and 162° respectively. Considering the
two circles for which the inclination is 90°, these arise from the planes —x + (e + e4)2 = 0,
= 0 respectively. The equations of the circles thus are (e + e4)O+ 2# = 0,
745] AND THE POLYHEDRAL FUNCTIONS. 211
2
ft — 2 (e 4- e4) x = 0, or recollecting that 2 (e + e4) = \/5 — 1 and therefore ^— = V5 4- 1,
the equations are
#2 + ?/2 - (V5 - 1) # - 1 = 0, a'2 + ?/2 + (V-5 + 1) # = 0 ;
hence for the first circle the ^-coordinate of the centre is £(\/5 — 1) and the radius is
= | V(10 — 2 V-5) 5 for the second circle the ^-coordinate of the centre is = -|-(v'5 + l),
and the radius = £ V(10 + 2 \/5). We have thus the construction of these two circles,
and consequently the construction of all the 12 circles.
125. For the B- form, after some easy reductions and attending to the relation
co — &>2 = i\/3, the equations of the 15 planes become
x =0,
ac . fca7
(- 3 + V5) y + 20 = 0,
aZ>. cd
(3 + \/5)y + 20 = 0,
ad. be
V3# + \/5 2/ + 20 = 0,
ac . be
- (1 + \/5) \/3a? + ( 3 - V5) 2/ + 40 = 0,
ab . ce
/ -| | / *" \ /O i / Q / K\ A f\
ae . be
a;+ V3 y =0,
ae . bd
-V3tf+ 7/ + (3 + V5) ^ = 0,
ad. be
V3*- 2/ + (3-V5) ^ = 0,
ab . de
— \/3a; + \/5 y + 20 = 0,
ac . de
(1 — /\/5) V3^ + (— 3 — \/5) y + 40 = 0,
ad. ce
(1 + V5) \/3# + (3 — \/5) y + 40 = 0,
ae . cd
»- V3 y =0,
bd . ce
V3*'+ 7/ + (3 + Vo) 0 = 0,
be . de
_ ^/3^ _ y + (3 _ ^5) z = o,
be . cd.
126. Of the 15 circles, 3 are the lines ac — y /v/3 = 0, a?=0, # + 2/\/3 = 0, viz.
these are lines at inclinations 30°, 90°, 150° to the axis of oc. The equations of the
remaining 12 circles are
ft + (3 - V5) y = 0,
H- (3 + V5)y = 0,
(3 + V5) H - 2 (y - x V3) = 0,
(3 + V5) H - 2 (y + x V3) = 0,
(3 - V5) ft 4- 2 (y + a V3) = 0,
27—2
212
ON THE SCHWARZIAN DERIVATIVE
[745
viz. these are pairs of circles having, for their common chords with O = 0, the diameters
at inclinations 0°, 60°, 120° respectively. And, lastly, we have the circles
2fi - [(- 1 + V 5) V3a? - (3 + V5) y} = 0,
5 y] = 0,
127. The first three of these have, for common chords with H = 0, the diameters
whose equations are
viz. these equations are y = (— 2 + \/5) # V3> y =
= (2 + V5)#\/3. If, as in a
/O /K /O
foregoing table, 0 = 37° 46', sin 0=^—.^, cos0 = ^^, and therefore tan#= ' ; then the
2i y 2i A y L y5
inclinations of these diameters to the axis of x are respectively 60° — 9, 6 and
120° -0, or say 30° -(0-30°), 30° + (0-30°) and 90° -(0-30°), where 0-30° = 7° 46',
i.e. the inclinations are 30° + 7° 46' and 90° - 7° 46'. And for the other three circles
the common chords are the diameters at the same inclinations taken negatively. The
geometrical construction of the fifteen circles for the .B-case in question is thus not
so simple as in the @- and A -cases.
The Regular Polyhedra as Solid figures. Art. Nos. 128 to 134.
128. I annex some results relating to the polyhedra considered as solid figures
bounded by plane faces ; or say results relating to the regular solids : s is in each
case taken for the length of the edge of the solid.
Tetrahedron.
Cube.
Octahedron.
Dodecahedron. Icosahedron.
Edge
Bad. of circum. sphere, R
Bad. of inters, sphere, p
Bad. of inscribed sphere, r
Bad. of circle circum. to face, R'
Bad. of circle inscribed to face, r'
Incl. of adjacent faces
Incl. of edge to adjacent face
s
s
s
s
s
v/3
S.W3
1
^3(^/5 + 1)
fo-\- >Jo
£2v/2
4
SV 8-
1
1
s . \
3 + v/o
l + N/5
"2^2
8 v/2
4
4
1
,.i
1
/25 + 11^/5
3 + v/o
S 2 J2 /3
Sv/2v/3
° V 40
4J3
1
•jl
1
/5 + J5
s V io"
1
1
8 '2^3
,.i
1
75 + 2^/5
8 V 20
1
cos-1 £ = 70° 28'
90°
cos-1 -^ = 109° 32'
cos"1 -4 = 54° 46'
90°
cos"1- 3 = 125° 44'
vrf
But we require further data in the cases of the dodecahedron and the icosahedron
respectively.
745]
AND THE POLYHEDEAL FUNCTIONS.
213
129. For the dodecahedron, taking the edge to be =s as before, then in the
pentagonal face
diagonal, g is = s . £ (V5 + 1),
altitude, k „ = s . \ \/(o + 2
segments of do., e „ = s .
where
130. The section through a pair of opposite edges is a hexagon, as shown in
the figure, viz. this is constructed by taking the four equal distances 0©, = p,
= s . | (3 + V5), meeting at right angles in 0 ; then drawing the double ordinates BB,
each =s, through ®1 and ©3 respectively, and joining their extremities with ®2 and
@4 : the sides ®2B and ®4B are then each = k, = s . | V(5 + 2 V5) ; and inserting
upon them the points A, 4> from the figure of the pentagon, we have several
geometrical relations; viz. the line A A cuts the parallel sides 502, B®4 at right
angles, and when produced passes through the intersection of B®1 and 5©4: we have
OA, OB, 0@ = r, R, p respectively: the four points 3> form a square, the side of
which is g, = s . \ (V5 + 1).
214 ON THE SCHWARZIAN DERIVATIVE
131. We find also
[745
AM=s
10 '
5 + 2 V5
/25_+JlV5
S\/ 5
25 + 11V5
03f_. /«ZV5,
/2 (
"V"
2 (5 + 2 V5)
It may be remarked that in the figure B®2, B®* are the projections of pentagonal
faces, at right angles to the plane of the paper, having their centres at the points
A, A, and the perpendicular distance between them =AA: the points Q, Q (only
one of them shown in the figure) determine the directions of the 5 + 5 sides which
abut on these pentagonal faces respectively ; and the 5 + 5 points B which are the
other extremities of these sides respectively form two pentagons, centres M, M in the
planes MB and MB respectively : the remaining 10 sides of the dodecahedron are the
skew decagon obtained by joining in order these 10 points B. We have thus the
means of making the perspective delineation of the dodecahedron.
132. The dodecahedron is built up from the cube, by placing on each face a
figure of two triangular and two quadrangular faces, the orthogonal projection of
which on the face of the cube is as in the figure : the side of the square is g>
= s.£(V5 + l): the slope-breadths of the triangular faces are e, = s. \ V(10- 2 \/5)»
and those of the quadrangular faces are /, = s . \ V(10 + 2 V^) 5 the lines represented
by the other lines of the figure are in actual length each = s. We have thus a
AND THE POLYHEDRAL FUNCTIONS.
215
745]
section which is an isosceles triangle, base = g, other sides each =/; and the square
of the altitude is thus =/2- i#2 = is2> or tne altitude =^s; viz. the altitude of the
ridge-line BB, above the face of the cube is =%s, the half-side of the dodecahedron.
We have in this result the most simple means of forming the perspective delineation
of the dodecahedron.
133. For the icosahedron the section through two opposite edges is a hexagon,
as shown in the figure (p. 216): to construct it, we take the four distances 0® each
— p = s,i(l + ^o) meeting at right angles; and then the distances A®.2, A®4 each
= £s; and complete the hexagon. This gives the sides A®1} A®3 each =s.^V3, the
altitude of the triangular face, side =s; and then, taking (Dj-B one-third of this,
= s , we have OB at right angles to A®lt and OA, OB, 0® = R, r, p respectively.
2 \/o
Moreover, joining
a point M : we find
., and OA.,, we have these lines cutting at right angles in
4(TJ =*A*M>
5_-V5
"10 '
134. It may be remarked that A^.,, A^ are the projections of two pentagons
in planes perpendicular to that of the paper, their centres being M, M: producing
OM, OM to the points A.2, A4 respectively, we have a pentagonal pyramid, summit
A.,, standing on the first pentagon, and an opposite pyramid, summit A4, standing on
216 THE SCHWARZIAN DERIVATIVE AND THE POLYHEDRAL FUNCTIONS. [745
the other pentagon : the 5 + 5 triangular faces of the two pyramids are ten of the
faces of the icosahedron, and the remaining ten faces are the triangles each having
for its base a side of the one pentagon, and for its vertex a summit of the other
pentagon, viz. the sides are the sides of the skew decagon obtained by joining in
order the angular points of the two pentagons. We have thus a convenient method
of forming the perspective delineation of the icosahedron.
746] 217
746.
HIGHEK PLANE CURVES.
[From Salmons Higher Plane Curves, (3rd ed., 1879) ; see the Preface.]
ONE chapter and a large number of articles, in the second edition of Salmon's Higher Plane Curves,
are due to Professor Cayley. Full reference to these is given by Dr Salmon in the preface.
c. xi. 28
218 [747
747.
NOTE ON THE DEGENERATE FORMS OF CURVES.
[From Salmon's Higher Plane Curves, (3rd ed., 1879), pp. 383—385.]
SOME remarks may be added as to the analytical theory of the degenerate forms
of curves. As regards conies, a line-pair can be represented in point-coordinates by an
equation of the form xy = 0 ; and reciprocally a point-pair can be represented in line-
coordinates by an equation ^ = 0, but we have to consider how the point-pair can be
represented in point-coordinates : an equation a? = 0 is no adequate representation of
the point-pair, but merely represents (as a two-fold or twice repeated line) the line
joining the two points of the point-pair, all traces of the points themselves being lost
in this representation : and it is to be noticed, that the conic, or two-fold line a? = 0,
or say (ax + fty + yzf = 0 is a conic which, analytically, and (in an improper sense)
geometrically, satisfies the condition of touching any line whatever ; whereas the only
proper tangents of a point-pair are the lines which pass through one or other of the
two points of the point-pair.
The solution arises out of the notion of a point-pair, considered as the limit of
u conic, or say as an indefinitely flat conic ; we have to consider conies certain of the
coefficients whereof are infinitesimals, and which, when the infinitesimal coefficients
actually vanish, reduce themselves to two-fold lines ; and it is, moreover, necessary to
consider the evanescent coefficients as infinitesimals of different orders. Thus consider
the conies which pass through two given points, and touch two given lines (four con
ditions) ; take y = 0, z=0 for the given lines, # = 0 for the line joining the given
points, and (x = 0, y — az — Q), (x = 0, y — fiz = Q) for the given points ; the equation of
a conic satisfying the required conditions and containing one arbitrary parameter 6, is
n-(y- OLZ) (y - #z) = 0 ;
747] NOTE ON THE DEGENERATE FORMS OF CURVES. 219
or, what is the same thing,
[x + By + 6 VO/3) z}' - 0s 0 + £) yz = 0 ;
and this equation, considering therein 0 as an infinitesimal, say of the first order,
represents the flat conic or point-pair composed of the two given points. Comparing
with the general equation
(a, b, c, f, g, h^x, y, zf = 0,
we have
ft = l, b = 6-, c = 62a/3, /=-£02(a+/3), <7 = 0V(a£), h=9,
viz. a being taken to be finite, we have g and h infinitesimals of the first order ;
b, c, f infinitesimals of the second order ; and the four ratios V(&) ' V(c) •' */(f) '• ff '• h
are so determined as to satisfy the prescribed conditions.
Observe that the flat conic, considered as a conic passing through the two given
points and touching the two given lines, is represented by a detwminate equation,
viz. considering the condition imposed upon 6 (6 = infinitesimal) as a determination of
6, the equation is a completely determinate one ; but considering the flat conic merely
as a conic passing through the two given points, the equation would contain two
arbitrary parameters, determinable if the flat conic was subjected to the condition of
touching two given lines, or to any other two conditions.
Generally, we may consider the equation of a curve of the order n; such equation
containing certain infinitesimal coefficients and, when these vanish, reducing itself to
a composite equation PaQ^ . . . = 0 ; the equation in its original form represents a curve
which may be called the penultimate curve. Consider the tangents from an arbitrary
point to the penultimate curve ; when this breaks up, the system of tangents reduces
itself to (1) the tangents from the fixed point to the several component curves
P = 0, Q = 0, &c. respectively ; (2) the lines through the singular points of these same
curves respectively ; (3) the lines through the points of intersection P = 0, Q = 0, &c.
of each two of the component curves; these points, each reckoned a proper number
of times, are called " fixed summits " ; (4) the lines from the fixed point to certain
determinate points called " free summits " on the several component curves P = 0,
Q = 0, &c. respectively. We have thus a degenerate form of the ?i-thic curve, which
may be regarded as consisting of the component curves, each its proper number of
times, and of the foregoing points called summits, and is consequently only inadequately
represented by the ultimate equation P*Q? . . . = 0 ; the number and distribution of the
summits is not arbitrary, but is regulated by laws arising from the consideration of
the penultimate curve, and there are of course for any given value of n various forms
of degenerate curve, according to the different ultimate forms P*Q? . . . = 0, and to the
number and distribution of the summits on the different component curves. The case
of a quartic curve having the ultimate form a?y- = 0 has been considered by Cayley,
Comptes Eendus, t. LXXIV. p. 708 (March, 1872), [515], who states his conclusion as follows:
28—2
220 NOTE ON THE DEGENERATE FORMS OF CURVES. [747
" there exists a quartic curve the penultimate of x-y2 = 0, with nine free summits,
three of them on one of the lines (say the line y = 0), and which are three of the
intersections of the quartic by this line (the fourth intersection being indefinitely near
to the point ao = 0, y = 0), six situate at pleasure on the other line x = 0 ; and three
fixed summits at the intersection of the two lines." Other forms have been con
sidered by Dr Zeuthen, Comptes Rendus, t. LXXV. pp. 703 and 950 (September and
October, 1872), and some other forms by Zeuthen ; the whole question of the degenerate
forms of curves is one well deserving further investigation.
The question of the number of cubic curves satisfying given elementary conditions
(depending as it does on the consideration of the degenerate forms of these curves)
has been solved by Maillard and Zeuthen ; that of the number of quartic curves has
been solved by Dr Zeuthen.
748]
221
748.
ON THE BITANGENTS OF A QUARTIC.
[From Salmons Higher Plane Curves, (3rd ed., 1879), pp. 387—389.]
THE equations of the 28 bitangents of a quartic curve were obtained in a very
elegant form by Riemann in the paper " Zur Theorie der Abel'schen Functionen fur
den Fall p = 3," Ges. Werke, Leipzig, 187G, pp. 456—472 ; and see also Weber's Theorie
der Abel'schen Functionen vom Geschlecht 3," Berlin, 1876. Riemann connects the
several bitangents with the characteristics of the 28 odd functions, thus obtaining for
them an algorithm which it is worth while to explain, but they will be given also
with the algorithm employed p. 231 et seq. of the present work*, which is in fact the
more simple one. The characteristic of a triple ^-function is a symbol of the form
a/3%
a'/3Y,
where each of the letters is = 0 or 1 ; there are thus in all 64 such symbols, but they
are considered as odd or even according as the sum era' + ft ft' + 77' is odd or even;
and the numbers of the odd and even characteristics are 28 and 36 respectively; and,
as already mentioned, the 28 odd characteristics correspond to the 28 bitangents
respectively.
We have x, y, z trilinear coordinates, a, @, 7, a', /3', 7' constants chosen at pleasure,
and then a", /3", 7" determinate constants, such that the equations
x+ y+ 2+ % + 77 + £ =0,
a ^j 7
[* That is, Salmon's Higher Plane Curves.]
222
ON THE BITANGENTS OF A QUARTIC.
[748
are equivalent to three independent equations; this being so, they determine f, 77, £,
each of them as a linear function of (as, y, z) ; and the equations of the bitangents of
the curve V(#£) + VCy?) + *J(Z%) = 0 (see Weber, p. 100) are
18
111
111
«-0,
28
001
Oil
y = o,
38
Oil
001
. = 0,
23
010
010
f-o, .
13
100
110
77 = 0,
12
110
100
r-o,
48
101
100
x + y + z = 0;
14
010
Oil
f+y + * = o,
58
100
101
OLX + /3y +yz = 0,
15
Oil
010
+ ySy + y2 = 0>
68
110
010
OLX + yS'y + y'z == 0,
16
001
101
£
— , + /3' ?/ + 7' z = i
a
78
010
110
<W,9-,+/,-<
17
101
001
4, + /3"j/ + y"z = I
24
100
111
a; +77 + z = 0,
34
110
101
* + y + ?=o,
25
101
110
ra + | + 7, = o,
35
111
100
cw? + ySy + - = 0,
7
748]
ON THE BITANGENTS OF A QUARTIC.
223
26
111
001
*„+! +y,-o,
36
101
Oil
a! x + ft'y + -, =0,
7
27
Oil
101
*••*£•••*•-«
37
001
111
£
av 4- fi"ti -1- - — 0
•*• i A-* y > it — "i
7
67
100
100
* y * 0
1 - /8y 1 - 7« l-a/8
57
110
Oil
\ y \ . . o
1 -/8'y 1 -7V + 1 -a'/3'~
56
010
111
x y z
1-^V' l-y'V ' 1-a'^""
45
001
001
fc t»
1 •(•
a (1 — /37) /8 (1 — 7a) 7 (1 — a/3)
46
Oil
110
*• ^
S ^ b
a' (1 - /3'7') yS' (1 - 7'a') y7 (1 - a'/3')
47
111
010
fc v»
H' 1
The whole number of ways in which the equation of the curve can be expressed
in a form such as V(#£) + V(2/^) + V(^) = 0 is 1260; viz. the three pairs of bitangents
entering into the equation of the curve are of one of the types
12 . 34, 13 . 24, 14 . 23 IS
12.34, 13.24, 56.78 Q \
13.23, 14.24, 15.25
No. is 70
„ 630
„ 560
1260.
It may be remarked that, selecting at pleasure any two pairs out of a system of
three pairs, the type is always D or \\ , viz. (see p. 233) the four bitangents are such
that their points of contact are situate on a conic.
224 [749
749.
SOLID GEOMETRY.
[From Salmon's Treatise on the analytic geometry of three dimensions, (3rd ed., 1874) ;
see the Preface.]
A considerable number of articles in the third edition of Salmon's Treatise are due to Professor Cayley.
Full reference to these is given by Dr Salmon in the preface.
750] 225
750.
ON THE THEOEY OF RECIPROCAL SURFACES.
[From Salmons Treatise on the analytic geometry of three dimensions, (3rd ed., 1874),
pp. 539—550.]
600. IN further developing the theory of reciprocal surfaces it has been found
necessary to take account of other singularities, some of which are as yet only
imperfectly understood. It will be convenient to give the following complete list of
the quantities which present themselves:
n, order of the surface.
a, order of the tangent cone drawn from any point to the surface.
8, number of nodal edges of the cone.
K, number of its cuspidal edges.
p, class of nodal torse.
ff, class of cuspidal torse.
6, order of nodal curve.
k, number of its apparent double points.
f, number of its actual double points.
t, number of its triple points.
;', number of its pinch-points.
q, its class.
c, order of cuspidal curve.
/;, number of its apparent double points.
0, number of its points of an unexplained singularity.
%, number of its close-points.
c. XL 29
226 ON THE THEORY OF RECIPROCAL SURFACES. [750
G>, number of its off-points.
r, its class.
/3, number of intersections of nodal and cuspidal curves, stationary points on
cuspidal curve.
7, number of intersections, stationary points on nodal curve.
i, number of intersections, not stationary points on either curve.
C, number of cnicnodes of surface.
B, number of binodes.
And corresponding reciprocally to these :
n', class of surface.
a', class of section by arbitrary plane.
S', number of double tangents of section.
K, number of its inflexions.
p, order of node-couple curve.
o-', order of spinode curve.
b', class of node-couple torse.
k', number of its apparent double planes.
/', number of its actual double planes.
t', number of its triple planes.
j', number of its pinch-planes.
q', its order.
c, class of spinode torse.
ti, number of its apparent double planes.
&, number of its planes of a certain unexplained singularity.
%, number of its close-planes,
a)', number of its off-planes.
r', its order.
/3', number of common planes of node-couple and spinode torse, stationary planes
of spinode torse.
7', number of common planes, stationary planes of node-couple torse.
t, number of common planes, not stationary planes of either torse.
G', number of cnictropes of surface.
B', number of its bitropes.
In all, these are 46 quantities.
750] ON THE THEORY OF RECIPROCAL SURFACES. 227
601. In part explanation, observe that the definitions of p and cr agree with
those already given. The nodal torse is the torse enveloped by the tangent planes
along the nodal curve ; if the nodal curve meets the curve of contact a, then a
tangent plane of the nodal torse passes through the arbitrary point, that is, p will
be the number of these planes which pass through the arbitrary point, viz. the class
of the torse. So also the cuspidal torse is the torse enveloped by the tangent planes
along the cuspidal curve ; and a- will be the number of these tangent planes which
pass through the arbitrary point, viz. it will be the class of the torse. Again, as
regards p' and cr' : the node-couple torse is the envelope of the bitangent planes of
the surface, and the node-couple curve is the locus of the points of contact of these
planes. Similarly, the spinode torse is the envelope of the parabolic planes of the
surface, and the spinode curve is the locus of the points of contact of these planes,
viz. it is the curve UH of intersection of the surface and its Hessian ; the two
curves are the reciprocals of the nodal and the cuspidal torses respectively, and the
definitions of p, a correspond to those of p and or.
602. In regard to the nodal curve b, we consider k the number of its apparent
double points (excluding actual double points) ; / the number of its actual double points
(each of these is a point of contact of two sheets of the surface, and there is thus at
the point a single tangent plane, viz. this is a plane f, and we thus have /' =/) ;
t the number of its triple points; and j the number of its pinch-points — these last
are not singular points of the nodal curve per se, but are singular in regard to the
curve as nodal/ curve of the surface ; viz. a pinch-point is a point at which the two
tangent planes are coincident. The curve is considered as not having any stationary
points other than the points 7, which lie also on the cuspidal curve ; and the
expression for the class consequently is q = b2 — b — 2k — 2f— 87 — 6t.
603. In regard to the cuspidal curve c, we consider h the number of its apparent
double points; and upon the curve, not singular points in regard to the curve per se,
but only in regard to it as cuspidal curve of the surface, certain points in number
6, %, w respectively. The curve is considered as not having any actual double or other
multiple points, and as not having any stationary points except the points /3, which
lie also on the nodal curve ; and the expression for the class consequently is
r = c2 - c - 2h - 3/3.
604. The points 7 are points where the cuspidal curve with the two sheets (or
say rather half-sheets) belonging to it are intersected by another sheet of the surface ;
the curve of intersection with such other sheet, belonging to the nodal curve of the
surface, has evidently a stationary (cuspidal) point at the point of intersection.
As to the points /3, to facilitate the conception, imagine the cuspidal curve to be
a semi-cubical parabola, and the nodal curve a right line (not in the plane of the
curve) passing through the cusp ; then intersecting the two curves by a series of
parallel planes, any plane which is, say, above the cusp, meets the parabola in two
real points and the line in one real point, and the section of the surface is a curve
with two real cusps and a real node ; as the plane approaches the cusp, these approach
29—2
228 ON THE THEORY OF RECIPROCAL SURFACES. [750
together, and, when the plane passes through the cusp, unite into a singular point in
the nature of a triple point (= node + two cusps) ; and when the plane passes below
the cusp, the two cusps of the section become imaginary, and the nodal line changes
from crunodal to acnodal.
605. At a point i the nodal curve crosses the cuspidal curve, being on the side
away from the two half-sheets of the surface acnodal, and on the side of the two
half-sheets crunodal, viz. the two half-sheets intersect each other along this portion of
the nodal curve. There is at the point a single tangent plane, which is a plane i! \ and
we thus have i = i'.
606. As already mentioned, a cnicnode C is a point where, instead of a tangent
plane, we have a tangent quadri-cone ; at a binode B, the quadri-cone degenerates into
a pair of planes. A cnictrope C' is a plane touching the surface along a conic; in
the case of a bitrope B', the conic degenerates into a flat conic or pair of points.
607. In the original formulas for a (n - 2), b(n — 2), c (n — 2), we have to write
K — B instead of K, and the formulae are further modified by reason of the singularities
6 and w. So, in the original formulas, for a (n — 2) (n — 3), b (n — 2) (n — 3), c (n — 2)(n — 3),
we have instead of 8 to write S — C — 3&>, and to substitute new expressions for
[ab], [ac], [be] ; viz. these are
[ab] = ab-2p -j,
[ac] = ac — 3<r — ^ — &>,
[be] =bc- 3/3 - 27 - i.
The whole series of equations thus is
(1) a' = a.
(2) /=/
(3) i' =i.
(4) a = n (n - 1) - 26 - 3c.
(5) K = 3n (n - 2) - Qb - 8c.
(6) 8' = \n (n - 2) (?i2 - 9) - (n2 - n - 6) (26 + 3c) + 26 (6 - 1) + 6bc + §c(c- 1).
(7) a (n - 2) = K - B + p + 2<r + 3&>.
(8) 6(n-2)= p + 2t3 + 3y + 3t.
(9) c (n - 2) = 2<r + 4/3 + 7 + 0 + a).
(10) a(«-2)(ft-3)-2(a-0-3«) + 3(a6~&r-x-8«)+2(a6-2p -j).
(11) 6(w-2)(w-3) = 4& + (ab-2p-j ) + 3(6c -3/8 - 27 -i).
(12) c(w-2)(n-3) = 6A + (ac-3o--x
(13) q = b*-b-2k- 2/- 87 - 6i.
(14) r = c2 - c - 2/i - 3/3.
750] ON THE THEORY OF RECIPROCAL SURFACES. 229
Also, reciprocal to these,
(15) a' = n (n - 1) - 26' - 3c'.
(16) K = 3»' (n' - 2) - 66' - Sc.
(17) a = in' (/*' - 2) (n'2 - 9) - (n'2 - n' - 6) (26' + 3c') + 26' (6' - 1) + 66V + f c' (c' - 1).
(18) a' («' - 2) = *' - 5' + p + 2o-' + 3ft)'.
(19) &'(n'-2)= p' + 2/3' + 37' + 3«'.
(20) c (n' - 2) = 2o-' + 4/3' + 7' + & + &>'.
(21) a' (n' - 2) (»' - 3) = 2 (V -C'- 3a>') + 3 (aV - 3o-' - %' - 3a>') + 2 (a'6' - 2p' -/).
(22) 6' (»' - 2) (n' - 3) = 4# + (a'b'-Zp'-f ) + 3(6V-3/3'-27W).
(23) c' (»' - 2) (»' - 3) = 6fc' + (aV - So-' - %' - 3a>') + 2(6'c'-3/3'-27W).
(24) g' = 6'2 - 6' - 2A;' - 2/ - 37' - 6t'.
(25) r = c'2 - c' - 2^' - 3/8',
together with one other independent relation: in all 26 relations between the 46
quantities.
608. The new relation may be presented under several different forms, equivalent
to each other in virtue of the foregoing 25 relations; these are
(26) 2(w-l)(7i-2)(?i-3)-120i-3)(6 + c) + 6? + 6r+24« + 42/3 + 307-f^ = S,
in each of which two equations 2 is used to denote the same function of the accented
letters that the left-hand side is of the unaccented letters.
(28) /3' + i<9'=
+ (_ 66n + 184) 6
+ (- 93?i + 252) c
+ 27(4/3+ 7 + <9)
+ 0 + W
- 240 - 285 - 27j - 38^ - 73<u
+ 4C" + 105' + 7/
Or, reciprocally,
(29) /3+|<9= 2»i/(w'-2)(lln'
+ (- 66n' + 184) 6'
+ (- 93??,' + 252) c'
+ 22 (2/8' + 37'
+ 27(4/3'+ y
- 246" - 285' - 27j' -
+ 4(7 + 105 + 7j +
230
ON THE THEORY OF RECIPROCAL SURFACES.
[750
The equation (26) expresses that the surface and its reciprocal have the same deficiency;
viz. the expression for the deficiency is
(30) Deficiency = %(n- 1) (w - 2) (n - 3) - (n - 3) (b + c) + £ (q + r) + 2t +l/3+f 7+;_i0,
= £ (n' - 1) (n' - 2) (n' - 3) - &c.
609. The equation (28) (due to Prof. Cayley) is the correct form of an expression
for /3', first obtained by him (with some errors in the numerical coefficients) from
independent considerations. But it is best obtained by means of the equation (26) :
and (27) is a relation presenting itself in the investigation. In fact, considering « as
standing for its value n (n— 1)— 26 — 3c, we have from the first 25 equations
6
+ 2
- 2
- 4
-6
+ 2
-3
_ 2
a
3w - c - K
a (n - 42) - K + B - p - 2o- - 3&>
6(w-2)-p-2/3-37-3*
c (n - 2) - 2<r - 4£ - 7 - 0 - w
= 2,
= 2,
= 2,
=2,
= 2,
+ K - a- - 2(7 - 4>B - 2j - 3^ - 3a> = 2,
=2,
&> =2;
multiplying these equations by the numbers set opposite to them respectively, and
adding, we find
- 2?i3 + 12w2 + 4w + b (Un - 36) + c (12n - 48)
-Qq-Qr- 46'- 105 - 4,1/3 - 307 - 24* - 7j - 8* + 20 - 4a> = 2,
and adding hereto (26) we have the equation (27); and from this (28), or by a like
process, (29), is obtained without much difficulty. As to the 8 2-equations or symmetries,
observe that the first, third, fourth, and fifth are in fact included among the original
equations (for an expression which vanishes is in fact = 2) ; we have from them
moreover 3n - c = 3a' - K, and thence 3?i - c - K = 3a' - K - K, which is = 2, or we have
thus the second equation; but the sixth, seventh, and eighth equations have yet to
be obtained.
610. The equations (15), (16), (17) give
n'= a (a -1)- 28 -3*,
c = 3a (a - 2) - 68 - SK,
From (7), (8), (9), we have
(a- b- c)(n-2) =K-B-Q/3-4ry-3t-0 + 2(o,
(a - 2b -3c)(n-2)(»- 3) = 2(8 -C)- 8k- I8h - Gbc + 180 + I2y + 6i - fa ;
750] ON THE THEORY OF RECIPROCAL SURFACES. 231
substituting these values for K and 8, and for a its value = n (n — 1) — 26 — 3e, we
obtain the values of n, c, b' ; viz. the value of n' is
n' - n (n - I)2 - n (76 + 12c) + 462 + 8b + 9c2 + 15c
- 8& - ISA + 18/3 + 12? + 12i - 9£
- 2(7 - 35 - 30.
Observe that the effect of a cnicnode C is to reduce the class by 2, and that of a
binode B to reduce it by 3.
611. We have
(n - 2) (n - 3) = n2 - n + (- 4>n + b') = a + 26 + 3c + (- 4n + 6) ;
making this substitution in the equations (10), (11), (12), which contain (n - 2) (n — 3),
these become
a (- 4>n + 6) = 2 (8 - C) - a2 - 4p - 9<r - 2j - 3X - low,
6 (- 4w + 6) = 4& - 262 - 9/8 - 67 - 3i - 2p - j,
c (- 4n + 6) = Qh - 3c" - 6/3 - 4>y - 2i - So- - % - 3&>,
which are the foregoing equations (6'); adding to each equation four times the corre
sponding equation with the factor (n — 2), these become
a2 - 2a = 2 (8 - C) + 4 (K - B) - a - 2j - 3* - 3a>,
262 - 26 = 4& - /3 + 67 + 12£ - 3i + 2p - j,
3c2 - 2c = Qh + 10/9 + 40 - Zi + 5a - x + a,.
Writing in the first of these a2- 2a = n' + 28 + 3/c — a, and reducing the other two by
means of the values of q, r, the equations become
n' - a = - 2(7-
3r + c + 2i + ^ = oo- + /3 + 40 + w,
which give at once the last three of the 8 2-equations.
The reciprocal of the first of these is
<r' = a-n + Kr- 2j - 3%' - 2C" - 4F - 3w',
viz. wiiting herein
a = n(n-l)-2b-3c and *'= 3n(n - 2) - 66 -8c,
this is
o-' = 4» (w - 2) - 86 - 1 Ic - 2j' - 3X' - 2C" - 4>B' - 3o>',
giving the order of the spinode curve; viz. for a surface of the order n without
singularities, this is =4n(w-2), the product of the orders of the surface and its
Hessian.
232 ON THE THEORY OF RECIPROCAL SURFACES. [750
612. Instead of obtaining the second and third equations as above, we may to
the value of 6 (— 4m + 6) add twice the value of 6 (n — 2) ; and to twice the value of
c (— 4>n + 6) add three times the value of c (n — 2), thus obtaining equations free from
p and a respectively; these equations are
b (- 2n + 2) = 4,k - 262 - 5£ - 3i + 6t -j,
c (- 5n + 6) = 12A - 6c- - 5y - 4t - 2X + 30 - 3a>,
equations which, introducing therein the values of q and r, may also be written
6(2n- 4) =2q+ 5/3 + 67 + Qt + 3i+j + 4/,
c (5n - 12) + 30 = 6r + 18/3 + 5y + 4i + 2X + 3o>.
Considering as given, ?& the order of the surface ; the nodal curve, with its singularities
b, k, f, t\ the cuspidal curve, with its singularities c, h ; and the quantities /3, y, i
which relate to the intersections of the nodal and cuspidal curves; the first of the
two equations gives j, the number of pinch-points, being singularities of the nodal
curve, quoad the surface ; and the second equation establishes a relation between
0, %, <u, the numbers of singular points of the cuspidal curve quoad the surface.
In the case of a nodal curve only, if this be a complete intersection P = 0, Q = 0,
the equation of the surface is (A, B, C%P, Q)2 = 0, and the first equation is
&(-2?i + 2) = 4&-2&2 + Gt-j;
or, assuming 2=0, say j= 2 (n — 1)6 — 262 + 4&, which may be verified; and so in the
case of a cuspidal curve only, when this is a complete intersection P = 0, Q = 0, the
equation of the surface is (A, B, CQP, Q)- = 0, where A C - B2 = MP + NQ ; and the
second equation is
c (- on + 6) = 12/i - 6c2 - 2% + 3(9 - 3o>,
or, say 2^ + 3<u = (5n — 6)c-6c2 + 12A + 30, which may also be verified.
613. We may in the first instance out of the 46 quantities consider as given
the 14 quantities
» : b, k, f,t : c, h, 0, X : 0, 7, i : C, B,
then of the 26 relations, ]7 determine the 17 quantities
a, 8, K, p, a- : j, q : r, a>
n' : a', B', K' : b',f : c' : i' •
and there remain the 9 equations
(18), (19), (20), (21), (22), (23), (24), (25), (28),
connecting the 15 quantities
p', a' : k', t', f, q' : h', 0', %', «', / : £', 7' : C", B'.
750] ON THE THEORY OF RECIPROCAL SURFACES. 233
Taking then further as given the 5 quantities j', %', w, C', B',
equations (18) and (21) give p, a-',
equation (19) gives 2/3' + 87' + 3£';
(20) ,, 4/3'+ 7+0',
(28) „ ff + W,
so that, taking also t' as given, these last three equations determine /3', 7', 6'; and
finally
equation (22) gives k',
(23) „ /,',
(24) „ q,
(25) „ r,
viz. taking as given in all 20 quantities, the remaining 26 will be determined.
614. In the case of the general surface of the order n, without singularities, we
have as follow :
n = n,
a = n (n — 1),
8 = \n (n - 1) (n - 2) (n - 3),
K = n (n — 1) (n - 2),
n' = n (n — I)'2,
a = n (n — 1),
K = Sn (n - 2),
I' = £ n (n - 1) (n - 2) (n3 - ?i2 + w - 12),
k' = in (n - 2) (nw - 6w9 + 16w8 - 54/i7 + 164?i6 - 288n5
+ 547w4 - 1058n3 + 1068n2 - 1214?i + 1464),
t' = £w (n - 2) (?i7 - 4?i« + 7?i5 - 45?i4 + 114n3 - 111??2 + 548w - 960),
q' = n (n - 2) (re - 3) (n2 + 2n - 4),
p' = ?^ (n — 2) (?i3 — ?i2 + n — 12),
c' = 4?i (/i - 1) (?i - 2),
A' = £»i (n - 2) (16?i4 - 64w3 + 80w2 - 108?z + 156),
r' = 2n (?i - 2) (3w - 4),
a-' = 4»i (w - 2),
7' = 4?z (71 - 2) (n - 3) (7i3 - 3n + 16),
the remaining quantities vanishing.
c. XL 30
234 ON THE THEORY OF RECIPROCAL SURFACES. [750
615. The question of singularities has been considered under a more general
point of view by Zeuthen, in the memoir " Recherche des singularity qui ont rapport
a une droite multiple d'une surface," Math. Annalen, t. iv. (1871), pp. 1—20. He
attributes to the surface :
A number of singular points, viz. points at any one of which the tangents form
a cone of the order fi, and class v, with y + 77 double lines, of which y are tangents
to branches of the nodal curve through the point, and z + £ stationary lines, whereof
z are tangents to branches of the cuspidal curve through the point, and with u double
planes and v stationary planes; moreover, these points have only the properties which
are the most general in the case of a surface regarded as a locus of points; and 2
denotes a sum extending to all such points. (The foregoing general definition includes
the cnicnodes /* = i; = 2, y = rj = z = ^=ii = v = Q, and the binodes /* = 2, 77 = 1,
i/ = y = &c. = 0.)
And, further, a number of singular planes, viz. planes any one of which touches
along a curve of the class p and order v, with y' + r\ double tangents, of which y'
are generating lines of the node-couple torse, / + f ' stationary tangents, of which z
are generating lines of the spinode torse, u' double points and v' cusps; it is, more
over, supposed that these planes have only the properties which are the most general
in the case of a surface regarded as an envelope of its tangent planes; and 2' denotes
a sum extending to all such planes. (The definition includes the cnictropes /// = v = 2,
y' = ,/ = / = f = u' = v' = 0, and the bitropes // = 2, if = 1, v' = y' = &c. = 0.)
616. This being so, and writing
« = v + 2i7 + 3£ x = v + 27;' + 3f,
the equations (7), (8), (9), (10), (11), (12), contain, in respect of the new singularities
additional terms, viz. these are
a (n - 2) = ... + 2 [x (UL - 2) - 17 - 2£],
a (n - 2) (n - 3) = ... + 2 [x (- 4>p + 7) + 277 + 4£j,
b (n - 2) (n - 3) = ... + 2 [y (- 4y*+ 8)] - 2' (4u' + 3t>'),
c (n - 2) (n - 3) = ... + 2 [2 (- 4^ + 9)] - 2' (2w'),
and there are of course the reciprocal terms in the reciprocal equations (18). (19),
(20), (21), (22), (23). These formulae are given without demonstration in the memoir
just referred to: the principal object of the memoir, as shown by its title, is the
consideration not of such singular points and planes, but of the multiple right lines
of a surface; and in regard to these, the memoir should be consulted.
751] 235
751.
NOTE ON RIEMANN'S PAPER "VERSUCH EINER ALLGEMEINEN
AUFFASSUNG DER INTEGRATION UND DIFFERENTIATION* "
[From the Mathematische Annalen, t. xvi. (1880), pp. 81, 82.]
THE Editors of Riemann's works remark that the paper in question was contained
in a MS. of his student time (dated 14 Jan. 1847) and was probably never intended
for publication: indeed that he would not in later years have recognised the validity
of the principles upon which it is founded. The idea is however a noticeable one:
Riemann considers zx+h, a function of tc + h, expanded in a doubly infinite, necessarily
divergent, series of integer or fractional powers of h, according to the law
zx+h=V~^kJd"xz.h", (2)
i/= —oo
where the meaning is explained to be that the exponents differ from each other by
integer values, in effect, that v has all the values a + p, a a given integer or fractional
value, and p any integer number from -co to + oo , zero included.
Riemann deduces a theory of fractional differentiation: but without considering
the question which has always appeared to me to be the great difficulty in such a
theory: what is the real meaning of a complementary function containing an infinity
of arbitrary constants ? or, in other words, what is the arbitrariness of the complemen
tary function of this nature which presents itself in the theory ?
I wish to point out the relation between the paper referred to, and a short
paper of my own "On a doubly infinite Series," Quart. Math. Journ. t. VI. (1851),
pp. 45—47, [102]: this commences with the remark "The following completely para
doxical investigation of the properties of the function T (which I have been in possession
* Werke, pp. 331—344.
30—2
236 NOTE ON RIEMANN'S PAPER. [751
of for some years) may perhaps be found interesting from its connexion with the
theories of expansion and divergent series." And I then give the expansion
where n is any integer or fractional number whatever, and the summation extends
to all positive and negative integer values (zero included) of r. And I remark that,
n being an integer, we have Cn = T (n), and hence that assuming that this is so in
general, or writing
T (n) .ex = Zr[n- l]r asn-1-r,
we have this equation as a definition of F (?i). The point of resemblance of course
is that we have a doubly infinite expansion of e* in a series of integer or fractional
powers of x, corresponding to Riemann's like expansion of zx+h in powers of h.
Cambridge, 10 Sept. 1879.
752] 237
752.
ON THE FINITE GROUPS OF LINEAR TRANSFORMATIONS OF
A VARIABLE; WITH A CORRECTION.
[From the Mathematische Annalen, t. xvi. (1880), pp. 260—263; 439, 440.]
IN the paper " Ueber endliche Gruppen linearer Transformationen einer Verander-
lichen," Math. Ann. t. xn. (1877), pp. 23 — 46, Prof. Gordan gave in a very elegant form
fl IT r\ / )
the groups of 12, 24 and 60 homographic transformations - — - ,-. The groups of 12
and 24 are in the like form, the group of 24 thus containing as part of itself the
group of 12 ; but the group of 60 is in a different form, not containing as part of
itself the group of 12. It is, I think, desirable to present the group of 60 in the
form in which it contains as part of itself Gordan's group of 12 : and moreover to
identify the group of 60 with the group of the 60 positive permutations of 5 letters:
or (writing abc for the cyclical permutation a into b, b into c, c into a, and so in
other cases) say with the group of the 60 positive permutations 1, abc, ab.cd and
abode.
Any two forms of a group are, it is well known, connected as follows, viz. if
1, a, /3, ... are the functional symbols of the one form, then those of the other form
are 1, ^a^"1, ^fi^1, ... (where in the case in question ^ is a functional symbol of
the like homographic form, SYB = ~ n). But instead of obtaining the new form in
\jffs "T~ _L//
this manner, I found it easier to use the values of the rotation-symbol
7T 7T
cos — h sin - (i cos X + j cos Y + k cos Z)
q q^
for the axes of the icosahedron or dodecahedron, given in my paper '"'Notes on
polyhedra," Quart. Math. Jour. t. vn. (1866), pp. 304—316, [375]; viz. if for any axes,
\ ft, v denote the parameters of rotation tan - cos X, tan - cos F, tan - cos Z, then,
q q q
238
ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE.
[752
by a formula which is in fact equivalent to that given in my note " On the
correspondence of Homographies and Rotations," Math. Annalen, t. XV. (1879),
pp. 238 — 240, [660], the corresponding homographic function of x is
(— v — i) ac + \ + ifi
(\ — ift) x + v — i
where i denotes V— 1 as usual.
The new formulae for the group of 60, or icosahedron group, of homographic
OLX I Q
functions - — ^ are contained in the following table, where the four columns show
the values of the coefficients a, /3, 7, 8 respectively : and where in the outside column,
the substitution is represented as a permutation-symbol on the five letters abcde:
moreover for shortness ® is written to denote Vo.
THE GROUP OF 60.
1
1
1
0
0
1
1
2
-1
0
0
1
ab . cd
3
0
1
1
0
ac . bd
4
0
-1
1
0
ad . be
5
2
-3+e-ri( i-e)
-3+0+j (-1 + 6)
-2
be . de
6
2
-s+0+i(-i+0)
- 3 + 0 + 1 ( 1-6)
-2
ae . be
7
2
3-e + i(-l + 0)
3-0 + i( 1-6)
-2
ad . ce
8
2
3-0 + /( 1-6)
3-0 + f (-1 + 0)
-2
ad, be
9
2
-i-e+i( 1-0)
-i-e+z(-i+e)
-2
ae . cd
10
2
-l-0 + i(-l + 0)
-l-0 + i( 1-6)
-2
ab . de
11
2
1 + 0 + f (-1 + 0)
i+e+?( i-e)
-2
be .cd
12
2
i+e+?( i-e)
l + 0+l(-l+0)
-2
ab . ce
13
2
-l-0 + i (-3-0)
-l-0 + f( 3 + 6)
-2
ac . be
14
2
-l-0 + f( 3 + 6)
-1-6 + ? (-3-6)
-2
bd . ce
15
2
l + 0 + f( 3 + 6)
1 + 0 + f (-3-0)
-2
ae . bd
16
2
1 + 0+ j (-5-0)
l + 6 + z( 3 + 6)
-2
ac . de
17
— i
i
1
1
a be
18
-1
i
1
i
acb
19
1
-i
1
i
adc
20
- i
-i
1
-1
acd
21
i
i
1
-1
adb
22
1
i
1
-i
abd
23
-1
-i
1
-i
bed
24
i
-i
1
1
bdc
752]
ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE.
239
25
-l-0 + i( 3 + 0)
2
-2
-1-0 + M-3-0)
aec
26
1 + 0 + i ( 3 + 0)
2
-2
1 + 0 + / ( - 3 - 0)
ace
27
l + e + i(-3-e)
2
-2
1 + 0 + i( 3+0)
bed
28
_l_0 + ;(_3-0)
2
-2
-l-e + i( 3 + 0)
Me
29
-3 + 0 + /( 1-0)
2
2
3-0 + -i( 1-0) ! bee
30
-8+6+<<-l+6)
2
2
3-0 + i(-l + 0)
bee
31
3-0 + i(_l + 0)
2
2
-3 + 0+t(-l + 0)
aed
32
3-0 + i( 1-0)
2
2
-3 + 0 + t( 1-0)
ade
33
2
-l-0 + i(-l + 0)
l + 0+i( -1 + 0)
cde
34
2
l + 0 + f( 1-0)
- 1 - 0 + i ( 1-0)
ced
35
2
-i-e+M 1-0)
l+0 + z( 1-0)
aeb
36
2
i+e+i(-i+e)
-l-0 + z(-l + 0)
abe
37
-l-0 + t(-3-0)
2
2
1 + 0 + ^-3-0)
abode
38
-l-0 + i( 1-0)
2
2
l + 0 + i( 1-0)
accbd
39
_i_e+z(-i+0)
2
2
l + 0 + i(-l + 0)
adbee
40
-l-0 + t( 3 + 0)
2
2
l + 0+i( 3 + 0)
aedcb
41
1 + 0 + i ( 3 + 0)
2
2
-l-0 + t( 3 + 0)
adceb
42
i+0+/(_i+e)
2
2
-l-0 + i(-l + 0) acbde
43
l + 0+(( 1-0)
2
2
-l-0 + z( 1-0) aedbc
44
1 + 0 + i ( - 3 - 0)
2
2
_l_e + i(-3-0)
abecd
45
-1-0 + i (-1 + 0)
2
-2
-l-0 + i( 1-0)
acbed
46
_3 + e + i(-l + 0)
2
-2
-3+0 + i( 1-0)
abdce
47
3_0 + t(-l+0)
2
-2
3-0 + i( 1-0)
aecdb
48
i+e+f (-1+0)
2
-2
l+0 + i( 1-0)
adebc
49
l + 0 + t( 1-0)
2
-2
l + 0 + i(-l + 0)
aecbd
50
3-0 + i( 1-0)
2
-2
3-0 + i(-l + 0)
acdeb
51
-3 + 0 + i( 1-0)
2
- 2
-3 + 0 + i(-l + 0)
abedc
52
- 1 - 0 + i ( 1-0)
2
-2
_i_e+i(-i+0)
adbce
53
2
-3 + 0 + ^-1 + 0)
3-0 + i(-l + 0)
2
aebdc
54
2
-l-0 + <( 3 + 0)
l + 0 + t( 3 + 0)
2
abced
55
2
l + 0 + t(-3-0)
_l_e+i(_3-0)
2
adecb
56
2
3-0 + t( 1-0)
-3 + 0 + i( 1-0)
2
acdbe
57
2
-3 + 0+t( 1-0)
3-0 + i( 1-0)
2
abdec
58
2
-l-e + t(-3-6)
l + e + i(_3_e)
2
adcbe
59
2
l + 0 + t( 3 + 0)
-l-0+i( 3 + 0)
2
aebcd
60
2
3-0 + i(-l + 0)
2
acedb
240
ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE.
[752
This contains (as one of five groups of 12) the group of the positive permutations
of abed ; and, completing this into a group of 24, we have
GROUPS OF 12 AND 24.
ft 7
1
1
0
0
1
1
2
-1
0 0
1
ab . cd
3
0
1
1
0
ac . bd
4
0
_ 1
1
0
ad . be
5
-i
i
1
1
abc
6
-1
i
1
i
neb
7
1
-i
1
i
adc
8
-i
— i
1
_ -^
acd
9
i
i
1
-I
adb
10
1
i
1
- i
abd
11
-1
-i
1
- i
bed
12
i
-i
1
1
bdc
13
i
0
0
1
adbc
14
- i
0
0
1
acbd
15
0
i
1
0
cd
16
0
i
-1
0
ab
17
1
-1
1
1
acdb
. 18
-i
-1
1
i
bd
19
i
1
1
i
abed
20
1
1
1
-1
be
21
-1
_ ^
1
-1
abdc
22
t
-1
1
- i
ac
23
-t
1
1
- i
adcb
24
-1
1
1
1
ad
The groups of 60 and 24 thus each of them contain the group of 12,
x — i
1 . 1 — x . 1 + x x + i
±as, ± - , ±1= --- , t » T > + -- •
X \+X \—X ~X — l
+
~
It may be remarked that, to verify the periodicities of the forms contained in the
group of 60, we have as the conditions that
.. may be periodic of the order 2, -
» » » "> »
» » » S> »
= 0, that is, a + 8 = 0,
= 1
752] ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE. 241
For instance, in the form
[_ i _ @ + j (_ 3 - 0)] x + 2
2a?+[l + ®+t(-3-®)] '
we have
a8 = _ (l+6)2-(3+@)2, =- 20-80,
a+8 =
and therefore
as it should be.
Cambridge, 11 Nov. 1879.
CORRECTION*, pp. 439, 440.
I erroneously assumed that the symbol adcb could be taken as corresponding
to the linear transformation ix: but this was obviously wrong, for it gave bd as
corresponding to the transformation —ix, and these are not of the same order, but
of the orders 4 and 2 respectively. The proper symbol is adbc, as given above, and
the remaining eleven symbols are then at once obtained.
Cambridge, 17 Feb. 1880.
[* The correction in the Table of the Groups of 12 and 24 has been inserted in the Table as now
printed on p. 240; it applies to the second half of the column of symbols on the extreme right-hand.]
C. XL 31
242 [753
753.
ON A THEOREM EELATING TO THE MULTIPLE
THETA-FUNCTIONS.
[From the Mathematische Annalen, t. XVII. (1880), pp. 115—122.]
I PROPOSE — partly for the sake of the theorem itself, partly for that of the
notation which will be employed — to demonstrate the general theorem (3'), p. 4, of
Dr Schottky's Abriss einer Theorie der Abel'schen Functionen von drei Variabeln,
(Leipzig, 1880), which theorem is there presented in the form :
,-,(«„...;, >'> 6 fa + *«/,...; fi, lO-e-^-'-ete,...; p + p! , v + „'), (3')
but which I write in the slightly different form
exp. [- H(u ; //, i/)] . 0 (u + 2«r' ; p,, v) = exp. [- Zirinv] .S(u; p, + p!, v + v').
I remark that the theorem is given in the preliminary paragraphs the contents
of which are, as mentioned by the Author, derived from Herr Weierstrass : and
that the form of the theta-function is a very general one, depending on the general
quadric function
of 2/> variables, p being the number of the arguments ul} ..., up (in fact, the periods
are not reduced to the normal form, but are arbitrary) ; and the characters vl , ..., vp\
//,!, ...,/ip, instead of having each of them the value 0 or 1, have each of them any
integer or fractional value whatever. The meaning of the theorem (u denoting a set
or row of p letters MJ, ..., up, and so in other cases), is that the function
@ (u ; A + //, v + v'}
753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 243
with the new characters /* + // and v + v' is, save as to an exponential factor, equal
to the function ® (u + 2«r' ; p, v) with the original characters //,, i>, but with the new
arguments u + 2or'.
Notation.
This is in some measure a development of the notation employed in my " Memoir
on the Theory of Matrices," Phil. Trans, t. CXLVIII. (1858), pp. 17 — 37, [152] I use
certain single letters u, etc. to denote sets or rows each of p letters, u = (ult ... , up):
or if, to fix the ideas p = 3, then u=(u-i, u2, u3\ and so in other cases.
But I use certain other letters a, etc. to denote squares or matrices each of pz
letters ; thus, if p = 3 as before,
a= ou, a13, ttj3 ,
and in any such case the transposed matrix is denoted by the same letter enclosed
in parentheses
(a) = elm ft21,
The sum w + v of the row-letters u, =(ult u2, u3) and v, —(v^ v2, v3) denotes the
row (MJ+VI, ?<2 + ^2» «3 + v3): and in like manner the sum a + b of the two matrices,
or square-letters a and b, denotes the matrix
and similarly for a sum of three or more terms.
The product uv, =(wj, u.2, ii3)(v1, v.2, v3), of the two row-letters u, v denotes the
single term u^ + u2vz + u3v3. We have uv = vu.
The product
au, = au, a12, al
l} u.2, u3),
of a preceding square-letter a and a succeeding row-letter u, denotes the set or row
On, «12, ais)(tti, '"2, MS). fa*, 0», O!B)(WI, «ai Ws), (asi, a32, att) (Wj, i<2, W,) ;
the notation ua is not employed.
31—2
244 ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS.
The product
[753
auv = < an, aI2,
HI, u2, u3) (vlt v.2> v3),
of a preceding square-letter a followed by the two row-letters u and v, denotes the
single term
(an, a12, a1
(a3l, a^, a33)(u1, uit u3)v3.
Observe that auv is not in general = avu ; but it is easy to verify that auv = (a)vu',
and hence if (a) = a, that is, if the matrix a be symmetrical, then auv = avu.
A product of two matrices
ab, =
denotes a matrix
(Ill, <Z12, (if 13
PHI &i2, ^13
J
#21, #22, #23
^21 > ^22 > 0<j3
^*J1 y ^32 > 33
0-n , Oi9 . bvt
Ol 7 «Kfi J «K>
6n, 621, b31), (b^, 622, &32), (bl3, 633, 633)
(a2i»
viz. the top-line of the compound matrix is
(an, a12, a13)(6u, b.21, &31), (an, aw
^, 32, au, a12,
and so for the other lines: or expressing this in words, we say that any line of
the compound matrix is obtained by compounding the corresponding line of the first
or further component matrix with the several columns of the second or nearer
component matrix.
Clearly ab is not in general =ba. We may easily verify that (ab) = (b)(a), that
is, the transposed matrix (ab) is that obtained by the composition of the transposed
matrix (6) as first or further matrix, with the transposed matrix (a) as second or
nearer matrix. Even if a and b are each symmetrical, we do not in general have
ab = ba, but only (ab) = ba, or what is the same thing, ab = (ba).
In a symbol such as abuv, we first combine a, b into a single matrix ab, and
then regard the expression as a combination such as auv : the expression denotes
therefore a single term. The theory might be explained in greater detail; but
the mode of working with row- and square-letters will be readily understood from
what precedes.
In all that follows, u, p, v, JJL, v, n, -or', % are row-letters; a, 6, h, &>, to', 77, 77'
are square-letters : a and b are symmetrical, viz. a = (a), b = (b).
753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 245
And I write
(*)O, i/)2, =(a, h, &)(«, i/)2
— aii? + 2huv + bv-
i n n (ti 11 11 V
ni > Ui2 , tli3 V "i > "2 , "3/
21 ' 22 '
+ 2 An, 7t12, A13
*^21 ) 22 ) 23
A31, //3.2, ^33
^21} ^22 > °23
^31, ^32, &33
to denote the general quadric function of the 2p letters u, v, with
>, =/t>(2p+l)
coefficients. It is assumed that the determinant formed with the ^p (p + 1) coefficients
6 is negative : this is the necessary and sufficient condition for the convergence of
the series.
Definition of ® (u ; ft, v}.
B (u ; fj,, v}, the general theta-function with p arguments u, and 2p characters
is the sum of a p-tuple series of exponentials
v,
(u ; /i, v) = 2 exp. [(*) (u, n + v)2 +
n + v)],
where each of the letters n, =(ni, ..,,np), has all integer values (zero included) from
— x to + oo .
The general theorem in regard to ®(u; n, v).
This is
exp. [- H (u ;
(u
/,, v) = exp. [-
. 0 (u ;
establishing a relation between the function ® (M ; p + p', v + v), with arbitrary character-
increments /A', i/', and the function ® (u + 2ar' ; p, v) with the original characters, but
with new arguments w+2tr'. Also #(w; ///, i/7) denotes a function, linear as regards the
arguments u, but quadric as regards p and v ; — Ziripv' is a single term depending
only on p and z/; and the theorem thus is that the two functions differ only by
an exponential factor. The relations between the constants will be obtained in the
course of the investigation.
246 ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. [753
Demonstration.
The truth of the theorem depends on the equality of corresponding exponentials
on the two sides of the equation : viz. substituting for the theta-functions their
values, and comparing the exponents or arguments of the exponentials : writing also
for convenience
G (u + 2w', n + v),
to denote the quadric function (*) (u + 2vr', n + v)2 ; we ought to have
- // (u ; fi, v'} + G (u + 2«/, n + v) + Strip (n + v}
= — 27rifj,v + G (u, n + v + v') + fai (/j, + //) (n + v + v),
or say
H(u ; p!, v'} = G(u + %•&', n + v) — G (u, n + v + v') — fai (n+v + v) /.
In this equation, if true at all, the terms containing n must destroy each other;
assuming that they do so, the equation becomes
H (u • fS, v'}=G(u + 2vr', v)-G (u, v + v'} - 2m (v + v'} //.
Consider first the terms in n: the right-hand side is
= a (u + 2«r')2 + 2/i (u + 2or') (n +v) + b(n + i/)2
- an* - 2hu (n + v + v'} - b (n + v + v )2 - fainp! ;
and the terms herein which contain n thus are
2A (u + 2-cr') n + bn2 + 2bnv
— 2kun — bn* — 2bn (v + v') - fain/*',
= kh-at'n — 2bnv — fainp!,
which, b being symmetrical, may be written
= 2 (2Aw - bv' - irifjf) n,
and these terms will vanish if, and only if
2}n&' — bv — Trip = 0,
a system of p equations connecting •or', //, v'.
Assuming them to be satisfied, the remaining relation,
H (u ; fi, v') = G (u + 2vr', v}-G (u, v+v)- fai (v + v) fj,',
becomes
H (u ; fi, v) = a (u + 2W-')2 + 2h (u + 2vr') n + bi>-
- «w2 - 2hu (v + v'}-b(v + vj - 2m (v + v'} p'.
Here, a and b being symmetrical, we have
a(u + %&')- = aii* + 4ara-'w + 4aCT/2, b (v + i/)2 = bv-
753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 247
and the value therefore is
= 4a (vr'u + TO'2) + 2A (2srV - uv) - b (2i/V + i/2) - ZTTI (v + v) /*'.
On the right-hand side, putting the term in h under the form
-2h(u+ TO') v + 2/m' (2;- + v'}, = - 2 (h) v (u + TO') + 2/tTOy (2i/ + i/),
and the last term under the form
— Trip (2v + P) — Trt'/iV,
the equation becomes
//" (-M ; //, z/) = (4aTO-' — 2 (A) i/') (M + TO-') — irip'v
+ (2/iTO' - 6i/' - irifi) (2i/ 4- i/'),
where the second line vanishes in virtue of the foregoing equation
2/iTO-' - bv' - irifi = 0 ;
the equation thus is
H(u; n', v') = (4aw' - 2 (h) v') (u + TV') - irip'v,
which equation, regarding therein TO-' as a linear function of // and v, shows that
H(u; p!, v) is a function linear as regards u (and containing this only through u + -&'),
but quadric as regards /*', v.
Introducing the new row-letter £', we may write
H (u ; fjt,', i/) = 2f (u + TO') — 7rt'/nV,
viz. the expression on the right-hand side is here assumed as the value of the
function
H (u ; /a', v'\ = G (u + 2w', v) — G (u, v + v')— 2?n' (v + v') p ;
and the theorem then is
exp. [- H (u ; p' ', v')] . B (u + Its' ; p, v) = exp. [- 2?n>z/] . © (u ; /* + /&', v + v'),
where, by what precedes,
2A/BJ-' — &Z/ — TTi'/Lt' = 0,
' - (A) v'-£ = 0,
2p equations for determining the 2p functions TO', £' as linear functions of p, v :
which equations depend on the p(2p + l) constants a, b, h.
Suppose that the resulting values of TO' and £' are
TO' = &>// + co'v',
!»/ / , / /
f = T//U, + 77 1/ ,
where o>, co', ?;, T;' are square-letters ; then, regarding a, b, h as arbitrary, the 4p2
new constants &>, o>', 77, ?;' cannot be all of them arbitrary, but must be connected
by 4/32 — p (2p + l), =p(2p — 1) equations.
248 ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. [753
We may regard a>, o>', 77, 77' as satisfying these p (2p — 1) equations, but as being
otherwise arbitrary ; the foregoing equations then are
2/m' — bv — Trip' = 0,
2avr'-(h)vf-? =0,
TX' = (a/jf + (a'v ' ,
j»' ' i ' '
r — «w* + *? " >
which lead to the equations connecting a, 6, A with a>, CD', 77, 77'.
The first and second equations, substituting for rar' and £' their values, become
(2A&> - iri) fi + (2ha>' -b)v = 0,
(2ao> - T; ) // + (Saw' - (A) - 77') v = 0,
or fji, v being arbitrary, we thus obtain the 4/>2 equations
2aco — 77 =0,
2A&) — iri = 0,
2aw' - 77' - (A) = 0,
2Aw' - b =0,
which are the equations in question. It is to be observed that m is, like the other
symbols, a matrix, viz. it is regarded as containing the matrix unity ; or, what is the
same thing, it denotes
1, 0, 0,...
0, 1, 0,
TTl
We can eliminate a, b, h from these equations and thus obtain the p (2p — 1)
equations before referred to, which connect the 4p2 constants ca, &>', 77, 77'. I give, but
without a complete explanation, the steps of the elimination.
The equation 2a&> — 77 = 0, may be written in the form
that is,
2 (») (a)- (77) = 0,
or since (a) = a] this is
from the original form, and the new form respectively, we find
2 (&)) aw - (&)) 77 = 0, 2 («) a (o>) - (77) o> = 0 ;
and comparing these
(o>) 77 — (77) &) = 0, (first result).
The equation 2a&>' - 77' - (h) = 0, or say (A) = - 77' + 2aw', may be written in the form
753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 249
that is, since a = (a),
A = -(77') + 2 (ft/) a;
and we thence deduce
Aft) = — (77') <w + 2 (a)'} aw.
But from the equation 2aa> — 77 = 0, we have 2 (&/) aw — (&/) 77 = 0, and the equation
thus becomes hco = — (77') w + (&>') 77 ; which, in virtue of 2/i&> — 7n' = 0, becomes
\TTI •= — (77') &) + (&)') 77, (second result).
From the equation above obtained, h = — (77') + 2 (a/) a, we have
Aeo' = _ (T/) a)' + 2 (to') aw' ;
in virtue of 2A&>' — 6 = 0, this becomes — 2 (77') &/ + 4 (&>') aa>' = 6 ; an equation which
may also be written — 2 ((77') &>') + 4 ((&/) aw') = (6), or, what is the same thing,
— 2 (&>') 77' + 4 (to') (a) G)' = (6) ; or since (a) = a and (6) = 6, this is
- 2 (to') 77' + 4 (&)') ato' = b:
and comparing with the original equation
-2 (77') to' + 4 (w') aw' =6,
we obtain
(w') 77' - (77') &>' = 0, (third result).
We have thus the three systems
(<o) 77 — (77) w = 0 , |p (p — 1) equations,
(&)') 77 —(if) to = i7rt, p2
(ft,') 77' -(77') ft,' = 0 , ip(p-l) „
in all p(2p — 1) equations. As to these systems, observe that (w)ij, (n)w, etc., are
all of them matrices of p2 terms; each of the three systems denotes therefore in the
first instance p2 equations, viz. the equations obtained by equating to zero the several
terms of such a matrix : but in the first system each diagonal term so equated to
zero gives the identity 0 = 0; and equating to zero the terms which are symmetrical
in regard to the diagonal we obtain twice over, in the forms P = 0, and — P = 0,
one and the same equation ; the number of equations is thus diminished from p2 to
£p (p — 1); and similarly in the third system the number of equations is =|p(p — 1):
but for the second system the number of equations is really =p2. It is hardly
necessary to remark that in this second system ^iri is as before regarded as a matrix.
The foregoing three systems of equations are in fact the equations (6) p. 4 of
Dr Schottky's work.
Cambridge, 12 July, 1880.
c. XT. 32
250 [754
754
ON THE CONNEXION OF CERTAIN FORMULAE IN ELLIPTIC
FUNCTIONS.
[From the Messenger of Mathematics, vol. ix. (1880), pp. 23 — 25.]
IN reference to a like question in the theory of the double ^--functions, it is
interesting to show that (if not completely, at least very nearly) the single formula
©'a , . , ®(u-a)
II ft*, a) = u .~ - + i log s-7 - ; ,
©a ° 0 (w + a)
that is,
sn acnadn asn2w du ©'a
f kz sn ac
I
Jo !-
— .11 ___ i_ 1 IOCT v
©a g©w
leads not only to the relation
r r
_^ du dusrfu,
Jo Jo
between the functions ©, sn, but also to the addition-equation for the function sn.
Writing in the equation a indefinitely small, and assuming only that sna, cna,
dn a then become a, 1, 1, respectively, the equation is
f a&'Q
sn2w du = u ^^ + \ log
Jo ©0 6
©"0 &u
= ua -pr-_- — a ^— - ,
©0 @M
that is,
©'% @"0 f .
75T = w "FT7T — ^'2 I <WI Sn2 U,
®u ©0 J0
or, integrating from u = 0, this is
©"o /' r
log Su=C + %u* n - A;2 du du sn2 it,
v^U ./ n .' n
754] ON THE CONNEXION OF CERTAIN FORMULAE IN ELLIPTIC FUNCTIONS. 251
which, except as regards the determination of the constants, is the required equation
for log ®u.
Next, differentiating twice the equation for II (u, a), and once the equation obtained
for ^— , we have
d
1 / -i, __ ft ) _ 1 (71 -L. n\
and
i* — _ __ 1/& gfi-2 7/
@a (M)Q
where, for shortness, ~ u is written to denote ^ , and the like in
©2 &2u
the first equation ; the right-hand side of the first equation therefore is
— ^ &2 {sn2 (u — a) — sn2 (u + a)},
or the equation becomes
, d sn2 u
2 sn a en a dn a -=- .,- — j— — = sn- (u + a) — sn- (u — a).
du 1 — k- sn2 tt sn2 a .
that is,
4 sn u sn' it sn a en a dn a
—T- Tn — , = sn- (M + a) - sn2 (ti - a).
(1 — k- sn2 M sn2 a)2
The numerator on the left-hand side must be a symmetrical function of u, a,
and hence (even if the value of sn' u were unknown) it would appear that sn' u must
be a mere constant multiple of en u dn u ; assuming, however, the actual value,
sn' u = en u dn u, the formula is
4 sn u en u dn u sn a en a dn a
(1 — k2 sn2 u sn2 of
= sn2 (u + a) — sn2 (u — a)
= {sn (u + a) + sn (u — a)} {sn (u + a) — sn (u — a)}.
The factor {sn (u + a) + sn (u — a)} becomes = 2 sn u for a = 0, and this suggests that
the factor snu on the left-hand side is a factor of {sn (u + a) + sn (u — a)}. That cnu
is not a factor hereof would follow from the properties of the period K ; viz. for
u = K, en u = 0, but {sn (u + a) + sn (u —a)}, = 2 sn (K + a) is not = 0 ; and, similarly, that
dn u is not a factor from the properties of the period iK ; hence, en u, dn u belong
to the other factor {sn (u + a) — sn(u — a)}, and by symmetry en a, dn a belong to the
first-mentioned factor. And we are thus led to assume
sn (u + a) + sn (u — a) = 2M sn u en a dn a,
sn (u + a) — sn (u — a) = 2M' sn a en u dn u,
where
denom. = l~k* sn2 a sn2 u,
and MM' = l. Some further investigation is wanting to show that M and M' are
constants, but assuming that they are so and each = 1, the formulas give at once the
ordinary expression for sn (u + a) ; that is, we have the addition-equation for the
function sn.
32—2
252
[755
755.
ON THE MATRIX ( a, b ), AND IN CONNEXION THEREWITH
c, d
ax + b
THE FUNCTION
cx + d '
[From the Messenger of Mathematics, vol. ix. (1880), pp. 104 — 109.]
IN the preceding paper, [due to Prof. W. W. Johnson,] the theory of the symbolic
/"/ /T* L| , /I
powers and roots of the function ^ is developed in a complete and satisfactory
manner; the results in the main agreeing with those obtained in the original memoir,
Babbage, " On Trigonometrical Series," Memoirs of the Analytical Society (1813), Note I.
pp. 47 — 50, and which are to some extent reproduced in my " Memoir on the Theory
of Matrices," Phil. Trans., t. cxLVin. (1858), pp. 17—37, [152]. I had recently
occasion to reconsider the question, and have obtained for the nth function <frnx, where
$x = — — -j , a form which, although substantially identical with Babbage's, is a more
CX -|- Ct
compact and convenient one ; viz. taking A, to be determined by the quadric equation
(A + l)2
the form is
ad — be'
(Xn+1 -
(\w-l)(cx+d)+(\n-\)( ex -aY
The question is, in effect, that of the determination of the nth power of the
matrix ( a, b ); viz. in the notation of matrices
c, d
c, d
755]
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH.
UNIV]
.CALIF
253
means the two equations xl = ax + by, y1 = cx + dy, and then if x2, y2 are derived in
like manner from as1} yi} that is, if xz=axl + byl, y2=cx1 + dy1, and so on, asn, yn will
be linear functions of x, y; say we have xn = anx + bny, yn = cnx + dny : and the nib.
power of ( a, b ) is, in fact, the matrix ( an, bn ).
) Cv
Cjj j
In particular, we have
a, b )2, =( aa, 6a ), =( a2
C, C
and hence the identity
(a, b )2 — (a
c, d
viz. this means that the matrix
( a2 — (a + d) a + ad — be, b2 - (a + d) b
a, b ) + (ad-bc)( 1, 0 ) = 0;
c, d
0, 1
) = ( 0, 0 ),
d-bc 0,0
or, what is the same thing, that each term of the left-hand matrix is =0; which is
at once verified by substituting for a2, b2, c2, d2 their foregoing values.
The explanation just given will make the notation intelligible and show in a
general way how a matrix may be worked in Jike manner with a single quantity:
the theory is more fully developed in my Memoir above referred to. I proceed
with the solution in the algorithm of matrices. Writing for shortness M=( a, b ),
! c, d
the identity is
M2 - (a + d) M+ (ad - be) = 0,
the matrix (1,0) being in the theory regarded as = 1 ; viz. M is determined by
0, 1
a quadric equation ; and we have consequently Mn = a linear function of M. Writing
this in the form
the unknown coefficients A, B can be at once obtained in terms of a, /3, the roots
of the equation
v? — (a + d) u + ad — bc — 0,
viz. we have
a.n - Aa + B = 0,
/3n - A/3 + B = 0;
or more simply from these equations, and the equation for Mn, eliminating a, /3, we
have
Mn, M, 1 =0;
an , a, 1
254
that is,
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH.
Mn(a-/3)-M (<xn - j3n) + a/3 (an~l - /S71"1) = 0.
[755
But instead of a, @, it is convenient to introduce the ratio X of the two roots,
say we have a = X/3 ; we thence find
(X + l)/3 = a + d,
X/32 = ad — be,
giving
(X+l)2 = (a + d)2
X ad — be
for the determination of X, and then
Q _a + d
P*X+1'
(a + d) X
a =
X+l
The equation thus becomes
Mn(\-
or we have
- X) £»+i = 0,
Mn = .
It is convenient to multiply the numerator and denominator by X + l, viz. we
thus have
The exterior factor is here
1 fa + d\n~l
X2 - 1 VX + 1
moreover (X + l)/3 is =a + c?: hence
M=( a, b ),
c,
and
the formula thus is
#*- —
= ( a, b )-( a + d, 0 ), =( -d, b );
c, d
0 , a + d
c , — a
a, 6 )+(x»-X)( -rf, 6 )
, |
c, d \ c , -a
viz. we have thus the values of the several terms of the ?ith matrix
M» = ( an, bn );
cn , dn
755] ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 255
and, if instead of these we consider the combinations anx + bn and cnx + dn, we then
obtain
(X» - X) (- dx + b)},
-
A/ ~~ JL
and in dividing the first of these by the second, the exterior factor disappears.
It is to be remarked that, if n = 0, the formulae become as they should do a^ + b0 = x,
CoX + d0=l; and if n = 1, they become a-jc + ^ = ax + b, c1x + d1 = cx + d.
If X'M —1=0, where m, the least exponent for which this equation is satisfied, is
for the moment taken to be greater than 2, the terms in { } are
(X - 1) (ax + b ) + (1 - X) (- dx + b),
and
(X - 1) (ex + d) + (1 - X) ( ex -a);
viz. these are (X — l)(a + d)x, and (X — l)(a + d), or if for (X— l)(a + d) we write
(X2— l)r— -., , the formulae become for n=m
A. ~t~ J.
i / CL ~T~ CL\
&mx + um = - - - 1 X,
\A,+ I/
viz. we have here
n v I
"'W'*' I
- j~ >
cmx + dm
or the function is periodic of the with order. Writing for shortness ^= — , s beino-
any integer not = 0, and prime to n, we have X = cos 2S- + i sin 2^-, hence
1 + X = 2 cos S- (cos § + i sin &),
n + x)2
or — - — =4cos2^; consequently, in order to the function being periodic of the nth
order, the relation between the coefficients is
„ STT (a + d)2
4 cos2 — = v -7 — ,-.
n ad — be
The formula extends to the case m = 2, viz. cos | (STT) = 0, or the condition is
a + d = Q. But here X + 1 = 0, and the case requires to be separately verified. Recurring
to the original expression for M2, we see that, for a + d = Q, this becomes
that is,
a2 + be,
0 ,
, =(a*+bc)
1, 0
0, 1
ow ^ „, = x>
or the result is thus verified.
256 ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. [755
But the case m = l is a very remarkable one; we have here X=l, and the
relation between the coefficients is thus (a + d)"2 = 4 (ad — be), or what is the same thing
(a — d)2 + 4<bc = 0. And then determining the values for X = 1 of the vanishing fractions
which enter into the formula?, we find
anx + 6n = — (a + d)11'1 {(n + 1) (ax + b) + (n - 1) (- dot + &)},
cnx + dn = 2- (a + d)n~l {(n + 1) (cos + d) + (n - 1) ( ex - a)\,
or as these may also be written
anx + 6n = — (a + dy1'1 {x [n (a - d) + (a + d)] + 2nb],
cnx + dn = — (a + d)n~l {x . 2nc + [- n (a -d) + a + d]},
which for n = 0, become as they should do a0# + b0 = x, c0x + d0=l, and for n=l they
become a1x + b1 = ax + b, c-p + dl = ex + d. We thus do not have a* 7* = x and the
c^x + d±
function is not periodic of any order. This remarkable case is noticed by Mr Moulton
in his edition (2nd edition, 1872) of Boole's Finite Differences.
If to satisfy the given relation (a - d)- + 4>bc = 0, we write 26 = k (a - d), 2c = - - (a - d).
K
then the function of x is
and the formula for the ?ith function are
anx + l>n = cp(a + dy1'1 {(a + d) x + n (a - d) (x + &)],
Cnx + dn = ^(a + d)n~i {(a + d) -n(a- d) (j +
^ ( \P
which may be verified successively for the different values of n.
Reverting to the general case, suppose n = oo , and let u be the value of ^°° (x).
Supposing that the modulus of X is not =1, we have \n indefinitely large or
indefinitely small. In the former case, we obtain
_
X (ex + d) + ( ex -a)' ' c(\,+
which, observing that the equation in X may be written
\a-d _6(X + 1)
cX+l~ \d-a '
755] ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 257
is independent of x, and equal to either of these equal quantities ; and if from these
two values of u we eliminate \, we obtain for u the quadric equation
CM2 — (a — d) u — b = 0,
that is,
au + b
u =
cu + d'
as is, in fact, obvious from the consideration that n being indefinitely large the nth and
O + l)th functions must be equal to each other. In the latter case, as \n is indefinitely
small, we have the like formulas, and we obtain for u the same quadric equation :
the two values of u are however not the same, but (as is easily shown) their product
is = — 6-r-c; u is therefore the other root of the quadric equation. Hence, as n
increases, the function <f>nx continually approximates to one or the other of the roots
of this quadric equation. The equation has equal roots if (a — d}2 + 4tbc = 0, which is
1 — 26
the relation existing in the above-mentioned special case ; and here u = — (a — d), = -, ,
£c d — d
which result is also given by the formulae of the special case on writing therein n = oo .
C. XL
33
258 [756
756.
A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY
QUANTITIES.
[From the Messenger of Mathematics, vol. x. (1881), pp. 1 — 3.]
LET A, B, C be given imaginary quantities, and let it be required to construct
the roots of the quadric equation
I
X-A^X-B ' X-C
The equation is
that is,
and we have therefore
3Z - (A + B + C) = ± J{(A + B+C)*-3(BC+CA + AB)},
or as this may be written
X = £ (A +B+ C) ± V(i(4 + Bo> + Ctf).%(A + Bo>- + Co.)},
where w is an imaginary cube root of unity,
= cos 120° + i sin 120° suppose.
Taking an arbitrary point 0 as the origin, let the imaginary quantity A, =a + a.'i
suppose, be represented by the point A, coordinates a and a' ; and in like manner
the imaginary quantities B and C by the points B and C respectively.
Then Bat, Bw2 are represented by points Blt B2, obtained by rotating the point
B about the origin through angles of 120° and 240° respectively; (7&>2, Ceo are repre-
756] A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY QUANTITIES. 259
sented by points C1} (72 obtained by rotating the point C about the origin through
angles of 240° and 480° (= 120°) respectively : and
$(A + B + C), i(A+Ba,+ C<o2), $(A+Be>*+Ca>)
are represented by the points G, Glt G2 which are the C.G.'s of the triangles ABC,
AB1C1, AB2C2 respectively. The formula therefore is
where, if a, a' are the coordinates of G, then OG is written to denote the imaginary
quantity a + a't; and the like as regards OGt, OG2. Taking ^'(OG1.OG2) = OH, we then
have H a point such, that the distance OH from the origin is = geometric mean of
the distances OG1} OG2, and that the radial direction* of the distance OH bisects
the radial directions of the distances OG1} OG2 respectively. Finally, measuring off
from G in the radial direction OH, and in the opposite radial direction, the distances
GX', GX" each = OH; we have the two points X', X" representing the two roots X,
The construction is somewhat simplified if we take for the origin the point G ;
for then OG = 0, and we have X = ± \/(GGl . GG2), so that the points X', X" are in
fact the point H, and the opposite point in regard to G.
The theory of the more general equation
P 9 r _0
X-A^X-B X-C~
(p, q, r real) is somewhat similar, but the construction is less simple ; we have
(p + q + r) X2 - {(q + r) A + (r + p) B + (p + q} C } X + pBC + qCA + rAB = Q.
Writing herein q + r, r+p, p + q = l, m, n, the equation becomes
(I + TO + n) X2 - 2 (IA + mB + nC) X + (- 1 + m + n) BC + (I - m + n) GA+(l + m- n) AB = 0,.
that is,
{(I + m + n} X -lA-mB- nC}2
= (IA + mB + nC)2 + {I2 - (m + w)2} BC + {m2 - (n + I)2} CA + {n2 -(1 + m)2} AB.
Here the right-hand side is
= 12A2 + m2& + n2C2 + (I2 - m2 - n2) BC + (-l2 + m2 - n2) CA + (-l2-m2 + n2) AB,
which is
= - I2 (C - A ) (A - B) - m2 (A - B) (B - C) - n2 (C - A) (A - B),
and consequently is a product of two linear factors; these, in fact, are
{PA + %(-l2-m2 + n2± VA) 5 + £ (- I2 + m2 - n2 + VA) C},
* Radial direction is, I think, a convenient expression for the direction of a line considered as drawn as
a radius of a circle from the centre, and not as a diameter in two opposite radial directions.
33—2
260 A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY QUANTITIES. [756
where
A = I4 + m4 + w4 - 2ra2tt2 - 2w2Z2 - 2Z2m2.
It is to be observed that A, = (I2 - m2 — ft2)2 — 4ra2w2, is negative ; hence, calling
the factors fA+gB + hC, f'A+g'B + h'C respectively, the coefficients /, g, h, and
/', g', h' are imaginary ; moreover f+ g + h = 0, /' + g' + h' = 0.
The values of X thus are
and then passing to the geometrical representation, we have — j— - represented
by the point which is the C.G. of weights I, m, n at the points A, B, C respectively;
on account of the imaginary values of the coefficients the construction is not immedi
ately applicable to the factors
fA+gB + hC, f'A+g'B + h'C;
but a construction, such as was used for the factors
A + wB + a>'-C, A + w2B + coC,
might be found without difficulty.
757] 261
757.
ON A SMITH'S PRIZE QUESTION, RELATING TO POTENTIALS.
[From the Messenger of Mathematics, vol. xi. (1882), pp. 15 — 18.]
A SPHERICAL shell is divided by a plane into two segments A and B, one of them
so small that it may be regarded as a plane disk: trace the curves which exhibit the
potentials of the two segments and of the whole shell respectively, in regard to a point
P moving along the axis of symmetry of the two segments.
Criticise the following argument :
The potential of the segment A in regard to a point P, coordinates (x, y, z), is
one and the same function of (x, y, z) whatever be the position of P ; similarly the
potential of the segment B in regard to the same point P is one and the same function
of (x, y, z) whatever be the position of P: hence the potential of the whole shell in
regard to the point P is one and the same function of (x, y, z) whatever be the
position of P.
The question is taken from my memoir "On Prepotentials," Phil. Trans, vol. 165
(1875), pp. 675—774, [607]; and the figure of the curves is given p. 689*. There is
no difficulty in tracing them by means of the expression for the potential of a plane
circular disk in regard to a point on its axis of symmetry: it was in order that
they might be so traced, that one of the segments was taken to be small ; but I
had overlooked the circumstance that the formula for the disk is in fact only a
particular case of a similar and equally simple formula for the spherical segment :
viz. (as was found in one of the papers) the potential of a spherical segment in
£\
regard to a point on the axis is = (p1 <- p2), where p, p-^ , p2 are the distances of
the attracted point from the centre of the sphere and from the centre and the circum
ference respectively of the segment. The segments might therefore just as well have
been any two segments whatever, or (to take the most symmetrical case) they might
have been hemispheres.
As to the argument : the assertion in regard to the potential of the segment
[* This Collection, vol. ix. p. 333.]
262 ON A SMITH'S PRIZE QUESTION, RELATING TO POTENTIALS. [757
A is based upon the consideration of this segment alone ; and, on the ground that
we can without crossing the segment pass from any one position of P to any other
position of P, it is inferred that the potential is one and the same function of the
coordinates, whatever be the position of P : it is therefore unassailable by any
considerations in relation to the non-existent segment B. Similarly the assertion in
regard to the potential of the segment B is based upon the consideration of this
segment alone, and it is unassailable upon any considerations in regard to the non
existent segment A : the potential of the whole sphere is certainly the sum of the
potentials of the segments A and B: it is therefore altogether off the purpose to
object that in the case of the whole sphere we cannot pass from a point outside
the sphere to a point inside the sphere without crossing one or other of the segments
A and B. I consider that the two assertions are each of them true, and that the
conclusion is a legitimate one, but it is true only in the sense in which a + x + \/[(a — a?)2]
is one and the same function of x whatever be the value of x : this is so, if
<v/[(& — x}2] denotes indifferently or successively the two functions ±(a — x): but if, a
and x being real, \/[(a — x)'2] is taken to mean the positive value, then the function
a + x + V[(a — #)2] ig = 2a or = 2x according as a — x is positive or negative.
Fig. 1.
In further illustration, let the dark line of fig. 1 represent the intersection of
an unclosed surface, or segment, by the plane of xz taken to be that of the paper,
and consider the potential of the segment in regard to a point P in the plane of
the paper, coordinates x, z. We have the potential V defined as a function of x, z
by an equation V= a definite integral, depending on the parameters x, z, and being in
general a transcendental function of (x, z); V is a real, one-valued, finite, continuous
function of x, z : in particular, if the point P, moving in any manner, traverses the
dark line, there is not any discontinuity in the value of V. There is however in
this case a discontinuity in the differential coefficients of V: if to fix the ideas we
imagine P moving parallel to the axis of x, so that z is taken to be constant and
V a function of x only, then when the path of P crosses the black line there is
dV
in general an abrupt change of value in -=- . Taking V as a coordinate y at right
dx
angles to the plane of the paper, a section by any plane parallel to that of xy is
(when the trace of the plane upon that of xz does not meet the dark line) a
continuous curve ; but when the trace meets the dark line, then for this value of x
there is an abrupt change of direction in the section.
757] ON A SMITH'S PRIZE QUESTION, RELATING TO POTENTIALS. 263
If (as may very well happen in particular cases) V is algebraically determinate,
then, qua one-valued function of (x, z}, V is not any root y at pleasure of an
algebraical equation <f> (x, y, z} — 0, but it is for any given values of (x, z}, some one
determinate root y± of this equation : and we thus see how in this case the before-
dV
mentioned discontinuity in the value of -7- must arise : viz. when the trace of the
plane meets the dark line the section is a curve having a double point; and, for
the positions of P on the two sides of the dark line, we have V the ordinate
belonging to different branches of the curve of section. If the path of P passes
through an extremity of the dark line, then the curve of section will, instead of a
double point, have in general a cusp ; and when the path of P does not cross the
dark line, then the curve of section is a continuous line without singularity. It may
be added that the surface $ (x, y, z) = 0 must have a nodal line which as to a certain
finite portion thereof is crunodal, giving the before-mentioned double points of the
sections, but as to the residue thereof is acnodal or isolated.
It may happen that (the surface being algebraical) any particular section thereof,
instead of being a single curve having a double point as above, breaks up into two
distinct curves, so that for the two positions of P, we have V the ordinate of two
distinct curves : and this is what really happens in the case of P a point on the
axis of a circular disk or a spherical segment : thus in the case of the disk, taking
c for the radius, and x for the distance from the centre of the disk, the formula
is V— 2-Tr {V(c2 + tf2) + x} ; or writing F-r-27r=;y, the section is made up of the two
distinct hyperbolas y (y — 2x) = c", and y (y 4- 2#) = c2.
It may be remarked that in each case, it is only for P on the axis that the
potential is algebraical.
In the case of the hemispheres, drawing OM a radius at right angles to the
axis, the formula for the potential of an axial point P is of the form
or writing F=27ry we have for the hemisphere A, the curve (1) or (2) according
as (x — a) is positive or negative ; and for the hemisphere B the curve (3) or (4)
according as x + a is positive or negative ; viz. the equations are
\ / t/ y»i*^ ' ^ y j '
x
/ C\\ f // 9 i *) \ I / \1
(2) y = - w(a + x~} + (* - *)}.
#
/ O \ f // 2 **\ / i \ I
264 ON A SMITH'S PRIZE QUESTION, RELATING TO POTENTIALS. [757
being four cubic curves. The whole curve (1) is shown in fig. 2. and the others are
Fig. 2.
y
equal or opposite curves: the rationalised equation of (1) is in fact
2a2 (y + a)
(y + <#-#>
and by writing — a for a, and in each equation — cc for x, we have the rational
equations of the other three curves.
But, drawing only the required portions of the curves, we have fig. 3 exhibiting
Fig. 3.
the potentials of the two hemispheres A and B ; and also the discontinuous potential
of the whole shell, the ordinate for this last being the sum of the ordinates for the
two hemispheres respectively.
758]
265
758.
SOLUTION OF A SENATE-HOUSE PROBLEM.
[From the Messenger of Mathematics, vol. xi. (1882), pp. 23 — 25.]
PROVE that, if a+b + c = 0 and x + y + z = 0, then
4 (ax + by + czf
- 3 (ax + by + cz) (a- + ft2 + c2) (x2 + y- + z2)
— 54iabcxyz = 0.
I do not know the origin of this identity, nor do I see any very simple way
of proving it: that which seems the most straightforward way is to transform the
third line, which, omitting the factor — 2, is
1, 1, 1 .1 1, 1, 1
a , b , c x , y , z
a2, b2, c2 x2, y2, z2
3, a +b +c , a- +b- +c-
x + y + z , ax + by + cz , d-x + b-y + c-z
x2 + y2 + z", ax- + by- + cz2, a-x- + b2y2 + c-z'
and therefore when a + b + c = 0 and x + y + z = 0, is
= 3 (ax + by + cz ) (a-x2 + b2y2 + c2z2)
— 3 (a-x + b2y + c-z} (ax- + by- + cz- )
- (ax +by +cz) (a- + b2 + c2) (x2 + y2 + z2) ;
c. xi. 34
266 SOLUTION OF A SENATE-HOUSE PROBLEM. [758
or, as this may be written,
= 6 (ax + by + cz ) (a2x* + fey + c2z2)
- (ax + by + cz) (or + b2 + c2) (a? + y2 + z2)
-3 (ax + by + cz) (a2x2 + fey + c2*2)
- 3 (a-x + b-y + c2z) (ax2 + by2 + cz2 ).
Here the third and fourth lines, omitting the factor - 3, are
2 (a?a? + fey + c3z3) + (ab2 + a-b) (xy* + x2y) + (ac2 + a*c) (as* + x-z) + (be- + b2c) (yz2 + ifz\
where, in virtue of the two relations, each of the last three product-terms is = abcxyz,
and the whole is thus
= 2 (a3#3 + fey + c3^)
+ Sabcxyz.
The product of the two determinants is thus
= 6 (ax + by + cz) (aW + fey + C2z-)
- (ax + by + cz) (a2 + b2 + c2) (x2 + y2 + z2)
- 6 (a3^3 + &y + c3*3)
— 9 abcxyz ;
and this being so the identity to be verified is
4 (ax + by + cz)3
+ (- 3 + 2 =) - 1 (ax + by + cz) (a2 + b2 + c2)
- 12 (ax + by + cz) (aW + fey
+ (18 - 54 =) - SQabcxyz = 0.
We have here the terms
12 (aW + fey + csz3 - Sabcxyz),
= I2(ax + by + cz) (aW + fey + c2^2 - bcyz - cazx - abxy),
so that the left-hand side is now divisible by ax + by + cz, and throwing out this
factor the equation becomes
4 (ax + by + cz)2
- (a2 + b2 + c2) (x2 + y2 + z2)
+ 12 (a2^2 + fey + c2z2 - bcyz - cazx - abxy) = 0 ;
758] SOLUTION or A SENATE-HOUSE PROBLEM. 267
or, as this may be written,
4 (a?a? + 62t/2 + cV — bcyz — cazx — abxy)
- (a2 + 62 + c2) (x- + y- + z-} = 0,
which under the assumed relations a + b + c = 0, x + y + z = 0 may be verified without
difficulty. It may be remarked that we have identically
8 (a2*-2 + 62y2 + c-z- — bcyz — cazx — abxy}
- 2 (a2 + b- + c2) O2 + f + 2*}
x( 3a2 - 62 - c2 4- 26c - 2ca - 2a6)
= (x + y + z) + ?/ (- a2 + 3fr- - c2 - 26c + 2ca - 2a6)
z - a2 - 62 + 3c2 - 26c - 2ca
/ a ( 3x2 — y- — 22 + 2yz — 2zx — Zxy) \
+ (a + b + c) J 4- 6 (— x- + 3y* - z2 — 2yz + 2zx — 2xy) L ,
4- c (— x- — y- + 82- — 2yz — Zzx + 2xy)
\ /
which is a more complete form of the last-mentioned theorem.
34—2
268
[759
759.
ILLUSTRATION OF A THEOREM IN THE THEORY OF
EQUATIONS.
[From the Messenger of Mathematics, vol. xi. (1882), pp. Ill — 113.]
THE knowledge of the value of an unsymmetrical function of the roots of a
numerical equation adds something to what is given by the equation itself; but it
may or may not add anything to what is given by the equation itself in regard to
each root separately. If, for instance, a, ft, 7 being the roots of a cubic equation,
it is known that a- ft + ft2y + y-ct = a given value k, then a, ft, 7 must denote the
roots, taken not in any order whatever, nor yet in a uniquely determinate order, but
with a certain restriction as to order, viz. if the roots in a certain order are a, b, c,
these roots being such that a-b + Ire + c~a = k, then clearly the relation in question
3? ft + /327 + 72a = k, will be satisfied if a, ft, 7 = a, b, c, or = b, c, a, or = c, a, b
(but not if a, ft, 7 = b, a, c, or = either of the remaining two arrangements) ; the
relation thus allows a to be = a, or = b, or = c ; that is, a is = any one at
pleasure of the roots of the cubic equation, and it is thus determined by the cubic
equation, and not by any inferior equation ; but a being known, the other two roots
ft and 7 will be uniquely, and therefore rationally, determined.
It is worth while to see how the result works out ; suppose, for greater simplicity,
the cubic equation is a? — 7x + 6 = 0 having roots (1, 2, — 3), and that the given
relation is a-ft + ft'2y + 7-0. — — 1, then the cubic equation gives
a + ft + 7 = 0, aft + 017 + £7 = - 7, afty = - 6,
and we have, besides, the relation in question
a.- ft + /327 + 72a = - 1 ;
eliminating 7 we have
a2 + a/3 + /32=7, «#(« + £) = 6, a3 + 3«-/3 - /33 + 1 = 0;
or, as it is convenient to write these equations,
ft- + aft + 1? - 7 = 0,
ft"-+ a/3-- =0,
B
ft3 - 3a2/3 - a* - 1 = 0.
759] ILLUSTRATION OF A THEOREM IN THE THEORY OF EQUATIONS. 269
If from these equations we eliminate ft, we obtain two equations in a, which it
might be supposed would determine a uniquely; but, by what precedes, a is any
root at pleasure of the cubic equation and can thus be determined only by the
cubic equation itself, and it follows that any equation obtained by the elimination
of ft must contain as a factor the cubic function a3 — 7 a + 6, and be thus of the form
M(a?— 7a+ 0) = 0, where M is a function of a; one result of the elimination is
a3 — 7a + 6 = 0, and every other result is of the form just referred to, M (a3 — 7a + 6) = 0 ;
hence we have definitely a3— 7a+6 = 0, viz. the roots of the equation J/ = 0 do not
apply to the question.
a
In verification, observe that the first and second equations give a2 — 7 = - , that
is, a2 — 6a + 7 = 0. To eliminate ft from the first and third equations we first find
a/32 + (4a2 - 7) ft + a3 + 1 = 0,
or say
/ 7\ r 1
ft- + 4a }ft + a2 + - = 0,
V a/ a
and combining herewith the first equation
we obtain
/ 7\ i
\ a.J a.
that is,
-3«2+7'
substituting in the first equation,
(7a + l)3
+ a(7a+l)(-3a2 + 7)
that is,
49 14 1
-21-3+49 +7
9 0-105 +343 -343
90-126-3+441 +21-342,
or, dividing by 3,
3a6 - 42a4 - a3 + 147a2 + 7a - 114 = 0,
which, in fact, is
(a? - 7 a + 6) (3a3 - 21a - 19) = 0,
of the form in question M (a3— 7a + 6) = 0. Thus a has any one at pleasure of the
7ff -i- 1
three values 1, 2. —3, but a being known we have 8 = — rr— — =. , and thence
— 3a2 + 7
-7a-l =3a3-14a-l
*" +-3a2+7' -3a2 + 7
in particular, as a = 1, then ft = 2 and 7 = — 3.
270
[760
760.
REDUCTION OF _ TO ELLIPTIC INTEGRALS.
[From the Messenger of Mathematics, vol. XL (1882), pp. 142, 143.]
WEITIXG s, c, d for the sn, en, and dn of u to a modulus k, which will be deter
mined, and denoting by 6 a constant which will also be determined, the formula of
reduction is
- 1 + Oscd
oc ^—
1 + Bscd '
To find from this the value of y, = ty(l - a?), putting for shortness X = 6scd, the
formula is # = —-—— and we thence have
l + A
where
1 + 3Z2 = 1 + 3(9V (1 - s2) (1 - k2s2),
= 1 + SPs- -3^(1+ k2) s4 +
may be put equal to (1 + #2s2)3, that is,
= 1
viz. this will be the case if
30' =
that is,
<92 = -l-A;2, ^ = 3^2;
these give
&-&+ 1=0;
that is, kz = w, if &> = — |- + ^' ^3, an imaginary cube root of unity ; and then
$2 = — 1 + (a, =o)- (QJ- — <w), = — io)2 \/3 ;
• f dx
7601 KEDUCTION OF 7- =*-* TO ELLIPTIC INTEGRALS. 271
J J (1 — #r)3
that is,
e= +
as may be verified by squaring.
Hence finally, 6 and k denoting the values just obtained,
_ - 1 + Oscd
X 1 + Oscd '
l + 0scd '
or, writing as before, X = Oscd, we have
dx.
~
whence
and then
dX = 0 jl - 2 (1 + #) s2 + 3^54} ^M, =0(1 + 08sa)3 dw ;
that is,
or say
the required formula.
272 [76
761.
ON THE THEOREM OF THE FINITE NUMBER OF THE
COVARIANTS OF A BINARY QUANTIC.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvn. (1881
pp. 137—147.]
GORDAX'S proof, the only one hitherto given, is based upon the theory of derivative
(Uebereinanderschiebungen). It is shown that the irreducible covariants of the binar
quantic f are included in the series
(f,fr.(f,fY, •••(/ h\ (/, h)*, ...
of the derivatives of the quantic upon itself or upon some other covariant, and tha
the number of the irreducible covariants thus obtained is finite. And not only sc
but for the quintic and the sextic the complete systems were formed, and the number
shown to be = 23 and 26 respectively.
It would seem that there ought to be a more simple proof based upon the con
sideration of the fundamental covariants : for the cubic (a, b, c, d$x, y)z, these an
the cubic itself (a, ...$>, y}*, the Hessian (ac- b2, ...$#, y)2, and the cubicovarian
(azd — 3a6c + 263, ...]£#, y)3; and so in general for the quantic (a, ...$#, y)n, we have i
series of fundamental covariants the leading coefficients whereof are the seminvariants
a, ac - b-, a-d - Sabc + 263, a3e - 4a26c£ + Qab2c - 364, &c.
It is known that every covariant can be expressed as a rational function of these, 01
more precisely that every covariant multiplied by a positive integral power of the
quantic itself can be expressed as a rational and integral function of the fundamenta
covariants, and we may for the covariants substitute their leading coefficients, or saj
the seminvariants ; hence, every seminvariant is a rational function of the fundamenta
seminvariants, and more precisely, every seminvariant multiplied by a positive integral
761] ON THE THEOREM OF THE FINITE NUMBER OF CO VARIANTS. 273
power of the first coefficient a is a rational function of the fundamental seminvariants.
Thus, in the case of the cubic, we have the discriminant V,
= a*d- - Qabcd + 4ac3 + 4>b3d - 362c2,
obtained from
a, ac-b2, a-d - Sabc + 263,
by the formula
«2 v = (a?d - Sabc + 263)2 + 4 (ac - 62)3,
and it is easily shown that this invariant V is the only new covariant thus obtainable,
and that every other covariant is thus a rational and integral function of the
irreducible covariants, the leading coefficients of which are
a, ac — b2, a-d — 3abc + 263,
and V. It appears a truism, and it might be thought that it would be, if not easy,
at least practicable, to show for a quantic of any given finite order n, that we can
in this manner, as rational functions of the n — 1 seminvariants, obtain only a finite
number of new seminvariants, so that all the seminvariants would be expressible as
rational and integral functions of a finite number of seminvariants; and, consequently,
all the covariants be expressible as rational and integral functions of a finite number
of irreducible covariants. But the large number, 23, of the covariants of the quintic
is enough to show that the proof, even if it could be carried out, would involve
algebraical operations of great complexity.
The theory may be considered from a different point of view, in connexion with
the root-form a (x — ay) (x — fty) . . . , or say (ac — a.) (x — 0) . . . of the quantic ; we have
here what may be called the monomial form of covariant, viz. the general monomial
form is
(a - /3)m (a - y)n (/3 - 7)? ...(as- a)3(x-/3)r ...,
where in all the factors (whether a — /3 or x — a) which contain a, in all the factors
which contain /3, . . . , and so for each root in succession, the sum of the indices has
one and the same value, = 0 suppose. Thus, for the cubic
we have the monomial covariants
<«-£)(« -7) (£ -7).
and so for the quartic
(x -a)(x- /3) (x -y)(x- B),
we have the monomial invariants
Observe that the monomial form is considered as essential ; a syzygetic function of
c. XL 35
274 ON THE THEOREM OF THE FINITE NUMBER [761
two or more monomials is not a monomial, and we are thus in no wise concerned
with identities such as
or
(a - 8) (/3 - 7) - (£ - 8) (a - 7) + (7 - 8) (a - /3) = 0 ;
notwithstanding these syzygies respectively,
(/3 — 7) (x — a), (a — 7) (as — /9), and (a — /3) (# — 7 )
are regarded as independent covariants of the cubic, and
(a_S)(/3_7), (/3-S)(a-7), and (7- 8) («_£),
as independent invariants of the quartic.
It is only when a monomial covariant is equal to a power or product of simple
or other powers of lower monomial covariants that it is regarded as a function of
these lower monomial covariants and therefore as not irreducible. Thus
is a reducible monomial covariant, expressible in terms of the lower irreducible
monomial covariants
-S and a-
The theorem of the finite number of the irreducible monomial covariants (as just
explained) of the root-quantic is a question of the same kind as, but entirely distinct
from, that of the finite number of the covariants of the quantic in the ordinary form ;
and there are thus the two questions ; (A), that of the finite number of the irreducible
monomial covariants of the root-quantic; and (C), that of the finite number of the
irreducible covariants of the ordinary quantic.
But we can pass from (A) to (C) by means of a lemma (B), which I have not
proved, but which seems highly probable, and which I enunciate as follows : (B) The
infinite system of terms X, rational and integral functions of a finite set of letters
(a, b, c, ...) which remain unaltered by all the substitutions of a certain group
(r(a, b, c, ...) of substitutions upon these letters, includes always a finite set of terms P
such that every term X whatever is a rational and integral function of these terms P.
In explanation of this lemma, observe that, if G (a, b, c, ...) denotes the entire
group of substitutions upon these letters, so that the functions which remain unaltered
by the substitutions of the group are in fact the symmetrical functions of (a, b, c, ...),
then the theorem is " The infinite system of rational and integral symmetrical
functions of (a, b, c, ...) includes always a finite set of terms P such that every such
rational and integral symmetrical function is a rational and integral function of the
terms P, viz. the terms P are here the several symmetrical functions
a + b + c + . . . , ab + ac + be + . . . , abc + . . . , &c." ;
761]
OF THE COVARIANTS OF A BINARY QUANTIC.
275
and so, if G (a, b, c, ...) is the group of all the positive substitutions, then we have
the like theorem for the rational and integral two-valued functions of (a, b, c, ...),
viz. the terms P are here the two- valued function (a — b)(a — c)(b — c) ..., and the
symmetrical functions
a + b + c + ..., ab + ac + bc + ..., abc + ..., &c.,
as before.
I return to the theorem (A), but instead of the covariants of a root-quantic of
any order, I consider first the invariants of a root-quantic of any even order. The
general form is
(«-£)w(a-7)»G8-7)P...,
where in all the factors which contain a, in all the factors which contain /3, and so
for each root in succession, the sum of the indices has one and the same value = 6.
Writing 12 for the index of a — (3, 13 for that of a — 7, and so in other cases, then
assuming always 12 = 21, 13 = 31, &c., the indices, taken each twice, form the square
21
31
12
13
23
the order of which, or number of its rows or columns, is equal to the order of the
quantic; the terms of the dexter diagonal are each =0, and the square is sym
metrical in regard to this dexter diagonal. Moreover, the square is such, that the
sum of the terms in each row (or column) has one and the same value = 6 ; and
conversely, every such square, say R6, represents an invariant.
Thus, for the quartic (as - a) (x - /3) (as - 7) (as - B), the square E9 is a square of
four rows (or columns) representing the invariant
(a-y3)12(a-7)13(a-S)14,
in which
(7-S)34,
12 + 13 + 14 = (9,
21 + 23 + 24 = 0,
31+32 + 34 = 0,
41 + 42 + 43 = 0.
35—2
276 ON THE THEOREM OF THE FINITE NUMBER
There are three squares Rl} viz. these are the squares
[761
1
representing the before-mentioned invariants (a - /3) (y - 8), (a - y) (/3 - 8), (a - 8) (/3 - 7)
respectively: say these are a, b, c, and every other invariant is a rational and
integral function of these; in fact, the ^-equations give easily 12 = 34, 13 = 24, 14 = 23,
so that the general form of the invariant is = a12b13c14, where 12, 13, 14 are each
of them a positive integer number (which may be = 0). Or, what is the same thing,
the square Re (0=12 + 13 + 14) is a sum
= 12. ^+13. E/ + 14. RS,
with positive integer coefficients 12, 13, 14, say for shortness it is a sum of squares
jRj. And so any like expression with a negative coefficient or coefficients may, for
shortness, be called a difference of squares Rlf
Observe that, in general, two squares Re, R^ are added together by adding their
corresponding terms, the result being a square Re+^', similarly, if each term of R^, be
less than or at most equal to the corresponding term of Re, then (but not otherwise)
the square R^ may be subtracted from Re, giving a square Re-^,.
In the case of the sextic
there are fifteen squares R1} which may be represented as follows:
12.34.56
12.35.46
12.36.45
13.24.56
13.25.46
2/2
13.26.45
*
14.23.56
#3
14.25.36
2/3
14.26.35
Z3
15.23.46
att
15.24.36
2/4
15.26.34
£4
16.23.45
#»
16.24.35
2/5
16.25.34
5
761]
OF THE COVARIANTS OF A BINARY QUANTIC.
277
viz. 12.34.56 here represents the square Rlt for which the terms 12, 34, 56 (and
of course the symmetrical terms 21, 43, 65) are each =1, the other terms all vanishing;
or, what is the same thing, it represents the invariant (a — /3)12 (7 — S)34 (e — £)56. But
it is not true that every square Re is a sum of squares Rl; this is not the case,
for the square R2,
= 12.13.23.45.46.56,
representing the invariant
(a - /3)12 (a - 7)13 (0 - 7)23 (S - e)45 (S - £)46 (e - £)56>
is not a sum of squares RI.
But the square last referred to is a difference of squares R^. it is in fact
= 12 . 36 . 45 + 13 . 25 . 46 + 14 . 23 . 56 - 14 . 25 . 36,
or, what is the same thing, the corresponding invariant is the product of the
invariants 12.36.45, 13.25.46, 14.23.56, divided by the invariant 14.25.36; viz.
it is a rational function of invariants RI.
It is required to show, first, that every square Re is a difference of squares J2a ;
and thence, secondly, that it is a sum of a finite number of squares Rk (being, in
fact, squares R! and R2).
For the first theorem we equate the general expression of R6 with the assumed
value
xl . 12 . 34 . 56 + 2/1 . 12 . 35 . 46 + zl . 12 . 36 . 45 + ... + zs . 16 . 25 . 34.
We thus obtain
fifteen equations satisfied by
xl = 34 - 26 + r + 8 - t,
#3 = 14 p S ,
12 = 2/1 + #1 + ^1
13 = #2 + 2/2 + ^2
14 = xs + 2/3 + z3
15 =#4 + ; 2/4 + ^4
16 = x5 + 2/5 + z5
^jO = #3 -J- X± T $/j
24 = x2 + 2/4 + ys
25 = 2/2 + 2/3 + *5
26 = z2 + z3 + z4
34 = X-L + z± + z5
35 = 2/1 + 2/5 + ^3
36 = 2/3 + 2/4 + ^1
45 = xs + zl + z2
46 = ar4 + 2/1 + 2/2
56 = xl + #2 + #s
= 15-26-3
= 45 — q — r ,
= 12-34 + 26 -q-r-s + t,
= 25 —p ,
P
2/3 =
2/4 = 36
y, = 16-45
-p-
-t,
. q
= 26
r
s
— r — s
278 OX THE THEOREM OF THE FINITE NUMBER [761
connecting xlt ylt zlt ...,zs with the terms 12, 13, etc. of Re (or indices of the
corresponding invariants). The fifteen equations are not independent, for regarding
them as giving the values of 12, 13, ... in terms of the xl, ylt z1} ..., zs, these
values satisfy identically the relations which ought to be satisfied by the terms 12,
13, etc., viz. the equations obtained by the elimination of 6 from the equations
12 + 13 + 14 + 15 + 16 = 0,
12 + 23 + 24 + 25 + 26 = 6,
16 + 26 + 36 + 46 + 56 =6.
The equations are thus insufficient to determine the values of a\, ylt zl} ..., zs, and the
general values given by the equations will contain five indeterminate quantities which are
taken to be p, q, r, s, t (these being in fact the values of ya, zlt z.2, za, zs respectively),
and we then have the equations all of them satisfied by the above-mentioned values
containing these indeterminate quantities ; taking them to be positive or negative
integers, then oc1, y1} z1}...,zs, will be all of them integers; but by what precedes,
it appears that they cannot all of them be made to be positive integers, so that we
have consequently Re>
= xl . 12 . 34 . 56 + 2/1 . 12 . 35 . 46 + zl . 12 . 36 . 45 + . . . + zs . 16 . 25 . 34,
equal in general to a difference of squares Rl.
Suppose in such difference of squares J^ we have any term, say — 12 . 34 . 56,
occurring with the coefficient — 1. Since the expression represents a square Re, we
must have among the positive terms, 12 . 35 . 46 or 12 . 36 . 45 to render possible the
subtraction of the 12; 15.26.34 or 16.25.34 to render possible the subtraction of
the 34 ; and 13 . 24 . 56 or 14 . 23 . 56 to render possible the subtraction of the 56 ;
that is, the expression must contain one of the eight combinations
12 . 35 . 46 + 15 . 26 . 34 + 13 . 24 . 56 - 12 . 34 . 56,
12 . 35 . 46 + 15 . 26 . 34 + 14 . 23 . 56 - 12 . 34 . 56,
12 . 35 . 46 + 16 . 25 . 34 + 13 . 24 . 56 - 12 . 34 . 56,
12 . 35 . 46 + 16 . 25 . 34 + 14 . 23 . 56 - 12 . 34 . 56,
12 . 36 . 45 + 15 . 26 . 34 + 13 . 24 . 56 - 12 . 34 . 56,
12 . 36 . 45 + 15 . 26 . 34 + 14 . 23 . 56 - 12 . 34 . 56,
12 . 36 . 45 + 16 . 25 . 34 + 13 . 24 . 56 - 12 . 34 . 56,
12 . 36 . 45 + 16 . 25 . 34 + 14 . 23 . 56 - 12 . 34 . 56.
The first of these is 35.46.15.26.13.24, viz. it is 13.15.35.24.26.46 which is a
square Rz (of the form mentioned above); the second is 35.46.15.26.14.23, which
is 15 . 23 . 46 + 14 . 26 . 35, a sum of squares ^ ; and similarly each of the other
combinations is either a square R2 or a sum of squares Rlt We have thus got rid
of the negative term —12.34.56, and in like manner if the negative term had been
- m . 12 . 34 . 56, = - 12 . 34 . 56 - 12 . 34 . 56 - &c.,
761] OF THE CO VARIANTS OF A BINARY QU ANTIC. 279
or, whatever the negative terms may be, we get rid one by one of each negative
term ; and thus ultimately express Re as a sum of squares R^ and Rz. Or, what
is the same thing, the invariant R0 originally expressed as a rational function of
invariants Rlt is finally expressed as a rational and integral function of invariants
R! and R2.
Similarly for a root-quantic of any even order n, we have the general square
Re expressed, first as a difference of squares R1} and then as a sum of squares
Rlt -R2, or it may be higher squares Rs, &c., but certainly as a sum of a finite
number of squares Rk. For a root-quantic of any odd order n, the investigation
would be of a somewhat different form, since here there are no squares R1, but the
lowest squares are squares R2 of a form such as 12.23.34.45.15; but the general
conclusion would still follow that every square Rg is a sum of a finite number of
squares Rk. And a like reasoning would apply to co variants instead of invariants :
viz. the reasoning (although for simplicity it has been given for a very particular
and special case) does, I think, really establish the theorem (A) in its generality,
viz. the theorem that for a root-quantic of any given finite order, the number of
irreducible monomial covariants is finite.
From any monomial covariant of the root-quantic, by taking the sum of the
forms belonging to the different roots, so as to obtain a symmetrical function of the
roots, that is, a rational and integral function of the coefficients, we obtain a covariant
of the quartic in its ordinary form (a, ...$#, y)n. Consider for a moment the before-
mentioned case of the invariants of the root-quartic
(x - ay) (x - fty) (x - 7y) (x - By),
now put
= -(a, b, c, d, e$x, 2/)4;
tl*
and to make the reasoning clearer, take a, b, c, f, g, h = (a — 8) (ft — 7), (ft - 8) (7 — a),
(7 — 8) (a — ft), (a — 8) (7— ft), (ft— 8) (a— 7), (y — B)(ft — a) respectively, these being,
with the signs +, the before-mentioned three monomial invariants. In the root-theory,
every monomial invariant is a rational and integral function of a, b, c, f, g, h. Every
invariant of (a, ...$#, y)*, qua rational and integral function of the coefficients, is,
when expressed in terms of the roots, a rational and integral function of the roots,
and then qua invariant is a sum of monomial invariants, and as such a rational and
integral function of a, b, c, f, g, h. But every such rational and integral function
of a, b, c, f, g, h is not a symmetrical function of a, ft, 7, B, and consequently not
in the present theory an invariant of (a, ...]£#, y)4; the invariants are those rational
and integral functions of a, b, c, f, g, h which are symmetrical functions of (a, ft, 7, B),
that is, which remain unaltered by every substitution whatever upon the roots
(a, ft, 7, B). Now each such substitution gives a substitution upon a, b, c, f, g, h,
and the 24 substitutions upon a, ft, 7, B give a group of 6, = £ . 24 substitutions
upon (a, b, c, f, g, h); the invariants are thus the rational and integral functions of
(a, b, c, f, g, h) which are unaltered by each of the substitutions of a certain group
G(a, b, c, d, e, f) of 6 substitutions. Theorem (B) asserts that, among the terms in
280 ON THE THEOREM OF THE FINITE NUMBER OF CO VARIANTS. [761
question, that is, among such rational and integral functions of (a, b, c, f, g, h), we
have a finite number of terms P, such that every one of the terms is a rational
and integral function of the terms P ; and recollecting that a + b + c = 0, these terms
P are in fact two terms be + ca -f- ab and (b — c) (c — a) (a — b) ; the conclusion being,
that the invariants of the quartic (a, b, c, d, e\x, y}* are all of them rational and
integral functions of the last-mentioned two functions, that is, of
/, = ae - 4>bd + 3c2, and J, = ace - adz - b*e + 2bcd - c3.
As regards the group G (a, b, c, f, g, h) of 6 substitutions upon a, b, c, f, g, h,
observe that the 24 substitutions of (a, /3, 7, 8} operating upon a, b, c, f, g, h give 6
substitutions taken each four times; for instance, the substitutions 1, 0/3.78, ay . (3&,
aS . j3y leave each of them a, b, c, f, g, h unaltered, that is, they each give the
substitution 1. And we thus find for the group Cr(a, b, c, f, g, h) the 6 substitutions
1,
abc . fgh,
acb . fhg,
af . bh . eg,
ah . bg . cf,
ag . bf . ch.
For the functions of a, b, c, f, g, h, which remain unaltered by the substitution of
this group, observe that we have f, g, h = — a, — b, — c ; so that any function of
the six letters may be represented as a function of a, b, c. An odd symmetrical
function, for instance abc, does not remain unaltered, for it is by any one of the last
three substitutions changed into fgh, that is, into —abc; on the other hand, the
two-valued function (b — c) (c — a) (a — b) does remain unaltered : the functions which
remain unaltered are therefore the even symmetrical functions of a, b, c (that is, the
symmetric functions a2 + b2 + c2, or ab + ac + be, &c., which are of an even order in
a, b, c conjointly), and the same even functions multiplied by (b — c) (c — a) (a — b);
and having regard to the relation a + b -I- c = 0, all these can be expressed as already
mentioned as rational and integral functions of be + ca + ab and (b — c) (c — a) (a — b).
The proof applies to the general case of the theorem (C'), viz. taking the theorem
(A) to be proved, and putting the root-quantic
O - ay) (ac - @y) . . . = - (a, ...$#, y)n,
Cv
then we have a, b, c, d, ... a system of monomial covariants of the root-quantic;
and all the covariants of (a, ...]£#, y) are rational and integral functions of (a, b, c, d, ...)
which remain unaltered by the substitutions of a certain group G (a, b, c, d, ...); hence,
assuming the theorem (B), they are rational and integral functions of a finite number
of irreducible covariants. And the demonstration thus depends upon that of the
theorem (B).
762] 281
762.
ON SCHUBERT'S METHOD FOR THE CONTACTS OF A LINE
WITH A SURFACE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvn. (1881),
pp. 244—258.]
I WISH to reproduce in part § 33, "Coincidenz von Schnittpunkten einer Geraden
mit einer Flache " of Schubert's very interesting work Calcul der abzahlende Geometric,
Leipzig, 1879, explaining in the first instance (but not altogether in the manner or from
the point of view of the author) the general principles of the theory.
We have to do with conditions relating to a subject; the subject is a geometrical
form or entity of any kind depending upon a certain number of constants; and the
condition is onefold, twofold, &c., according as it imposes a onefold, twofold, &c.,
relation upon these constants. The number of constants is the Postulandum of the
subject, and the manifoldness of the condition is called also its Postulation. A condition
is incomplete when its postulation is less than postulandum of subject, complete when
its postulation is equal to postulandum of subject; two or more incomplete conditions,
making up a complete condition, are supplementary to each other. The case where
the postulation exceeds the postulandum, or say that of a more than complete
condition, is not in general considered ; it may however sometimes present itself.
For instance, the subject may be a line with n points upon it ; the number of
constants is here = n + 4. A condition that the line shall meet a given line, or that
a certain one of the n points shall lie on a given plane, is a onefold condition;
the condition that such point shall lie upon a given line is a twofold condition: and
so in other cases.
Conditions are denoted by letters, and simultaneous conditions by a product; for
instance, the subject is a line carrying the n points 1, 2, ..., n; g is the condition
that the line meets a given line; p1 the condition that the point 1 lies on a given
plane; then gpl is the twofold condition that the line meets a given line and that
C. XI. 36
282 ON SCHUBERT'S METHOD FOR THE [762
the point 1 lies on a given plane ; pf is the twofold condition that the point 1
lies on each of two given planes (in fact, on their line of intersection). The letters
p, g, e are used as the initials of Punkt, Gerade, Ebene.
The letter or combination of letters denoting an incomplete condition, or, say,
the incomplete condition itself, has no numerical value ; but for a complete condition
there exists a definite number of subjects satisfying the condition, and the condition
is regarded as having this number as its value. A more than complete condition
has the value 0.
Conditions of the same postulation may be connected by the sign + ; for instance,
subject a line,
fje the condition that it lies in a given plane,
gp the condition that it passes through a given point,
then ge+gp is the condition that the line shall either lie in the given plane or
else pass through the given point.
I abstain from attempting any definition in regard to the sign — .
Conditions of the same postulation may be connected by an equation or equations;
for instance,
subject a point,
p the condition that the point shall lie in a given plane,
pg the condition that the point shall lie in a given line,
then p2=pg.
This equation has (so far) no numerical signification ; it has the logical significa
tion that the condition that a point shall lie on each of two given planes is equivalent
to the condition that the point shall lie on a given line.
Second example. Subject a line,
g the condition that the line meets a given line,
ge the condition that it lies in a given plane,
gp the condition that it passes through a given point,
then g'2=ge
This equation has (so far) no numerical signification, and I regard it as having
no logical signification. Schubert, however, gives it a logical signification by means of
his " Princip der speciellen Lage " (Principle of Special Situation), viz. the condition
of the line meeting each of two given lines is, in the particular case where the
two given lines meet, equivalent to the condition, that the line shall either lie in
the plane of the two given lines or else pass through their point of intersection.
762] CONTACTS OF A LINE WITH A SURFACE. 283
Third example. Subject a line bearing upon it the points 1 and 2,
e the condition of the coincidence of the two points,
p „ that the point 1 shall He on a given plane,
q » » " » »
g „ that the line shall meet a given line,
then e = p + q — g.
This equation has (so far) no numerical signification, and it does not appear to
have any logical signification. In fact, in the actual form of the equation we have
a sign — which has not had given to it any logical interpretation ; and if we write
the equation in the form e + g = p + q, there seems to be no logical signification in
the assertion, the condition that either the points shall coincide, or else the line
meet a given line, is equivalent to the condition that either the first point, or else
the second point, shall lie in a given plane.
Any equation connecting complete conditions is a numerical equation ; and to
render a condition complete, we have only to join to it a supplementary condition X
of the proper postulation. Thus, in the last example the postulandum is = 6 ; e, p,
q, g are onefold conditions, and joining to each of them one and the same fivefold
condition X, we have Xe = Xp + Xq — Xg. And, taking X to be an arbitrary fivefold
condition, the original equation e=p+q—g has in fact the meaning
Xe = Xp + Xq - Xg.
For instance, the fivefold condition X may be that the line shall belong to a
given regulus (scroll or developable surface), and that the points 1, 2 upon the line
shall be the intersections of the line with given surfaces S1} S2 respectively. The subject
is the line of the given regulus with its two points ; and the meaning of the equation
is that the number of subjects with two coincident points is equal to the number
of subjects with the point 1 on a given plane, phis the number of subjects with the
point 2 on a given plane, minus the number of subjects for which the line meets
a given line. Although for the moment concerned only with the meaning of the
theorem, not with its truth, I stop to show d posteriori that the theorem is in fact
true: take k for the order of the regulus; m,, ra2 for the orders of the surfaces
$! , $2 respectively ; then it is to be shown that X e, Xp, Xq, Xg are each = km^m^
(values which satisfy the equation). First X e : the points 1 and 2 here coincide at
a point of the curve of the order m^n^, which is the intersection of S1 and S.2 ;
the regulus meets this curve in km-^m^ points, and through each of these we have a
line of the regulus having upon it the two coincident points; that is, Xe = kmlmz.
Next Xp: the point 1 is here on the plane curve of the order mlt which is the
intersection of Sl with the corresponding given plane ; the regulus meets this plane
curve in kml points; through each of these we have a line of the regulus intersecting
$2 in m2 points, any one of which may be taken for the point 2 ; that is, the
number of subjects is Xp, = km}.m2. Then Xq: in precisely the same manner we
have Xq = km2.ml. Lastly Xg : the given line meets the regulus in k points, and
36—2
284 ON SCHUBERT'S METHOD FOR THE [762
through each of these there is a line of the regulus meeting $x in m^ points, any
one of which may be taken for the point 1, and meeting S2 in m, points, any one
of which may be taken for the point 2 ; the number of the subjects Xg is thus
Xg, = k . m1 . m2.
The general theorem Xe = Xp + Xq — Xg is proved by means of Chasles' theorem
of united points as follows: the subject is a line, or say, for convenience, an axis £,
bearing upon it the two points 1 and 2 ; we consider in conjunction therefore a given
line X, and through this draw the planes Plt P2 passing through the points 1 and
2 respectively.
Suppose that when 2 lies in a given plane there are a' positions of the axis,
and on each of these /3' positions of the point 1 ; and, similarly, that when 1 lies
on a given plane there are a positions of the axis, and on each of these /3 positions
of the point 2 ; then, 1 lying in a given plane, the number of subjects is a/3, or
we have Xp = a/3 ; and, similarly, Xq = cc'/3'. Take now for the point P! an arbitrary
plane through X; then, 1 lying on this plane, the number of the points 2 is =a/3,
or, since each of these determines with X a position of the plane P2, the number
of these planes is = ay9, that is, it is = Xp ; and, similarly, taking P2 an arbitrary
plane through X, the number of the planes Pj is a'/3', that is, it is = Xq ; viz. the
two planes Pa, P2 through the line X have an (Xp, Xq) correspondence ; hence, by
Chasles' theorem, the number of united planes is = Xp + Xq.
But we have a united plane, 1°, if the points 1 and 2 coincide, that is, if the
condition Xe be satisfied, and the number of these united planes is Xe ; 2°, if the
axis | meet the arbitrary line X, that is, if the condition Xg be satisfied, and the
number of these united planes is = Xg ; hence the whole number is = Xe + Xg ; or
we have Xp + Xq = Xe + Xg, that is, Xe = Xp + Xq — Xg, which is the theorem in
question.
The conclusion is that the equation e = p + q — g, which in this, its original form,
has neither a numerical nor a logical signification, is to be understood as meaning the
numerical equation Xe = Xp+Xq — Xg, the truth of which numerical equation has just
been proved. Or we may, without explicit introduction of the condition X, understand
the equation e=p + q—g as a numerical equation as follows, viz. taking for the subject
a line with two points, which line and points are regarded as satisfying a given fivefold
condition, then
e is the (additional onefold) condition that the two points shall coincide,
p „ „ „ that the point 1 shall lie in given plane,
q „ „ „ that the point 2 shall lie in given plane,
g „ „ „ that line shall meet given line.
The conditions e, p, q, g are thus in effect complete conditions, having values which may
be connected by an equation ; there, in fact, exists between them the relation
762] CONTACTS OF A LINE WITH A SURFACE. 285
The like remarks would apply to the before-mentioned equation (subject a point)
P*~Pg' either adding to it a onefold condition X, and so taking it in the form
Xp* = Xpg, or understanding it in its original form pt=pg as belonging to a point which
satisfies already a onefold condition, the equation is true as a numerical equation; and
this in fact follows at once from its truth as a logical equation. But observe the
difference: the equation in question pz=pg has, the equation e=p+q—g has not, a
logical signification.
I regard as the fundamental notion of the theory the existence of equations between
conditions such as the foregoing equation e=p + q — g; equations which in their original
form have not (of necessity) any logical signification, and have not any numerical signi
fication; but which, when we adjoin to them a supplementary condition X of the proper
postulation, become numerical equations, which are true, independently of the form of
the supplementary condition X and whatever this condition may be. And this being
so, it seems to follow at once that such equations may be treated and worked with as
ordinary algebraical equations. For instance, let M be any condition of less postulation
than X : then if from the equation e = p + q — g, assumed to be true, we deduce
Me = Mp + Mq — Mg, this (like the original equation e=p+q—g) is in its actual form an
equation without logical or numerical signification ; but if we adjoin to it a supplementary
condition K, such that postulation of K + do. of Jf=do. of X (or, what is the same
thing, that the condition KM shall be supplementary to the several conditions contained
in the original equation e = p + q—g), then the equation in question, Me — Mp + Mq — Mg,
is to be interpreted as meaning
KM e = KMp + KMq - KMg,
that is,
Xe = Xp + Xq - Xg,
which is numerically true. We thus see that the original equation e=p + q — g implies
the new equation
Me = Mp + Mq - Mg,
which is its algebraical consequence. And if we regard, for instance, A + B as the
condition that either the condition A shall be satisfied or else the condition B shall
be satisfied, then A + B is a condition, and as such we have
It is going a step further to say that if we have, for instance, an equation
M = A + B — C between conditions M, A, B, C, then that, instead of
Me = Mp + Mq - Me,
we may write
(A + B - C ) e = ( A + B - C) p + ( A + B - C) q - (A + B - C) e ;
this is, in fact, treating A + B — C as being to all intents and purposes a condition such
as M, or an alternative condition A + B. It is, in fact, assumed that the step is per
missible ; and we thus make such deductions as
(e + p + q - g) (e - p - q + g) = 0 ;
286 ON SCHUBERT'S METHOD FOR THE [762
that is,
e*-(p + q-g)* = 0,
or
e2 = (p + q - g)-, =p* + 2pq + q2 - 2pg - 2qg + g" ;
viz. this is an equation such as the original equation e = p + q — g, acquiring a numerical
signification when we adjoin to it a supplementary condition X of the proper postulation.
The section above referred to deals with the question to determine the number of
lines which satisfy the several relations of contact in regard to a given surface F of the
order ??, without point-singularities, that is, the surface represented by the general
equation (*$#, y, z, w)n = 0.
The chief results are contained in the following table, the notation of which will be
explained :
1. €2gs = n(n-l),
2. esb«ge = n,
3. e3ge = 3w(w-2),
4. €sgp = n(n-l)(n-2),
5. es&32 = 2n,
6. €.»ge = in(?i-2)(w-3)(n + 3),
7. €»gp = iw(n-l)(n-2)(n-3),
8. e2-A2 = n (n - 3) (n + 2),
9. e^Co = tt(n3-2n2 + 2?i-6),
10. e4g = 2?i(n-3)(3ro-2),
11. e464 = ??(ll?i-24),
12. e32g = n (n - 3) (n - 4) (if- + 6n - 4),
13. €3,63 = n (n - 4) (3rc2 + 5?i - 24),
14. 63.,62 = ??(?l_2)(?i-4)(?i2 + 2n + 12),
15. e^g = ^n(n-3)(n-4)(n~5)(n»+3n-2),
16. 622363 = £M(w,-2)(?i-4)O-5)02 + 5/i+ 12),
17. e5 = 5w(w-4)(7w-12),
18. e42 = 2n (n - 4) (w - 5) (n + 6) (3n - 5),
19. e;53 = i« (n - 4) (n - 5) (n3 4- 3/i2 + 29n - 60),
20. 63,2 = %n(n-4>)(n-o)(n-6)(n3+9n- + 20n-60),
21. eoooo =TVrc(ri-4)(w-o)(H-6)(?i-7)(y + 6n2 + 7n-30),
22. 622-6! = J»(n-4)(n-5)(n-6)(n»+8n*-2ii-12X
23. 636^ = (n-3)(n2+2),
24. eo^Cj^ = ?i2 (w - 4) (2n2 - '3n - 3).
762] CONTACTS OF A LINE WITH A SURFACE. 287
In the foregoing formulae the suffixes of the e refer to the contacts, viz. e2 denotes a
2-pointic intersection, e32 a 3-pointic and a 2-pointic intersection. The letters b, c, d refer
to the points of contact or intersection, thus e3263, b3 is the point of 3-pointic intersection ;
€222^1, &i is one of the points of simple intersection; 6j is also the condition that the point
in question lies on a given plane ; g, gs, ge, gp have their ordinary signification explained
a little further on. Thus (15) £222*7 denotes the number of triple tangents which can be
drawn to meet a given line; or, what is the same thing, it is the order of the regulus
formed by the triple tangents.
The following are elementary formulas used in the investigation of the foregoing
results.
Subject a line having upon it a point,
Postal.
p the condition that point is in a given plane 1
pg „ „ line 2
g „ line meets a given line 1
ge „ ,, is in a given plane 2
gp „ „ passes through a given point 2
gs „ „ lies in a given plane and passes through a
given point of that plane 3
G „ „ coincides with a given line 4
We have (p. 22 et seq.)
Postul.
pg = p2 2 (logical)
Pg =P2 + 9e 2
gs = gge 3 (logical)
9s = ggp 3 (logical)
Pffp = P3 + 9s 3 (demons, infra)
p4 = 0 4
gegp = 0 4
ge2 = G 4
.) s~y A
^Jf) — ^Jf J&
p3g = p2ge 4 (demons, infra)
Pffs = p*ge+G 4
P3ffe =0 5
p3gp=pG 5
p*gs = pG O
288 ox SCHUBERT'S METHOD FOR THE [762
Ps + 9s', we have Q=ge+pz—pg, 0=ge+gp — g-, and thence
o = (p + g) [ge +p(p- g)} -p [ge + gp- g2}
= P9e + 99 e + P3 ~ Pg*
-P9e-P9P
P3g = p-ge- from pg=p*+ge, we have p2g = p* + p*ge = p*ge , since p* = 0,
Pffs = p2ge + 0,, gs=gge „ pge =pgge = (p- + ge)ge=p"ge + G,
Pffs = p*gp „ gs = ggP „ pg* = pggp = (P- + ge)gp= p*gp , since gegp = o ;
and in a similar manner we prove the last three equations.
For the demonstration of the formulae of the table we take the subject to be a line
bearing upon it the points 1, 2, ..., n, which are its intersections with a given surface of
the order n. The symbols plt p», ... refer to these points respectively; thus, pl is the
condition that the point 1 may lie on a given plane ; and then, writing
=pi+p3-g,
it appears that e will denote the condition of the coincidence of the points 1 and 2 ; e'
that of the points 1 and 3, &c. Hence also, ee' will denote the twofold condition of the
coincidence of the points 1, 2, 3 ; and so in other cases. But, according to the notation
above explained, e is also denoted by e2« ee' by e3, ee" by e22, &c.
We thus have
*2 = pi + p-2-g,
ea =(PI+ p-2 - g} (PI +PS- g\
2*22 = (PI + p., - g) (p3 + p,- g),
£4 = (PI +p-2-g) (PI + p3-g) (PI +pt - g),
**> = (P! +p-2-g) (PI +ps-g) (P* +PS- g),
Bezs = (p, +p2-g) (p3 +p^-g) (p5 +PS- g),
e5 = (PI +p2-g) (pi+pt-ff) (Pi+p* - g) (PI + p5- g),
e42 = (PI + p-2 - g} (PI + p3-g) (PI +p*-g) (ps + ps- g},
2e322 = (p, + p, - g) (p, +p.-g) (p, +p5-g) (ps +p7- g),
7 +ps -g}-
J.1C1 J I ,t
*~S
762] CONTACTS OF A LINE WITH A SURFACE. 289
We can now, by a mere analytical process of development and reduction, express
each of the foregoing values as a linear function of
Pi2p22, p*pap», PiP&spi, and G.
(Schubert says, as a linear function of these four symbols and pip.2ge ; but in fact Pip.2ge
s =
Observe, first, that we may, p. 287, in all the general equations instead of p write
p1} p2, &c. ; and, further, that any symbol containing for instance p^ is =0. For the
symbols now belong to the intersections of the line with a given surface ; pf is the
condition that a certain one of these intersections shall lie in three given planes, that is,
that it shall coincide with a given arbitrary point; this cannot be the case, for the
arbitrary point is not on the surface F; and therefore p1a = Q.
We thus have Pig=pi2 + ge, thence Pi*g=pi*+Pige, that is, pfg=p1ge; and thence
further Pisg=pizge, that is, p12ge = Q.
Again, from p2g=p.22 + ge, Pi9=p* + ge> we have
Pi*(p*+ge)=pip*(pi>
which, in virtue of p?ge = 0 and pfp^ = 0, becomes
As a simple instance of the reductions, take
%ff,, = (pi+p-2
Here
Pigs, =p2gs, = pi*ge + G, = G, since pl*ffe = 0',
and
99 B = g*ge = (ge + gp) ge = g<? + gegp = G, since g/ = 0, gegp = G ;
whence the value is
e2gs=G + G-G, = G.
As a more complicated example, take
Observe that, after the multiplication is effected we may, in any way we please,
interchange the suffixes, pl2p3pi=p12p2ps, p^pf^p^pj, &c. ; the suffixes serve only to
distinguish from each other symbols in the same product (thus pf is different from
PipzpspJ, but there is nothing to distinguish one point of intersection from another.
Thus the foregoing expression containing the terms (p, +p3 +p4 + p5) (p^ -g)3, these may
be combined into the single term 4>p2(p1 — g)3; expanding in powers of pl-g and
reducing in this manner, the value of e5 is, in fact, found to be
= (Pi ~ g}" + 4^2 (p1 - g)3 + 6p2p3 (Pl - <7)2 + 4>p,p3p. (Pl -g)+ Plpap3pt.
Developing this in powers of g, omitting the terms containing p.? which vanish, and
further reducing, the value is
t) - Sp.g3
c. XL 37
290
ON SCHUBERT S METHOD FOR THE
[762
We have
g4 = 2G, p^g3 =pig* =pi9e + G, =G.
Next for the terms in g*t from p\g—p\-\-ge. we have
Pi 9= Piffe,
and thence
or, since p^gs = G as before, the whole term is = 18pfp2g + 24>G. The terms in g thus
become = g(6p12p2— IQp^ps), and from the same equation Pig = p^ + ge we find
P*P*ff =PiP»
The value is> thus finally found to be
The whole series of like results is
7>iV
1- «2 9s
+ 1
2. ,, b,,ffe
+ 1
1
3- e3 #e
3
- 3
4- „ gp
+ 1
5. „ &,«
— 2
+ 1
+ 1
6. 2«ffl <7e
+ 4
- 3
7 o „
•^ >? i/j)
+ 1
8. „ &,«
- 3
+ 2
+ 1
9. e, 62c2
— 2
+ 1
+ 1
10. C4 0
- 2
+ 4
- 2
11. „ b,
- 6
+ 1
+ 4
12. €30 #
3
+ 6
- 2
13. „ bs
7
1
+ 2
+ 4
14. „ 62
6
- 3
+ 3
+ 4
15. 6e222 <jr
4
+ 8
- 2
16. 2,, b.
7
4
+ 4
+ 4
17. €5
- 10
- 10
+ 5
+ 10
18.
- 10
- 16
+ 8
+ 10
19. 2e33
9
- 18
+ 9
+ 10
20. 2C322
9
- 24
+ 12
+ 10
21. 24C0222
- 8
-32
+ 16
+ 10
22. ee^ 6j
- 6
- 12
+ 8
+ 4
^O 7 2
3
+ 3
+ 1
24. c2 ftjCjC?!
- 1
- 1
+ 2
762] CONTACTS OF A LINE WITH A SURFACE. 291
But in these formulae pip%, Pi2p2p3, Pip^psjh, G have numerical values which are
different according to the number of points of intersection presenting themselves in
the several formulae; viz. this number being called i, we have for the formula? in
62 €3 €33 £4 e32 e222 e5 ^42 ^33 £322 e2222 ^222^1 e3^1
i= 23445656678 7 4 5,
and the values of the symbols are
pfp* =ri>(n- 2) (n - 3) ...(n-i + 1),
= n»(TO-l)(TO-3) ...(n-i + 1),
* = n* (2n* - 6n + 3) (TO - 4). . .(n - i + 1),
G = n(n-l)(n-2) ...(n-i+1).
Thus, suppose i = 4, the subject is a line bearing the points 1, 2, 3, 4, which are
intersections of the line with the surface F; we have then G as the condition in
order that this line (or, say, the line of the subject) may coincide with a given line,
which given line intersects the surface in n points; any four of these (their order
being attended to) may be regarded as being the points 1, 2, 3, 4 ; or there are
»(n — l)(n — 2)(n— 8) subjects satisfying the prescribed condition (that the line of the
subject may coincide with the given line). Hence here G = n (n — 1) (n — 2) (TO -3);
and so in general G = n(n — l)(n — 2). . .(n - i + 1).
Next, for pfpf. Here pf is the condition that the point 1 shall lie in each
of two given planes, that is, in a given line, say L^; and, similarly, pf is the condition
that 2 may lie in a given line L2. We take any one of the n intersections of L:
with F for the point 1, and any one of the n intersections of L2 with F for the
point 2; this determines the line of the subject, but the i-2 points 3, 4, ..., i are
then any i - 2 of the remaining n - 2 intersections of this line with F; that is,
2) (n - 3). ..(n-i+1) as above.
Again, for p^p^p,. Here pf is the condition that 1 shall lie in a given line Zx;
we therefore take for 1 any one of the n intersections of L^ with F; ps is the condition
that 2 may lie in a given plane P2, it lies therefore in the curve of intersection
of P2 with F; and, similarly, 3 lies in the curve of intersection of a plane P3 with
F; the two planes intersect in a line meeting F in n points <r, and the two cones,
vertex 1, which stand upon the plane curves respectively, intersect in the n lines
joining 1 with the n points <r, and in TOS-TO other lines. The line of the subject is
then any one of these n*-n lines, or, since the vertex is any one of n points, the
line is any one of TO(TOS-TO), = ri>(n-l) lines; the remaining points 4, 5, ..., i are
any i-3 of the remaining TO -3 intersections of the line with F ; hence the formula
Pi*p2p3 = n* (n -l)(n- 3) (n - 4). ..(n - i + I).
For p!p9papt. We have here 1, 2, 3, 4 lying in given plane sections of the surface
F, and we have consequently to find the number of lines which can be drawn to meet
each of these four sections. Observing that any two of the sections meet in the n
37—2
292 ON SCHUBERT'S METHOD FOB THE [762
intersections with F of the line of intersection of their planes, the order of the scroll
generated by the lines which meet three of the sections is 2n3 — 3w2; this scroll meets
the fourth section in n (2n3 — 3w2), = 2w4 — 3n3 points ; or we have this number of lines
meeting each of the four sections. But among these are included 3%2 (n — 1) lines
which have to be rejected, viz. the sections 1 and 4 meet in n points, each of which is
the vertex of cones through the sections 1 and 2 respectively; these cones meet in n
lines, which are to be disregarded, and in n2 — n other lines, and we have thus n (n2 — n),
= n" (n — 1) lines ; and similarly from the intersections of 2 and 4, and from the inter
sections of 3 and 4, nz(n— 1) and nz(n— 1) lines, in all 3?i2(?i — 1) lines. Hence the
number of lines meeting the four sections is
2n4 - 3w3 - 3n3 + 3w2 , = 2n* - Qn3 + 3n? ;
taking any one of these for the line of the subject, the remaining points 5, 6, ..., i are
any i — 4 of the remaining n — 4 intersections, or we have the required formula
* = n2 (2rc2 - Qn + 3) (n - 4). . .(n -i + l).
The four numbers p^p/, Pi2p2p3, Pip2p3p4, G for any line of the table being now
known, we can at once calculate the required values e2gs, &c., as the case may be ; for
instance,
- lOrc2 (TO - 1) (n - 3) (n - 4)
+ ona (2n2 - Qn + 3) (n - 4)
= on (»-4)(7w-12).
In fact, throwing out n(n — 4), the remaining terms give
-10n3 + 50n2- 60n,
-10n3+40?i2- 30n
+ 10w3 - 30wa + I5n
+ Wn3 - 60w2 + 11 On - 60
35n-60, = 5(7n-12).
And we obtain in like manner the other formulae of the table.
The remainder of § 33 contains investigations of less systematically connected
theorems, and I quote the results only.
25. If on the surface Fn there is a curve order r, then of the tangent planes of Fn
along this curve there pass r(n-l) through an arbitrary point of space; editor,
class of torse is =r(n— 1).
In particular, for curve of 4-pointic contact, r = n(lln — 24), class of torse is
No. of tangent planes through line, or class of surface, =n(n-\)-.
762] CONTACTS OF A LINE WITH A SURFACE. 293
26. e3b3g = e3b32 + €3ge = 2n + 3n (n - 2), = n (3n - 4).
€3^3 ff, = n (3w — 4), is the order of curve of contact of the 3-pointic (chief) tangents
which meet a given line.
Parabolic tangents are coincident chief tangents.
No. of 4-pointic parabolic tangents = 2n (n — 2) (lln — 24).
27. Order of parabolic curve = 4n (n — 2).
Order of regulus formed by parabolic tangents
= 2n(n-2)(3w-4).
The parabolic curve and curve of contacts of an e4 tangent meet in
4»(*-2)(lln-24)
points, i.e., they touch in 2n (n — 2) (lln — 24) points.
28. Umbilici. No. is =2w(5na- 14m + 11).
29. No. of points at which the chief tangents being distinct are each of them
4-pointic, or, what is the same thing, No. of actual double points of
curve e4,
= 5n (7w2 - 28w + 30),
n = 3, No. is 15 (63 - 84 + 30), = 135, viz. this is the number of points of
intersection of two of the 27 lines ; or, what is the same thing, the number
of triple tangent planes is =45.
30. No. of parabolic tangents which have besides a 2-pointic contact is
= 2n (n - 2) (n - 4) (3w2 + on - 24).
31. No. of double tangent planes such that line through points of contact is at one
of these points 3-pointic
= n (n - 2) (n - 4) (n3 + 3?i2 + 13n - 48).
32. No. of points where one chief tangent is 4-pointic, the other 3-pointic and (at
another point of the surface) 2-pointic is
= n (n - 4) (27w3 - 13w2 - 264n + 396).
33. No. of points where chief tangents being distinct are each of them at another point
of the surface 2-pointic is
= n(n- 4) (4w5 - 4%4 - 95w3 + 99na + 544ft - 840).
34. The curve of contacts b3 of an e32 tangent has with the parabolic curve 2-
pointic intersections only, and these are at the points for which the chief
tangent is (at another point of the surface) 2-pointic.
35. The curve of contacts b3 of an €32 tangent has, with the curve of contacts of
an e4 tangent, 2-pointic intersections at the contacts of an e5 tangent ; and
has also simple intersections with the same curve, 1° at the contacts 64 of
an e42 tangent, 2° at the points where the chief tangents are e4 and e32.
294 [763
763.
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvn. (1881),
pp. 258—276.]
A SUM of 2 squares multiplied by a sum of 2 squares is a sum of 2 squares ; a
sum of 4 squares multiplied by a sum of 4 squares is a sum of 4 squares ; a sum
of 8 squares multiplied by a sum of 8 squares is a sum of 8 squares; but a sum
of 16 squares multiplied by a sum of 16 squares is not a sum of 16 squares. These
theorems were considered in the paper, Young, " On an extension of a theorem of Euler,
with a determination of the limit beyond which it fails," Trans. R. I. A., t. xxi. (1848),
pp. 311 — 341 ; and the later history of the question is given in the paper by Mr S.
Roberts, " On the Impossibility of the general Extension of Euler's Theorem &c.," Quart.
Math. Jour. t. xvi. (1879), pp. 159 — 170; as regards the 16-question, it has been
throughout assumed that there is only one type of synthematic arrangement (what this
means will appear presently); but as regards this type, it is, I think, well shown that
the signs cannot be determined. It will appear in the sequel, that there are in fact
four types (the last three of them possibly equivalent) of synthematic arrangement ; and
for a complete proof, it is necessary to show in regard to each of these types that the
signs cannot be determined. The existence of the four types has not (so far as I am
aware) been hitherto noticed ; and it hence follows., that no complete proof of the
non-existence of the 16-square theorem has hitherto been given.
For the 2 squares the theorem is of course
For the 4 squares (for which the nature of the theorem is better seen) it is
(x? + x?- + ac32 + x?) (y? + y2- + ys* + y?} = (x^ + x%* + xsys + xtyt)2
- #42/i + #22/3 -
763] ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 295
or, as this may be written,
(X2 + x22 + #32 + x/} (y-c + y22 + yf + ?/42) - (x^ + x2y2 + x3y3 + x.y,)-
= (12 + 34)2
+ (13 - 24)2
+ (14 + 23)2;
where 12 is used to denote x^j* — x^, &c., and the truth of the theorem depends
on the identity 12.34-13.24 + 14.23=0. Clearly, the first step for forming the
equation is to arrange the duads in a synthematic form
12.34
13 . 24
14.23,
and then to determine the signs : such an arrangement exists in the case of 8, and
the signs can be determined; it exists also in the case of 16, but the signs cannot
be determined to satisfy all the necessary relations.
In the case of 8, we have the synthematic arrangement
12.34.56.78
13.24.57.68
14.23.58.67
15.26.37.48
16.25.38.47
17.28.35.46
18.27.36.45,
being the only type of synthematic arrangement. This is, in fact, important as regards the
16-question, and it will appear that the case is so ; but in the 8-question, starting from
this arrangement, we have to show that there exists an equation which, for convenience,
I write as follows :
(x* + . . . + x<?) (y? + . . . + */82) - (awi + - - - + fftfs)8
= (12 + 34 + 56 + 78)2
+ (13 + 24 + 57 + 68)3
+ (14 + 23 + 58 + 67)s
+ (15 + 26 + 37 + 48)2
+ (16 + 25 + 38 -4- 47)2
296 ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. [763
but in which it is to be understood that each duad is affected by a factor + 1
which is to be determined; say the factor of 12 is e12, that of 34, e^; and so in
other cases. It is however assumed that 612> e^, ex, e78; eu, eu, el5, e16, e17, e18 are
each =+1.
We have then on the right-hand side triads of terms such as, 2 into
6,3634 12 . 34 4- 613e24 13 . 24 + €„€„ 14 . 23,
which triad ought to vanish identically, as reducing itself to a multiple of
12.34- 13.24 + 14.23;
viz. we ought to have
€12 e34 == ^13e24 = e14^23 j
or, using now and henceforward when occasion requires, 12, 34, &c. to denote e12, 634, &c.
respectively, we have
12.34-+*,
13 . 24 = - k,
where k, = + 1, has to be determined (in the actual case we have 12 = 4-1, 34 = 4-1,
13 = 1, 14=1; and therefore the first equation gives k=l, and the other two then give
24 = - 1, 23 = 4- 1).
We have in this way triads of values corresponding to the different tetrads
1234
. 1256
1278
1357
1368
1458
1467
2358
2367
2457
2468
3456
3478
5678,
which can be formed with the several lines of the formula. Thus we have from the
first line 1234, 1256, 1278; then from the second line (not 1324 which in the form
1234 has been taken already) 1357, 1368, ...; and finally from the last line 5678.
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
297
We might consider each line as giving 6 tetrads, but the tetrads would then be
obtained 3 times over ; the number of tetrads is thus 6 x 7 -f- 3, =14 as above.
And observe, that the systems of values for the coefficients e = ± 1 are obtained
directly from the tetrads, without the employment of any other formula.
We thus obtain the system of signs as follows :
12
+ 1
13
+ 1
14
+ 1
15
+ 1
16
+ 1
17
+ 1
18
+ 1
23
+ 1
24
-1
25
+ 1
26
-1
27
+ 1
28
-1
34
+ 1
35
a
-e
36
b
e
37
— a
e
38
-b
-e
45
c
e
46
d
e
47
-d
-e
48
— c
-e
56
+ 1
57
a
-e
58
c
e
67
d
e
68
b
e
78
+ 1
C. XI.
38
298 ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. [763
viz. the original assumptions 12 = + 1, &c., and the tetrads 1234, 1256, 1278 give all
the signs +1 up to 34 = + 1 ; from the tetrad 1357 we have
13.57 + 1 a,
15.37 - 1 a,
17.35 + 1 a,
that is, 35 = a, 37 = — a, 57= a, where a, = ± 1, is still undetermined; and similarly,
the tetrads 1368, 1458, 1467 give the remaining signs b, c, d. The tetrad 2358 then
gives
23.58 + 1 c,
25.38 - 1-6,
28.35 +-1 a,
that is, — a = b = c ; and similarly the tetrads 2367, 2457, 2468 give —a = b = d,
— a = c = d, b = c = d respectively ; the four tetrads thus give — a = b = c = d, say each
of these = 6. But retaining for the moment a, 6, c, d, the tetrad 3456 then gives
34.56 +11,
35.46 - a d,
36.45 + b c,
that is, 1 = — ad = be, and similarly the last two tetrads 3478 and 5678 give
1 = — ac = bd and 1 = — ab = cd respectively ; substituting the values in terms of 6,
the several equations give only &* — 1, that is, 6 = ± I at pleasure ; and the series of
signs for the 8-formula, containing this one arbitrary sign #= + 1, is thus determined.
Passing to the case of 16, we have in like manner to form a synthematic arrange
ment of the numbers 1, 2, ..., 16 in 15 lines, each containing the 16 numbers in 8 duads
(no duad twice repeated), and this containing all the 120 duads. And, using for the
moment letters instead of numbers, the necessary condition is, that ab . cd occurring in one
line, ac . bd must occur in another line, and ad .be in a third line. Observe that as well
the order of the letters in a duad as the order of the duads is thus far immaterial; so
that a line containing bd . ca may be considered as containing ac . bd.
Considering any such combination ab . cd, the line which contains it may be
taken to be the first line ; and the line which contains ac . bd may be taken to be
the second line. And then writing 1, 2, 3, 4 in place of a, b, c, d respectively, the
first line will contain 12.34, and the second line will contain 13.24. Let e be any
other symbol occurring in the first line, say in the duad ef, and in the second line
say in the duad eg; then g must occur in the first line in some duad gh, or the
first line will contain ef.gfi, and then the second line as containing eg will contain
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
299
also fh, that is, it will contain eg . fh. And then writing 5, 6, 7, 8 in place of e, f,
g, h respectively, the first line will contain 56 . 78 and the second line will contain
57 . 68. And continuing the like reasoning, it appears that the first line and the second
line may be taken to be
and
12.34.56.78.9 10. 11 12. 13 14. 15 16,
1 3. 2 4. 5 7. 6 8. 9 11. 10 12. 13 15. 14 16,
respectively. There will then be a line containing 1 4 which may be taken for the
third line, a line containing 1 5 which may be taken for the fourth line, and so on ;
viz. the successive lines may be taken to begin with 1 2, 1 3, 1 4, ..., 1 16 respectively.
Proceeding to form the synthematic arrangement, and starting with the first and
second lines and first column as above, it appears that in each of the remaining
lines there are three duads which occur of necessity, and putting these in the second,
third, and fourth places (the order of the duads in any line being immaterial), it is
seen that the second, third, and fourth columns can be filled up in one, and only
one way ; see the annexed first-half :
First-half common to all.
1 2
3 4
5 6
7 8
1 3
2 4
5 7
6 8
1 4
2 3
5 8
6 7
1 5
2 6
3 7
4 8
1 6
2 5
3 8
4 7
1 7
2 8
3 5
4 6
1 8
2 7
3 6
4 5
1 9
2 10
3 11
4 12
1 10
2 9
3 12
4 11
1 11
2 12
3 9
4 10
1 12
2 11
3 10
4 9
1 13
2 14
3 15
4 16
1 14
2 13
3 16
4 15
1 15
2 16
3 13
4 14
1 16
2 15
3 14
4 13
38—2
300
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
Four forms of second-half.
[763
I.
9 10
11 12
13 14
15 16
9 11
10 12
13 15
14 16
9 12
10 11
13 16
14 15
9 13
10 14
11 15
12 16
9 14
10 13
11 16
12 15
9 15
10 16
11 13
12 14
9 16
10 15
11 14
12 13
5 13
6 14
7 15
8 16
5 14
6 13
7 16
8 15
5 15
6 16
7 13
8 14
5 16
6 15
7 14
8 13
5 9
6 10
7 11
8 12
5 10
6 9
7 12
8 11
5 11
6 12
7 9
8 10
5 12
6 11
7 10
8 9
III.
9 10
11 12
13 14
15 16
9 11
10 12
13 15
14 16
9 12
10 11
13 16
14 15
9 15
10 16
11 13
12 14
9 16
10 15
11 14
12 13
9 13
10 14
11 15
12 16
9 14
10 13
11 16
12 15
5 15
6 16
7 13
8 14
5 16
6 15
7 14
8 13
5 13
6 14
7 15
8 16
5 14
6 13
7 16
8 15
5 11
6 12
7 9
8 10
5 12
6 11
7 10
8 9
5 9
6 10
7 11
8 12
5 10
6 9
7 12
8 11
II.
9 10
11 12
13 14
15 16
9 11
10 12
13 15
14 16
9 12
10 11
13 16
14 15
9 14
10 13
11 16
12 15
9 13
10 14
11 15
12 16
9 16
10 15
11 14
12 13
9 15
10 16
11 13
12 14
5 14
6 13
7 16
8 15
5 13
6 14
7 15
8 16
5 16
6 15
7 14
8 13
5 15
6 16
7 13
8 14
5 10
6 9
7 12
8 11
5 9
6 10
7 11
8 12
5 12
6 11
7 10
8 9
5 11
6 12
7 9
8 10
IV.
9 10
11 12
13 14
15 16
9 11
10 12
13 15
14 16
9 12
10 11
13 16
14 15
9 16
10 15
11 14
12 13
9 15
10 16
11 13
12 14
9 14
10 13
11 16
12 15
9 13
10 14
11 15
12 16
5 16
6 15
7 14
8 13
5 15
6 16
7 13
8 14
5 14
6 13
7 16
8 15
5 13
6 14
7 15
8 16
5 12
6 11
7 10
8 9
5 11
6 12
7 9
8 10
5 10
6 9
7 12
8 11
5 9
6 10
7 11
8 12
And it is to be noticed that in this first-half the upper part, or first seven
lines, give in fact the synthematic arrangement for the 8-question ; so that (as
remarked above) in this 8-question there is but one form of synthematic arrangement.
Proceeding to fill up the remaining columns, the duad 59 cannot be placed in
any line which contains a 5 or a 9 ; that is, it must be placed in some one of the
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
301
last 4 lines; and placing it successively in each of these, it appears that the columns
can be filled up in one, and only one, way ; we have thus the above " four forms
of second-half," each of which, taken in conjunction with the common first-half, gives
a synthematic arrangement of the 16 numbers.
Each of these synthematic arrangements may be converted into a square, the
first line of which is formed with the numbers 1 to 16 in order, and the other
fifteen lines of which are derived from the fifteen lines of the synthematic arrange
ment respectively : thus the line
1 2. 3 4. 5 6. 7 8. 9 10. 11 12. 13 14. 15 16
gives the second line of
1 2. 3 4. 5 6. 7 8. 9 10. 11 12. 13 14. 15 16,
2 1. 4 3. 6 5. 8 7. 10 9. 12 11 . 14 13. 16 15,
and so in other cases. And conversely, by comparing with the first line of the
square each of the other fifteen lines respectively, we have the fifteen lines of the
synthematic arrangement ; we thus obtain the four squares presently given. These
squares are not required in the sequel, but they serve to put in a clearer light
the construction of the synthematic arrangements ; by converting in like manner into
a square the formula p. 332 of Young's paper, it appears that his arrangement is in
fact the first of the foregoing four arrangements. The squares are
1234
2143
3412
4321
5678
6587
7856
8765
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
5678
6587
7856
8765
1234
2143
3412
4321
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
1234
2143
3412
4321
5678
6587
7856
8765
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
5678
6587
7856
8765
1234
2143
3412
4321
302
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
II.
12345
6
7
8
9
10
11
12
13
14
15 16
21436
5
8
7 10
9
12
11
14
13
16 15
3412
7
8
5
6 11
12
9
10
15
16
13 14
4321
8
7
6
5 12
11
10
9
16
15
14 13
5678
1
2
3
4 14
13
16
15
10
9
12 11
6587
2
1
4
3 13
14
15
16
9
10
11 12
7856
3
4
1
2 16
15
14
13
12
11
10 9
8765
4
3
2
1
15
16
13
14
11
12
9 10
9 10 11 12
14
13
16
15
1
2
3
4
6
5
8 7
10 9 12 11 ; 13
14
15
16
2
1
4
3
5
6
7 8
11 12 9 10 16
15
14
13
3
4
1
2
8
7
6 5
12 11 10 9
15
16
13
14 4
3
2
1
7
8
5 6
13 14 15 16
10
9
12
11
6
5
8
7
1
2
3 4
14 13 16 15
9
10
11
12 5
6
7
8
2
1
4 3
15 16 13 14
12
11
10
9 ! 8
7
6
5
3
4
1 2
16 15 14 13
11
12
9
10
7
8
5
6
4
3
2 1
III.
1234
2143
3412
4321
5678
6587
7856
8765
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
5678
6587
7856
8765
1234
2143
3412
4321
15 16 13 14
16 15 14 13
13 14 15 16
14 13 16 15
11 12 9 10
12 11 10 9
9 10 11 12
10 9 12 11
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
15 16 13 14
16 15 14 13
13 14 15 16
14 13 16 15
1234
2143
3412
4321
7856
8765
5678
6587
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
11 12 9 10
12 11 10 9
9 10 11 12
10 9 12 11
7856
8765
5678
6587
1234
2143
3412
4321
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
IV.
303
1234
2143
3412
4321
5678
6587
7856
8765
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
5678
6587
7856
8765
1234
2143
3412
4321
16 15 14 13
15 16 13 14
14 13 16 15
13 14 15 16
12 11 10 9
11 12 9 10
10 9 12 11
9 10 11 12
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
16 15 14 13
15 16 13 14
14 13 16 15
13 14 15 16
1234
2143
3412
4321
8765
7856
6587
5678
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
12 11 10 9
11 12 9 10
10 9 12 11
9 10 11 12
8765
7856
6587
5678
1234
2143
3412
4321
The foregoing investigation of the synthematic arrangements is exhaustive : it
thereby appears that there are at most four types, viz. that every synthematic
arrangement is of the type of one or other of the four arrangements above written
down. The real nature of these is perhaps more clearly seen by means of the
corresponding squares; and it will be observed, that there is in the first square a
repetition of parts without transposition, which does not occur in the other three
squares; this seems to suggest, that while the first square (and therefore the first
synthematic arrangement) is really of a distinct type, the other three squares (or syn
thematic arrangements) may possibly belong to one and the same type. If this were
so, it would be sufficient to prove the 16-theorem (viz. the non-existence of the
16-square formula) for the first and for any one of the other three synthematic
arrangements ; but I provisionally assume that the four types are really distinct, and
propose therefore to prove the theorem for each of the four arrangements separately.
The process is the same as for the 8-theorem ; we require the tetrads 1234, &c.,
contained in the synthematic arrangements. In any one of these, each line gives
£8.7, =28 tetrads, and the 15 lines give therefore 15.28, =420 tetrads: but we thus
obtain each tetrad 3 times, or the number of the tetrads is 420 -=- 3, = 140.
For the four arrangements respectively, these are as follows : the word " same "
means same as in column I.
304
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
1234
same
same
same
5 6
7 8
9 10
11 12
13 H
15 16
1357
6 8
9 11
10 12
13 15
14 16
1458
6 7
9 12
10 11
13 16
14 15
1 5 9 13
1 5 9 14
1 5 9 15
1 5 9 16
10 14
10 13
10 16
10 15
11 15
11 16
11 13
11 14
12 16
12 15
12 14
12 13
1 6 9 14
1 6 9 13
1 6 9 16
1 6 9 15
10 13
10 14
10 15
10 16
11 16
11 15
11 14
11 13
12 15
12 16
12 13
12 14
1 7 9 15
1 7 9 16
1 7 9 13
1 7 9 14
10 16
10 15
10 14
10 13
11 13
11 14
11 15
11 16
12 14
12 13
12 16
12 15
1 8 9 16
1 8 9 15
1 8 9 14
1 8 9 13
10 15
10 16
10 13
10 14
11 14
11 13
11 16
11 15
12 13
12 14
12 15
12 16
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
305
2358
same
same
same
6 7
9 12
10 11
13 16
14 15
2457
same
same
same
6 8
9 11
10 12
13 15
14 16
2 5 9 14
2 5 9 13
2 5 9 16
2 5 9 15
10 13
10 14
10 15
10 16
11 16
11 15
11 14
11 13
12 15
12 16
12 13
12 14
2 6 9 13
2 6 9 14
2 6 9 15
2 6 9 16
10 14
10 13
10 16
10 15
11 15
11 16
11 13
11 14
12 16
12 15
12 14
12 13
2 7 9 16
2 7 9 15
2 7 9 14
2 7 9 13
10 15
10 16
10 13
10 14
11 14
11 13
11 16
11 15
12 13
12 14
12 15
12 16
2 8 9 15
2 8 9 16
2 8 9 13
2 8 9 14
10 16
10 15
10 14
10 13
11 13
11 14
11 15
11 16
12 14
12 13
12 16
12 15
3456
same
same
same
7 8
9 10
11 12
13 14
15 16
C. XI.
306
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. IT. III. IV.
3 5 9 15
3 5 9 16
3 5 9 13
3 5 9 14
10 16
10 15
10 14
10 13
11 13
11 14
11 15
11 16
12 14
12 13
12 16
12 15
3 6 9 16
3 6 9 15
3 6 9 14
3 6 9 13
10 15
10 16
10 13
10 14
11 14
11 13
11 16
11 15
12 13
12 14
12 15
12 16
3 7 9 13
3 7 9 14 3 7 9 15
3 7 9 16
10 14
10 13
10 16
10 15
11 15
11 16
11 13
11 14
12 16
12 15
12 14
12 13
3 8 9 14
3 8 9 13
3 8 9 16
3 8 9 15
10 13
10 14
10 15
10 16
11 16
11 15
11 14
11 13
12 15
12 16
12 13
12 14
4 5 9 16
4 5 9 15
4 5 9 14
4 5 9 13
10 15
10 16
10 13
10 14
11 14
11 13
11 16
11 15
12 13
12 14
12 15
12 16
4 6 9 15
4 6 9 16
4 6 9 13
4 6 9 14
10 16
10 15
10 14
10 13
11 13
11 14
11 15
11 16
12 14
12 13
12 16
12 15
4 7 9 14
4 7 9 13
4 7 9 16
4 7 9 15
10 13
10 14
10 15
10 16
11 16
11 15
11 14
11 13
12 15
12 16
12 13
12 14
4 8 9 13
4 8 9 14
4 8 9 15
4 8 9 16
10 14
10 13
10 16
10 15
11 15
11 16
11 13
11 14
12 16
12 15
12 14
12 13
5678
same
same
same
9 10
11 12
13 14
15 16
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
307
7 9 11
10 12
13 15
14 16
9 12
10 11
13 16
14 15
7 9 12
10 11
13 16
14 15
6 8
9 11
10 12
13 15
14 16
9 10
11 12
13 14
15 16
9 10 11 12
13 14
15 16
9 11 13 15
14 16
9 12 13 16
14 15
10 11 13 16
14 15
10 12 13 15
14 16
11 12 13 14
15 16
13 14 15 16
same
same
39—2
308 ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. [763
As regards the signs, observe that the first line may always be written
ab + cd+ e/+&c.,
with the signs all of them + ; and then writing a, b, c, ... = 1, 2, 3, ..., 16 respectively,
the first line will be
1 2 + 3 4 + 5 6 + 7 8 + 9 10 + 11 12 + 13 14+15 16,
with the signs all of them + ; that is, we may assume e12, eM, &c., or say
1 2, 3 4, 5 6, 7 8, 9 10, 11 12, 13 14, 15 16,
<
all of them = + 1. And in the other lines, the signs of all the terms of any line
may be reversed at pleasure, that is, we may assume e13, e14, &c., or say 1 3, 1 4,
1 5, 1 6, 1 7, 1 8, 1 9, 1 10, 1 11, 1 12, 1 13, 1 14, 1 15, 1 16, all of them
= + 1.
Making these assumptions, then for any one of the synthematic arrangements the
several tetrads give as before relations between the signs ; among these are included
the results already obtained for the 8-question, and taking as before
we have the signs of the several terms belonging to the 8-question given as = ± 1
or + 6 as before. The remaining tetrads up to 1 8 12 13 then serve to express all
the remaining signs in terms of the as yet undetermined signs e, f, g, h, i, j, k, I,
m, n, o, p, q, r, s, t, u, v, w, x, y, z, a, /3, for instance
1 3. 9 11+ 1 e,
1 9. 3 11- 1 e,
1 11. 3 9+ 1 e,
that is, 3 9 = e, 3 ll=-e, 9 11 = e; and then the tetrads up to 2 8 9 15 serve to
express these signs in terms of the undetermined signs X, p, v, p, a; r; for instance
2 3. 9 12+ 1 i,
2 9. 3 12- I-/,
2 12. 3 9 + -1 e,
that is, -e=f=i; and in like manner 2 3 10 11, 2 4 9 11 and 2 4 10 12 give
respectively - e =f=j, -e = i =j, /= i =j ; that is, we have - e =f= i =j, = X suppose.
And in this way we have, for each of the four synthematic arrangements the signs
of all the terms expressed in terms of the undetermined signs 6, X, p, v, p, a, r,
as shown in the following table; where observe that the results apply to the four
synthematic arrangements separately, viz. the e, f, g, &c., and the 6, X, //,, v, p, <r, r
in each column are altogether independent of the like symbols in the other three
columns.
Signs for the four synthematic arrangements :
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
309
1 2
+ 1
same
same
same
3
+ 1
4
+ 1
5
+ 1
6
+ 1
7
+ 1
8
+ 1
9
+ 1
10
+ 1
11
+ 1
12
+ 1
13
+ 1
14
+ 1
15
+ 1
16
+ 1
2 3
+ 1
same
same
same
4
- 1
5
+ 1
6
- 1
7
+ 1
8
- 1
9
+ 1
10
_ i
11
+ 1
12
- 1
13
+ 1
14
_ i
15
+ I
16
- I
3 4
+ 1
same
same same
5
-$
6
e
7
e
8
- e
9
e - \
10
f A.
11
— e A
12
-/ -A
13
y - **
14
h p.
15
-9 f-
16
— h — ft.
310
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
4
5
e
same
same
same
6
6
7
-e
8
-e
9
i
X
10
3
X
11
-3
-X
12
— i
-X
13
k
P-
14
I
P-
15
V
- P-
16
-k
- P-
5
6
+ 1
+ 1
+ 1
1
7
-6
-6
-e
-6
8
e
e
e
e
9
m
— V
m
V
m —
V
m
V
10
n
V
n
V
n
V
n
V
11
o
- p
o
p
0 —
p
o
P
12
P
p
P
p
P
p
P
P
13
— m
V
— n —
V
— 0
p
-P
P
14
— n
— V
— 171 —
V
-P
p
— o —
P
15
— 0
p -p -
p
— m
V
— n —
V
16
-P
- p
— O —
p
— n —
V
— m —
V
6
7
e
e
0
9
8
e
e
e
0
9
<i
V
q.
V
q
V
q
V
10
r
V
r —
V
r
V
r
V
11
s
p
s
p
s
p
s
p
12
t
p t
p
t
p t
p
13
— r
— V
-q
V
- t
p — s —
p
14
~ ?
— V
- r
V
— s —
p
- t
p
15
- t
- p
— s —
P
— r —
V
- q —
V
16
~s
- p
- t
P
-q
V
— r
V
7
8
+ 1
+ 1
+ 1
+ 1
9
u
— cr
u —
cr
u —
cr
u
cr
10
V
cr
V
cr
V
cr
V
cr
11
w
— r
w
T
w —
T
w
T
12
X
T
X
T
X
T
X
T
13
— w
T
— X —
T
— u
cr
— V
or
14
— X
— T
— w —
T
— V —
cr
— u —
cr
15
— u
cr
— V —
cr
— w
T
/v*
r
16
— V
— cr
— u —
cr
— X —
T
— w —
T
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
311
8 9
V
(T
y
9
y
cr
y
cr
10
z
CT
%
— cr
t*
cr
z
— cr
11
a
T
a
T
a
T
a.
T
12
ft
ft
ft
T
ft
-T
13
-ft
— T —a
— T
— %
— cr
- y
— cr
14
— a
-ft
T
-y
— cr
— z
cr
15
~~ Z
-a- -y
— (T
-ft
— r
— a
— T
16
-y
— cr
— Z
a
— a
— r
-ft
T
9 10
+ i
+ 1
+ 1
+ 1
11
e
-X
e
- X
e
-X
e
-X
12
i
A.
i
X
•
X
i
X
13
m
— v
1
V
u
— cr
y
<r
14
9
V
in
V
y
cr
u
cr
15
it,
— cr
y
cr
in
— v
9
V
16
y
CT
u
CT
1
V
in
V
10 11
j
X
j
X
j
X
3
X
12
f
X
f
X
f
X
f
X
13
r
V
n
V
z
cr
v
cr
14
n
V
r
- v
V
cr
f"
— cr
15
z
<T
V
cr
r
V
n
v
16
V
cr
m
— cr
n
v
r
— v
11 12
+ 1
+ 1
1
+ 1
13
w
— r
a
r
0
~ P
s
P
14
a
T
W
T
s
P
0
P
15
o
~ P
s
P
w
— T
a
T
16
I
P
o
P
a
T
w
T
12 13
ft
T
X
T
t
P
P
P
14
X
T
ft
— T
P
P
t
~ P
15
t
P
p
P
ft
r
X
T
16
p
P
t
~ P
X
T
ft
— T
13 14
+ 1
+ 1
+ 1
+ 1
15
9
~ P-
9
~ P-
9
- P-
9
~ P*
16
k
P-
k
P-
k
P-
k
P-
14 15
I
P-
I
P-
I
P'
I
P-
16
h
P-
h
P-
h
P-
h
P-
15 16
+ 1
+ 1
+ 1
+ 1
312 ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. [763
We have now for the four arrangements respectively, by means of hitherto unused
tetrads, the following determinations of sign : these being in each case inconsistent with
each other.
First arrangement.
3 5 9 15 +-0.-0- that is,
3 9 515 X. p 0<r= Xp = — fj,v,
3 15 5 9 + 11. -v
3 5 10 16 +-0. a-
3 10 5 16 - \.-p -6(7=
3 16 5 10 + -fi. v
3 5 11 13 +-0.-T
3 11 5 13 - X. v 0T = -\v=
3 13 5 11 + -p.-p
3 5 12 14 +-0. r
312 514--X.-V
3 14 5 12 + ft. p
Second arrangement.
3 5 9 16 +-0. a-
3 9 5 16 X.-p
3 16 5 9+-ft. v
3 5 11 14 + -0. T
3 11 5 14 - \.-v
3 14 5 11 + At. p
3 5 12 13 +-0. r
3 12 5 13 --X.-y
3 13 5 12 +-ft. p
3 5 10 15 +-0. a-
3 10 5 15 - X.-p - 00- = Xp =
3 15 5 10 + ft. i;
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
Third arrangement.
3 5 9 13 +-6>.-<r
3 9 5 13 X. p 0o- = -Xp= pv,
3 13 5
313
3 5 10 14 +-B. a-
3 10 5 14 - X.-p
3 14 5 10 + p. v
3 5 11 15 +-0. -T
3 11 5 15 - X. v
3 15 5 11 + /*.-p
3 5 12 16 +-0. r
3 12 5 16 --\.-v
3 16 5 12 + -/i. p
Fourth arrangement.
3 5 9 14 + _,9. o-
3 9 5 14 __x._p
3 14 5 9 + /*. j,
3 5 10 13 +-0. a-
3 10 5 13 - \.-p
3 13 5 10 +-At. i;
3 5 11 16 +-0. T
3 11 5 16 - X.-i/
3 16 5 11 +-/i. p
3 5 12 15 +-6. -r
3 12 5 15 --x.-z,
3 15 5 12 +-/A. p
0r =
= — fj,p,
= -\v= pp.
0<r = \p = —
0T = - \V =
0T = — \V = — pp.
And it hence finally appears, that we cannot, in any one of the four arrange
ments, determine the signs so as to give rise to a 16-square theorem; that is, the
product of a sum of 16 squares into a sum of 16 squares cannot be made equal to
a sum of 16 squares.
C. XI.
40
314 [764
764.
THE BINOMIAL EQUATION a?-l=0: QU1NQUISECTION.
[From the Proceedings of the London Mathematical Society, vol. xn. (1881), pp. 15, 16.
Read December 9, 1880.]
THE theory should be precisely analogous to those for the trisection and quarti-
section (see my paper, " The Binomial Equation a? — 1 = 0, Trisection and Quartisection,"
Proceedings of the London Mathematical Society, vol. XL (1879), pp. 4 — 17, [731]),
only I have not been able to carry it so far. We have in the present case five
periods X, Y, Z, W, T, the actual expressions for which, X = rjl+..., Y=rj3+..., etc.,
with Reuschle's selected prime root g, can be (for the primes 5?i + 1 under 100) at
once written down by means of the table given, pp. 16, 17, of that paper; [see this
volume, pp. 95, 96]. The relations between the periods are of the form
X Y Z W T
X2 = a b c d e
XY=f g h i j
XZ =k I m n 0 ;
that is, we have
X* = (a, b, c, d, e\X, Y, Z, W, T},
and thence, by cyclical permutations,
Y2=(e, a, b, c, dQ „ }, etc.;
viz. from the value of X2 we have those of Y2, Z2, W2, T2; from the value of XY
those of YZ, ZW, WT, TX ; and from the value of XZ those of YW, ZT, WX, TY.
764] THE BINOMIAL EQUATION Xp — 1 = 0 I QUINQUISECTION. 315
From the equation X+Y + Z+W+T= — I, multiplying by X and then substi
tuting for X2, XY, &c., their values, we obtain
-a=l+f+k +m + g,
— b = g + I + n + h,
— c = h + m + o +i,
— d = i -f n + k +j,
-e= j + o + I +f,
which determine (a, b, c, d, e) in terms of (/, g, h, i, j) and (k, I, m, n, o). It is,
moreover, easy to prove that
f + g + h +i+j= UP-1\
k+l+m + n + o= i(p - 1),
whence also
a + b+ c + d + e = -I-±(p-l).
We obtain other relations between the coefficients by considering the two triple
products XYZ and XYW: these are all that need be considered, since the other
triple products are deducible from them by cyclical permutations. From the first of
these we have
X.YZ = Y.XZ =Z .XY,
and from the second
X.YW=Y.XW=W.XY;
and if we herein substitute for YZ, XZ, &c., their values, and then in the resulting
equations for X2, XY, &c., their values as linear functions of X, Y, Z, W, T, we
obtain in all 5.2.2=20 quadric relations between the 15 coefficients; or if we
substitute for (a, b, c, d, e} their foregoing values, in all 20 relations between the 10-
coefficients (/, g, h, i, j) and (k, I, m, n, o). These are at most equivalent to 8
independent equations, since we have, besides, the sums f+g+h+i+j and k+l+m+n+o
each = i(p — 1); but I have not succeeded in finding the connexions between them,
or even in ascertaining to how many independent equations they are equivalent.
For any given prime p=5n + I, the values of the coefficients, and also the
coefficients of the quintic equation for the periods, could of course be calculated
directly from the expressions of the periods; but for the primes under 100, that is,
for the values 11, 31, 41, 61, 71, they are at once obtained from Reuschle. We
have thus the two Tables, the former giving the coefficients a, b, ...,n, o, and the
latter the coefficients of the quintic equations.
40—2
316
THE BINOMIAL EQUATION Xp — I = 0 : QUINQUISECTION.
[764
TABLE 1.
a
b
c
d
e
p
f
9
h
i
J
k
I
m
n
o
11
- 2
1
- 2
- 2
- 2
1
0
0
1
0
0
0
0
1
1
31
- 4
6
- 6
- 4
- 5
0
1
2
1
2
0
2
2
1
1
41
- 8
- 5
6
- 6
- 8
3
0
2
1
2
2
2
2
1
1
61
- 10
9
- 12
-8
- 10
3
2
2
3
2
0
2
4
3
3
71
- 14
- 10
- 12
- 9
- 12
4
2
3
2
3
2
3
5
2
2
TABLE 2 OF THE QUINTIC
EQUATIONS.
COEFFICIENTS OF
,
,<
rf
T
f1
1
11
1
1
_ 4.
- 3
+ 3
+ 1
31
1
1
- 12
_ 2
+ 1
+ 5
41
1
1
- 16
+ 5
+ 21
- 9
61
1
1
- 24
- 17
+ 41
-23
71
1
1
- 28
+ 37
+ 25
+ 1
= 0.
765] 317
765.
ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE.
[From the Proceedings of the London Mathematical Society, vol. xn. (1881), pp. 103 — 108.
Read March 10, 1881.]
THE skew surface is taken to be such that the strip between two consecutive
generating lines is rigid, and that the flexure takes place by the rotation of the
strips about the generating lines successively. The theory of the flexure is well known,
but I am not aware that the theory of the equilibrium of such a surface, when acted
upon by any given forces, has been considered; it is, however, a question which
presents itself naturally in connexion with those relating to other continuous bodies
treated of in the Mecanique Analytique, and forms a good example of the principles
made use of.
To begin with the mechanical theory: we may regard the forces as acting on
the generating lines regarded as material lines ; and if for an element of mass dm,
coordinates (x, y, z} of a particular generating line G, the forces parallel to the axes
are X', Y', Z', then the corresponding term in the equation of equilibrium is
and observing that there are (as will afterwards appear) Jive geometrical conditions, which
I represent by U1 = 0, U2 = 0, . . . , U5 = 0, the equation of equilibrium is
8{(X/Ba: + Y'Sy + Z'Sz)dm+T1SU1+TzSUs+ T38U3 + T4SU, + T5SU5} =0,
where T1} T2, ..., T5 are the indeterminate multipliers, representing colligation-forces
which correspond to the five geometrical conditions respectively.
Taking (f, 77, £) for the coordinates of a particular point P on the generating
line ; p, q, r for the cos-inclinations of the line (whence ZTj = p2 + q* + r2 — 1=0 is one
of the geometrical relations), and p for the distance of dm from P, we have
«> V> z= £+ pp, V + pq, C+ pr,
Bx, By, Sz = 8% + pSp, Srj + pSq, 8% + pSr.
318 ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE. [765
The summation S extends first to the different points of the generating line, and
then to the different generating lines; applying it first to the particular generating
line, we write
SX'dm, SY'dm, SZ'dm, SX'pdm, SY'pdm, SZ'pdm
= X, Y, Z, I, M, N,
where X, Y, Z are the whole forces, and L, M, N the whole moments about the
point P, for the generating line G ; retaining the same summatory symbol 8, as now
referring to the different generating lines, the equation becomes
We have now to consider the geometrical theory of the flexure. Taking on the
skew surface an arbitrary curve cutting each generating line G in a point P, coordinates
(£> ^j £)» and taking a for the distance along the curve of the point P from a fixed
point of the curve ; also p, q, r, as before, for the cos-inclinations of the generating
line G, then when the surface is in a determinate state, £, 77, f, p, q, r are given
functions of cr ; but these functions vary with the flexure of the surface, with, however,
certain relations unaffected by the flexure; and the problem is to find first these
relations. As already mentioned, one of them is p" + (f + r2 — 1 = 0.
Taking P' as the consecutive point on the curve, so that the direction of the
element PP' is that of the tangent PT at P, it is convenient to write I, m, n for
the cosine-inclinations of the tangent ; we have, it is clear,
The conditions in order to the rigidity of the strip, are that the angles GPP.
G'P'P (=180°-£'PT), and the inclination G'P' to GP, shall have given values^
P P' T
variable it may be from strip to strip — that is, these values must be given functions
of a: Taking Z GPT = I, the value of G'P'T can differ only infinitesimally from that
of GPT, and we take it to be G'P'T = I - Slda-; also the inclination GP to G'P'
is an infinitesimal, = %dv : we have /, H, © given functions of cr. It is to be
remarked that these conditions imply, inclination of G'P' to tangent plane GPT at P
has a given value Ada- ; in fact, if through P we draw a line Py parallel to FG't
then, if P is regarded as the centre of a sphere which meets PG, Pj, PT in the
765] OX THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE. 319
points g, g', t respectively, we have a spherical triangle gg't, the sides of which are
/ — Cider, I, and Sdcr, and of which the perpendicular g'm is = Ada- ; we have thus
an infinitesimal right-angled triangle, the base and altitude of which are Cider, Ada,
g tlda-
and the hypothenuse is ®do- ; whence ®2 = H2 4- A2. In the case of the developable
surface A = 0 and @ = O. It may be remarked that, when the curve on the skew
surface is the line of striction, we have H = 0 ; in fact, taking P to be on the line
of striction, the line
qr — qr rp — rp pq —pq
through (£, rj, £) at right angles to the two generating lines, meets the consecutive
generating line X, Y, Z = % + pp', if + pq', g + pr' ; and the condition that this may
be so is easily found to be O = 0.
Take, for a moment, p', q', r' for the cos-inclinations of the consecutive generating
line P'G'\ we have
Ip + mq + nr = cos /,
lp' + mq + nr' = cos (/ — Cider),
' + <<' + rr> = cos
and then writing p', q, r' =p + dp, q + dq, r + dr, and observing that the equation
p'2 + q'2 + r'2 = 1 gives
pdp + qdq + rdr=-% (dp* + dq2 + dr2),
these equations and the before-mentioned two equations become
(Z70 p* + q2 +r2-l = 0,
( £/,) I3 + ra2 + n2 - 1 = 0,
( U3) lp + mq + nr — cos 7 = 0,
( U4) Idp + mdq + ndr — O sin Ida- = 0,
( U5) dp2 + dq* + drn- - ®2dcr2 = 0,
which equations, considering therein I. m, n as standing for their values -^ , -? > ^-r ,
da- da- da-
are the geometrical relations which connect the six variables f, 77, £", p, q, r, considered
as functions of a: And in these equations /, O, @ denote given functions of a-,
invariable by any flexure of the surface.
To complete the geometrical theory, it is to be observed that we can by flexure
bring the generating lines of the surface to be parallel to those of any given cone
320 ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE. [765
C(p, q, r) = 0, where C (p, q, r) denotes a homogeneous function of (p, q, r). Hence,
joining to the foregoing five equations this new equation
C(p, q, r) = 0,
these six equations determine £, 77, £, p, q, r as functions of a. To make the
solution completely determinate, we have only to assume for the point P, which
corresponds, say, to the value cr = 0, a position in space at pleasure, and to take the
corresponding generating line PG parallel to a generating line, at pleasure, of the cone.
As an example, writing 7 to denote an arbitrary constant angle, if the invariable
conditions are
7 = 7, @ = sin 7, H = 0,
then the five equations are
p*+ 22 + r*- I = 0,
12+ m? + n-- 1 =0,
Ip + mq + nr — cos 7 =0,
dp2 + d(f + dr- - sin2 7 da* = 0..
Idp + mdq + ndr = 0.
We assume first
and secondly
C (p, q, r) = r, = 0.
Then, in the former case, we find the solution
p, q, r = — sin 7 sin cr, sin 7 cos cr, cos 7 ;
£, 77, £ = cos cr, sin cr, 0 ;
giving
x, y, z = cos o- — p sin 7 sin cr, sin cr + p sin 7 cos cr, cos 7 ;
and consequently
#2 -f- y2 — z- tan2 7 = 0,
the hyperboloid of revolution. And, in the latter case,
p, q, r = cos (cr sin 7), sin (a sin 7), 0,
£. *?> f = cot 7 sin (cr sin 7), — cot 7 cos (cr sin 7), a sin 7,
that is,
x, y = cot ysin.z + p cos £, — cot 7 cos 0 + p sin 2,
whence
# sin z — y cos z = cot 7,
a skew helicoid generated by horizontal tangents of the cylinder #2 + f = cot2 7. This
is a known deformation of the hyperboloid.
765] ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE. 321
Returning now to the mechanical problem, we have to consider the terms
8 . T^Ky + 22 +?'2-l)
+ T2S|(Z2 +ms + wa_i)
+ T3B (lp + mq + nr — cos 7)
m * f7dp dq dr _ J\
+ T48 (I f- -4- m -j^- + n -j- fl sin 1}
\ do- da- da /
The first term gives, under the sign 8,
T^pSp + qSq + rSr). (*)
The second term gives, in the first instance,
or, since in general
= n"Sf " - n'Sf + ^ (-
then, attending only to the terms under the sign 8, these are
rp i $f- @> rn 5. f" rn Sn*
= — ~T~ J. nl . o£ — --J- J.2in . or) — -j- ZjJl . oc.
The third term giVes
where the second line,
attending only to the terms under the sign 8, gives
T »•> s>- ** /IT _ a d
The fourth term gives
/ df) do dT
T4 ( ..- ol + j- 8m + -j- B
\(ta- da a<r
rn f I W1 n \
+ Tt ( -j- dop + j— doq + -j - dor } ,
\ao~ da- da- J
where the first line, written under the form
(dp . c?g . d?' . \
-f- a6^ + 3* 056?; + -j- aoc
\da- da- da- *J
j
ao- \da-
and attending only to the terms under the sign S, gives
d (T dp\ d ( dq\ d ( dr
--
C. XI.
322
ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE.
[765
and the second line, attending in like manner only to the terms under the sign 8,
gives
71 / Xr» _ Y7 vn f)n T . 11 $r (*\
j — J. 4b . OIJ — , J- 4 III, . Ul[ 7 J- 4" . C" . V, ^
The fifth term, written under the form
da
and attending only to the terms under the sign S, gives
d rr dp , d efy ^._ d „ dr
(•)
(*)
where in each case I have marked with an asterisk the lines which present them
selves in the final result.
Hence, joining to the foregoing the force-terms
XBt; + Yfy + ZS£ + LBp + MBq + NBr, (*)
and equating to zero the coefficients of S£, Brj, B£, Bp, Bq, Br respectively, we have
d T j — T r> -— T&-
— , J. f)V ' ~ 7 J. 9jJ 7 -*-4
J- da r J"-
da-
0=7
0 = L + T,p
da
d Tdq
~~J •* 4 7 >
da- da-
A T n - d Tr - -d- T —
da 2 da da da '
+ T,l -
d
dp
j J- 4 ' ~ J *l J >
da da da
+ Tsm -
m
T3n -
,
da
d rfl dr
, TI-J-
da da
where it will be recollected that I, m, n stand for ,
, the variables being
£ V, £, P> <1> r> and °"- Tne elimination of Tlt T2,...,T5 from the six equations
should lead to a relation between f, TJ, £ p, q, r, which, with the foregoing five
relations, would determine the six variables £, 77, £, p, q, r in terms of a.
In particular, the forces and moments X, Y, Z, L, M, N may all of them
vanish; assuming that T1} TZ,...,TS do not all of them vanish, we still have the
sixth relation, which (with the foregoing five relations) determines £, 77, £ p, q, r in
terms of a ; and it is to be remarked that the problem in question, of the figure
of equilibrium of the skew surface not acted upon by any forces, is analogous to
that of the geodesic line in space ; only whilst here the solution is, curve a straight
line, the solution for the case of the skew surface depends upon equations of a
complex enough form ; in the case of the developable surface, the required figure is
of course the plane.
766] 323
766.
ON THE GEODESIC CURVATUEE OF A CURVE ON A
SURFACE.
[From the Proceedings of the London Mathematical Society, vol. xil. (1881), pp. 110 — 117.
Read April 14, 1881.]
THERE is contained in Liouville's Note II. to his edition of Monge's Application
de I'Analyse a la GeomJtrie (Paris, 1850), see pp. 574 and 575, the following
formula,
1 _ di _1 dG ._ _1 dE . .
p~~ds + 2G</E duc< 2E^G dv S1
di cos i sin i
— _ _| — — ^ .
as p2 p!
which gives the radius of geodesic curvature of a curve upon a surface when the
position of a point on the surface is defined by the parameters u, v, belonging to
a system of orthotomic curves; or, what is the same thing, such that
ds* = Edu2 + Gdv".
Writing with Gauss p, q instead of u, v, I propose to obtain the corresponding formula
in the general case where the parameters p, q are such that
ds* = Edp2 + ZFdpdq + Gdq2.
I call to mind that, if PQ, PQ' are equal infinitesimal arcs on the given curve
and on its tangent geodesic, then the radius of geodesic curvature p is, by definition,
a length p such that 2p . QQ' = PQ2. More generally, if the curves on the surface
are any two curves which touch each other, then p as thus determined is the radius
of relative curvature of the two curves.
41—2
324 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766
The notation is that of the Memoir, " Disquisitiones generales circa superficies
curvas " (1827), Gauss, Werke, t. in. ; see also my paper " On geodesic lines, in
particular those of a quadric surface," Proc. Lond. Math. Society, t. iv. (1872),
pp. 191 — 211, [508]; and Salmon's Solid Geometry, 3rd ed., 1874, pp. 251 et seq.
The coordinates (x, y, z) of a point on the surface are taken to be functions of
two independent parameters p, q; and we then write
dx + \d?x = adp + a'dq + ^ (adp- + 2<z' dpdq + a" dqz),
dy + %d-y = bdp + b'dq + \ (/3dp- + 2/3' dpdq + /3"dq-),
dz + \<&z = cdp+ c'dq + % (ydp2 + 2 7' dpdq + 7" c&f ) :
E, F, £=a2 + 62 + c2, aa' + bb'+cc, a'2+6/2 + c'-; V* = EG-F2:
and therefore
ds2 = Edp- + 2Fdpdq + Gdq-,
where E, F, G are regarded as given functions of p and q.
To determine a curve on the surface, we establish a relation between the two
parameters p, q, or, what is the same thing, take p, q to be functions of a single
parameter Q ; and we write as usual p', p", q', etc., to denote the differential
coefficients of p, q, etc., in regard to 6 ; we write also Elt E2, etc., to denote the
J pi JTji
differential coefficients -=- , -=— , etc. In the first instance, 6 is taken to be an
dp dq
arbitrary parameter, but we afterwards take it to be the length s of the curve from
a fixed point thereof.
First formula for the radius of relative curvature.
Consider any two curves touching at the point P, coordinates (x, y, z) which
are regarded as given functions of (p, q); where (p, q) are for the one curve given
functions, and for the other curve other given functions, of 6.
The coordinates of a consecutive point for the one curve are then
x+dx+^d^x, y+dy + ^d^y, z + dz + ^d*z,
where
dp = p'dd + %p"d62, dq = q'dO + $ q"d6- ;
hence these coordinates are
x + (ap1 + a'q') dd + $ (ap'- + Za'p'q' + a'Y2) d&- + £ (ap" + a'q") d&,
and for the other curve they are in like manner
x + (ap + a'q'} dd + \ (a/2 + Za'p'q' + a"q'*) dffi + t (aP" + a'Q") dfr,
766] ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. 325
the only difference being in the terms which contain the second differential coefficients,
p", q" for the first curve, and P", Q" for the second curve. Hence the differences
of the coordinates are
i {a (p" - P") + a' (q" - Q")} dP, * [b (p" - P") + V (q" - Q")} d&,
i{c(p"-P") + c'(q"-Q")}dP,
and consequently the distance QQ' of the two consecutive points Q, Q' is
^GW^P*, q"-Q'J M*.
The squared arc PQ2 is
= (E, F, G~&p', qjdfr;
and hence, if as before 2p.QQ' = PQ2, that is, - = 2Q$ + PQ>, then
P
I _ VQg, F, ~G$p" - P", g'^Q^'
p~ (E, F, G\p',qJ
the required formula for p.
Second formula for the radius of relative curvature.
We now take the variable 9 to be the length s of the curve measured from a
fif) fj 2/i-j
fixed point thereof, so that p, p", etc. denote ~~ , ,~ , etc. We have therefore
ds ' ds"
l=(E, F,
and the formula becomes
But, differentiating the above equation as regards the curve, we find
0 = 200, f,
where JS, F, G are used to denote the complete differential coefficients E^' + E^q', etc.
And similarly, differentiating in regard to the tangent geodesic, we obtain
0=2(#, F, G\p\ q'%P", Q") + (E, F, G%p', qj-
and hence, taking the difference of the two equations,
0 = (E,F,
Hence, in the equation for -, the function under the radical sign may be written
(E, F, &&, qJ.(E, F, 0&>"-P", q"-Q'J-{(E, F,
326 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766
which is identically
= (EG - F*) [p (q" - Q") - q' (p" - P")}2-
Hence, extracting the square root, and for \ lEG — F* writing V, we have
1 , H
or say
which is the new formula for the radius of relative curvature.
Formula for the radius of geodesic curvature.
In the paper " On Geodesic Lines, etc.," p. 195, [vol. vm. of this Collection, p. 160],
writing EG - F* = V'2, and P", Q" in place of p", q", the differential equation of the
geodesic line is obtained in the form
(Ep1 + Fq'} {(2^ - Es)p'* + 2GlP'q' + GJ*}
- (Fp + Gq'} {E#'* + 2E.2p'q' + (2F2 - GJ q*}
or, denoting by fl the first two lines of this equation, we have
The foregoing equation gives therefore, for the radius of geodesic curvature,
which is an expression depending only upon p', q', the first differential coefficients
(common to the curve and geodesic), and on p", q", the second differential coefficients
belonging to the curve.
Observe that O is a cubic function of p', q : we have
n = (2l, 33, (5, £>$/, qj,
the values of the coefficients being
2- FEly
1-3FE2- GE,,
3)= FG2-2GF2+ GG,.
766] ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. 327
The Special Curves, p = constant and q = constant.
Consider the curve p = const. For this curve p' — 0, p" = 0 ; therefore also Gq'" = I,
and, if R be the radius of geodesic curvature, then
Similarly for the curve q = const. Here q' = 0, ^" = 0; therefore Ep'- = l, and, if S
be the radius of geodesic curvature, then
1-4^ .*_*_
8 Y*p ~
These values of R and S are interesting for their own sakes, and they will be
introduced into the expression for the radius of geodesic curvature p of the general
curve.
Transformed Formula for the Radius of Geodesic Curvature.
From the values of -^ , -~ , we have
H b
where the term in { } is
, 21 3)
= 2lp 3 — j=j p + xjp'^q + (&p q2 + \£)q'3 — -^ q'.
Jll (jT
The terms in 21 are
= ~ §/ C1 - EP'2\ = ~ § P' (ZFp'q' + Gq'2),
and those in 3) are
Hence the whole expression contains the factor p'q', and is, in fact,
or substituting for 51, 53, (S, 3) their values, this is
= M P' ~ GE, + EG,
' - GE2 + EG, - + 2FF,
328 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766
say this is
= p'
and the formula thus is
_ _= _p + ?
Taking <£, # to be the inclination of the curve to the curves q = const., p = const.,
respectively, and w (= <f> + 6) the inclination of these two curves to each other, then
cos
Fp'+Gqf a Ep' + Fq' F
— __•*._ r*r\Q rt — _— i— f*r\Q f*\ —
j L-Uo I/ — ~~ , V/Uo tt/ — , _
Vp' Vq' V
sin <p = — *- , sin P = — i sin u> = — — — ;
hence —. — — =»' JE, —. - = q'\/G, and the formula may also be written
smtu smo)
1 sin 6 1 sin <t> I ,,. , ,, „ ,. i- , ,._ , ,, ,.
--- - - •» - - - v=V W ~ P 9) + TrP<l (LP + Mq ').
p sin to R sin co S
The Orthotomic Case F=0, or ds* = Edp" + Gdq2.
The formula becomes in this case much more simple. We have
1 = Ep'* + Gq"2, V= \f~EG, 6) = 90°, sin^ = cos</);
and the term Lp' + Mq' becomes = EG — EG, if, as before, E, G denote the complete
differential coefficients E^p' + E.2q' and G±p + Gzq. The formula then is
cos <> sn
-.r , . ,. ,, ,, , „, -^~.
R — = (pq ~pq^ + F( ~ }>
1 1 l-C1 — ^Jf
where the values -^ and -^ are now = -^—r--^, and ?T ,J , respectively. But we have
ri o (r v-«* ** v^
moreover <f> = tan~1^-r . ~ , and thence
= - V(p'q"-P"q) ~ M (EG - EG) ;
or the formula finally is
1 cos</> sin <£.,,_ A
P ~ ~S~ ~^ " f ^ :
which is Liouville's formula referred to at the beginning of the present paper. It
will be recollected that <6' is the differential coefficient -*- with respect to the arc s
ds
of the curve.
7f)6] ON THE GEODESIC CURVATI7KE OF A CURVE ON A SURFACE. 329
ADDITION. — Since the foregoing paper was written, I have succeeded in obtaining
a like interpretation of the term
V(p'q"-p"q} + ±p'q' (Lp' + Mq'\
which belongs to the general case. I find that these terms are, in fact, = — <£ + &>1p';
or, what is the same thing (since w = (f> + 6 and therefore w^p + &>2</ = <j> + 0), are
= 0 — co2q. It will be recollected that <£ is the inclination of the curve to the curve
q = c, which passes through a given point of the curve, <j> is the variation of (f>
corresponding to the passage to the consecutive point of the curve, viz., <f> + <j>ds is
the inclination at this consecutive point to the curve q = c + dc, which passes through
the consecutive point ; co is the inclination to each other of the curves p = b, q — G,
which pass through the given point of the curve, wl the variation corresponding to
the passage along the curve q = c, viz., w + ^ds is the inclination to each other of
the curves p = b + db, q = c; and the like as regards 0 and o>2.
For the demonstration, we have, as above,
Vp' V
4> = iaU~1W^Gq" W = tan->
where
and moreover Ep''2 + ZFp'q + Gq'- = 1. In virtue of this last equation,
Vy*
and we have
where
D = (Fp1 + Gq) p'V- Vp' (Fp' + Gq'} •
or, since VZ = EG-F'2, and thence 2FF = GE -2FF '+ EG, we have
D =^r {(Fp'+ Gq')(GE-<2FF+EG)- 2(EG-F*)(Fpf + Gq%
Substituting herein for E, F, G their values E^' + E^q', F.p'+F^q', G^'+Gft', the
term in { } becomes
where
/ = FGE1 - 2EGF, + EFG,,
J = G^ - 2FGFl + (-EG + 2^2) G1 + FGE, - 2EGF2 + EFG,,
K = GPE,_ - 2FGF, + (-EG+ 2Fa) G,.
But from the equation &> = tan"1 ^ , differentiating in regard to p, we obtain
c. xi. 42
330 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766
or, for p writing
p' (E^ + ZFp'q' + Gq'*), = Ep' (> + 2 J p'q' + | q'^ ,
we have
1'
4> - «y = - V(p'q' - p'q)
The terms in p'3 destroy each other, and the form thus is
where
and, upon substituting herein for /, «/, -K" their values, we find
9 //"" W
L = -GE1 + EG, + p l - 2FFl - FE* + 2EF, -
M =-GE, + EG, - + 2FF, + FG1-2 GF1
viz., these are the values denoted above by the same letters L, M. The final result
thus is
-p a
= 6 — o)2q',
where the meanings of the symbols have been already explained. A formula sub
stantially equivalent to this, but in a different (and scarcely properly explained)
notation, is given, Aoust, "The'orie des coordonne'es curvilignes quelconques," Annctli
di Matem., t. n. (1868), pp. 39 — 64; and I was, in fact, led thereby to the foregoing
further investigation.
As to the definition of the radius of geodesic curvature, I remark that, for a
curve on a given surface, if PQ be an infinitesimal arc of the curve, then if
from Q we let fall the perpendicular QM on the tangent plane at P (the point
M being thus a point on the tangent PT of the curve), and if from M, in the
tangent plane and at right angles to the tangent, we draw MN to meet the
osculating plane of the curve in N, then MN is in fact equal to the infinitesimal
arc QQ; mentioned near the beginning of the present paper, and the radius of geodesic
curvature p is thus a length such that 2p . MN = PQ\
767] 331
767.
ON THE GAUSSIAN THEORY OF SURFACES.
[From the Proceedings of the London Mathematical Society, vol. xn. (1881), pp. 187 — 192.
Bead June 9, 1881.]
IN the Memoir, Bour, "Theorie de la deformation des surfaces" (Jour, de VEc.
Polyt., Cah. 39 (1862), pp. 1—148), the author, working with the form ds" = dv2 + g*du-
as a special case of Gauss's formula ds2 = Edp- + ZFdpdq + Gdq-, obtains (p. 29) the
following equations which he calls fundamental : —
1 7 _,
1 m.-> TT TT
ffdv =
[IV.]...
where gl is written to denote -j- , and where (see p. 26)
H is the curvature of the normal section containing the tangent to the curve
v = constant,
H1 is the curvature of the normal section at right angles to the preceding,
containing the tangent to the (geodesic) curve u = constant,
T is the torsion of the same geodesic curve ;
or, what is the same thing (see p. 25), the quadric equation for the determination
of the principal radii of curvature at the point of the surface is
du dv
dv du
42—2
332
ON THE GAUSSIAN THEORY OF SURFACES.
[767
Writing for greater convenience K in place of the suffixed letter Hl, also V
instead of g, so that the differential formula is ds2 = dv2 + V2du2, the equations become
Lay m
dT d.HV
du dv
dv
d.TV2
j
dv
'£ =«;
,Vn = V(T2-HK),
or, if we use the suffix 1 to denote differentiation in regard to v, and the suffix 2
to denote differentiation in regard to u, then the equations are
_J1 — 712_ 77 ir
V ~ '
or, what is the same thing,
= 0.
I wish to show how these formulae connect themselves with formulae belonging
to the general form ds2 = Edp2 + 2Fdpdq + Gdq2. These involve not only Gauss's coefficients
E, F, G, but also the coefficients E', F', G' belonging to the inflexional tangents ;
and, for convenience, I quote the system of definitions, Salmon's Geometry of Three
Dimensions, 3rd ed., 1874, p. 251, viz.
dx, dy, dz = adp + a'dq, bdp + b'dq, cdp + c'dq ;
d2x = adp2 + 2a'dpdq + af'dq2,
d-y = /3dp* + 2/3'dpdq + @"dq2,
d?z = ydp2 + fy'dpdq + y"dq2 ;
A, B, C=bcf- b'c, ca? - c'a, aV - ab ; F2 = EG - F* ;
E' = Aa + Bp + Cy, F' = Aa' + B/3' + Cy', G' = A*" + B/3" + Cy",
so that E', F', G' are, in fact, the determinants
a, b , c
a', b', c'
«, A 7
i
a, b , c
a', b', c'
«', &, 7
>
a, b , c
a', b' , c'
«", 0", y"
The equation for the determination of the principal radii of curvature is
(E'p - EV}(G'P -
767] ON THE GAUSSIAN THEORY OF SURFACES. 333
which, in the particular case F=0 (and therefore V2 = EG), becomes
(E'p - EV) (G'p -GV)- F'2P2 = 0,
or, as this may be written,
I _JE'\ A _ &' \ _ F'2
EV)\ GV) EGV2~"'
p EVp
an equation which corresponds with Bour's form
and becomes identical with it, if
G' = GVH,
But, making p, q correspond to Bour's variables, p to v, and q to u, it is
necessary to show that the foregoing values (and not the interchanged values
E' = GVH, G' = EVK) are the correct ones. We have, Salmon, p. 254,
dq, pE'-VE, pF'-VF
- dp, PF' - VF, PG' - VG
or, putting herein F= 0, the equations may be written
d_E^( _
~" ~
= 0;
_. — - -
-dp~f" ~ pE' ~ G pG''
i f
or, we see that to dq = Q corresponds the value -=^ and to dp = 0 the value
p Jit V
1 C" 1
- = p^fr - Hence the former of these values of - corresponds to Bour's du = 0, that
p (JTV p
is, to his - = K ; and the latter to Bour's dv = 0, that is, to his ~ = H: or the
P P
values are, as stated,
E' = EVK, G'=GVH.
The formula ds2= Edp2 + 2Fdpdq + Gdq1 agrees with Bour's ds2 = dv2 + g^dii2, if
p = u> q = v, E=l, F = Q, G = g\ With these values, V2 = EG-F2 = y2, or say g= V,
and Bour's equation is, as it was before written, ds2 = dv2 + V2du2. And we have to
find the three equations which, putting therein p=u, q = v, E=l, F=0, G = F2,
E' = VK, F' = - V2T, G' = VSH, reduce themselves to Bour's equations.
The first of these is nothing else than the equation for the measure of curvature,
viz. Salmon, p. 262 (but, using the suffixes 1 and 2 to denote differentiation in
regard to p and q respectively), this is
4 (E'G' - F'2) = E (E2 G, - 2F, G, + G,2)
4 F (E,G, - E,G, - 2E,F, +
2 (EG - F2) (E,2 - 2^12 + 0U). /<
UNIVERSITY
•^CALIFOB^
334
ON THE GAUSSIAN THEORY OF SURFACES.
[767
In fact, writing herein E=l, F=0, and therefore the differential coefficients of E
and F each =0, the equation becomes
'//•>/ jT"2\ f*1 2 f)C<ri
Or — £ "V = Oi — ^tr(rn,
which is
or finally it is
The other two of Bour's equations are derived from equations which give
respectively the values of E2' — F^ and F2' — G/ ; viz. starting from the equations
E' = Aa. 4-5/3 + Cy ,
F'=Aa' +5/3' + <77',
G' = Aa." + B$" + Cy",
we see at once that E2 and FI contain, E2 the terms Aa,2 + B(32 + Cy«, and FI the
terms J.«1' + .B/31/ + Cy^, which are equal to each other (a, = a/ since a and «' are
the differential coefficients xn , #12 of x, and so /32 = fti and y2 = 7/). Hence
and similarly
Fs' - G,' = A,a' + Bsff + C2y' - A.a." - B^" - Cl7".
Here, from the values of -A, B, C, we have
A = be' - cb' ; A, = j3c - 76' + by' - c/3' ; A, = /3V - 7'6' + by" - c/3" ;
B = ca — ac' ; Bt = 7^ — ac' + ca' — ay' ; 5., = 7'a' — a'c' + ca" — ay" ;
C=ab'~ la' ; (7, = a&' - /3a' + a/3' - ba ; Cz = a'6' - /S'
and, substituting, we find
E2' - FI = Za'aa.' + aa"a ,
if, for shortness, a'aa denotes the determinant
a, a, a ,
and so for the other like symbols. Observe that, with
a, a, a., a', a"
b, b', ft, ff, &
c, c', 7, 7', y"
767]
ON THE GAUSSIAN THEORY OF SURFACES.
335
we have in all 1 0 determinants, viz. these are aa'a, = E' ; aa'a, = F' ; aa'a", = G' ;
aa'a"; and the six determinants aaa', aa'a", aa"a; a'aa, a'a' a", a'a" a. The foregoing
expressions of E2' — F^ and F.,'—Gi respectively, substituting therein for the determinants
a'aa', aa"a, aa'a", a'a"a their values as about to be obtained, are the required two
equations. We have
aa +bb +cc = E,
a' a + b'b + c'c = F,
aa +/3b + yc =\Elt
a'a + &b + y'c = £#„
a" a + /3"b + 7"c = Fa-
aa + bb' + cc = F,
a'a' + b'bf + c'c' = G,
aa' + /3b' + yc' = F1
a'a' + j3'b' + y'c' = $
a'a + $"V + 7"c' = %
and if from the first five equations, regarded as equations linear in (a, b, c), we
eliminate these quantities, and from the second five equations, regarded as linear in
(a , b', c), we eliminate these quantities, we obtain two sets each of five equations,
a, a,
a,
«',
a"
= 0, and
a, a',
a, i
*', a"
b, b',
&
ff,
r
6, 6',
A /
3', /S"
c, c,
7>
7'>
7"
c, c',
7,
/ //
y> 7
E F
•*-*) •*• }
i#i,
i#3,
FO-^
J1, »,
F.-^E.,, \t
ri, %G2 1
= 0.
These may be written,
Fa a' a" -
and
'tt a" - %E,a'a"a -(F2-% G,) a'aa = 0,
- Ea a a" + %E1a a' a" + | E2aa"a +(F,-± G,} aaa' = 0,
Ea'a'a"- F aa'a"+ \E,G' - (F,- ^G,)F' =0,
Ea'a"a - F a a"a - \E&' + (F, - ^GJ E' =0,
Ea'a a - F aa a' + ^E,F' - \E,E' = 0 ;
Ga a' a" - (F, - $ES) a a' a" - $ G^'a - \ G,a'aa' = 0,
- Fa a' a" + (F, -±EJa a' a" + 1 G,aa"a + 1 G.aaa' = 0,
Fa'a'a." - G aa'af' + ^G -%G,F' =0,
Fa'a"a - G aa'a -(F.-^E,) G' + %G,E' =0,
Fa' a. a - G a a a + (F, - \ E3) F' - % G^' = 0.
Attending in each set only to the third, fourth, and fifth equations, and combining
these in pairs, we obtain
Wa'a" + (
V*a a"a + (-
FWa + (-
V«-a a a' + (
a' +
EF, - \
FF1 -
G' + (-
G' + -
,-^ GE2)
GF, -
E' + ( 1>FE1 - EF, +
E' + ( \GE,- FF, +
G' = 0,
G' = 0;
J E' = 0,
' = 0;
' = 0,
.> F' = 0.
336 ON THE GAUSSIAN THEORY OF SURFACES. [767
We thus obtain
E,' - F; = * K - i^i + *G^«) E' + (-
2' - G,' = ,
or, finally,
EJ -Fi=y-, {(- %FG> + GE, - FFt + $EG2) E'
+ (- GE1 + 2FF, - FFZ) F' + ($FEl - EFl + %EEZ) G'},
F» - G>' = -, {(- i GG, + GF, - %FGZ) E'
which are the required formulae ; and which may, I think, be regarded as new formulae
in the Gaussian theory of surfaces.
Writing herein as before, the first of these becomes
( VK), + ( V"-T\ = ^ {i ( V^ VK], =
that is,
V2K+
or finally
which is Bour's third equation. And the second equation becomes
that is,
- F2r2
or finally
which is Bour's second equation.
768] 337
768.
NOTE ON LANDEN'S THEOREM.
[From the Proceedings of the London Mathematical Society, vol. xm. (1882), pp. 47, 48.
Read November 10, 1881.]
LANDEN'S theorem, as given in the paper "An Investigation of a General Theorem
for finding the length of any Arc of any Conic Hyperbola by means of two Elliptic
Arcs, with some other new and useful Theorems deduced therefrom," Phil. Trans.,
t. LXV. (1775), pp. 283—289, is, as appears by the title, a theorem for finding the
length of a hyperbolic arc in terms of the length of two elliptic arcs; this theorem
being obtained by means of the follo\ving differential identity, viz., if
where
_ m2 -
9~ ~tf
then
(this is exactly Landen's form, except that he of course writes sb, i in place of dx,
dt respectively): viz., integrating each side, and interpreting geometrically in a very
ingenious and elegant manner the three integrals which present themselves, he arrives
at his theorem for the hyperbolic arc; but with this I am not now concerned.
Writing for greater convenience m=l, n = k', and therefore g = k2, if as usual
kz+k'* = I, the transformation is
C. XL
43
338 NOTE ON LANDEN'S THEOREM. [768
leading to
(i+kj-t2
The form in which the transformation is usually employed (see my Elliptic
Functions, pp. 177, 178) is
leading to
(l+k')dx = dy
where
x = 1 - k'
If, to identify the two forms, we write y = = r/ and in the last equation
J. ~~~ /C
introduce t in place of y, the last equation becomes
dx dt
have
Vl - a?. 1 - tea? V{(1 - kj - 12} {(1 + kj - 12} '
Comparing with Landen's form, in order that the two may be identical, we must
/• 1 -kj-
g
kj-t2 ^(l + kj-t2,
viz., this is
kj - 12}2,
that is,
I -tea? = i [1 + ^ - ^2 + V{(1 - ^')2 - £<} {(1 + ^')2 - 12}],
where the function under the radical sign is
(1 - k'*-)2 -2(1 + k'2) t2+V(=T suppose) ;
and this must consequently be a form of the original integral equation
In fact, squaring and solving in regard to x2 with the assumed sign of the radical,
we have
_ A;2 + 12 -
'~~
768] NOTE OX LAXDEX'S THEOREM. 339
corresponding to an equation given by Landen. And we thence have
which is the required expression for 1 — te
The trigonometrical form sin (2<£' — <£) = c sin <£ of the relation between y and x
does not occur in Landen; it is employed by Legendre, I believe, in an early paper,
Mem. de I'Acad. de Paris, 1786, and in the Exercices, 1811, and also in the Traite
des Fonctions Elliptiques, 1825, and by means of it he obtains an expression for the
arc of a hyperbola in terms of two elliptic functions, E (c, <£), E(c, <£')> showing that
the arc of the hyperbola is expressible by means of two elliptic arcs, — this, he observes,
" est le beau theoreme dont Landen a enrichi la geometric.' We have, then (1828).
Jacobi's proof, by two fixed circles, of the addition-theorem (see my Elliptic Functions,
p. 28), and the application of this (p. 30) to Landen's theorem is also due to Jacobi,
see the " Extrait d'une lettre adressee a M. Hermite," Crelle, t. xxxn. (1846),
pp. 176 — 181 ; the connection of the demonstrations, by regarding the point, which
is alone necessary for Landen's theorem as the limit of the smaller circle in the
figure for the addition-theorem is due to Durege (see his Theorie der elliptischen
Functionen, Leipzig, 1861, pp. 168, et seq.).
43—2
340 [769
769.
ON A FORMULA RELATING TO ELLIPTIC INTEGRALS OF
THE THIRD KIND.
[From the Proceedings of the London Mathematical Society, vol. xm. (1882),
pp. 175, 176. Presented May 11, 1882.]
THE formula for the differentiation of the integral of the third kind
in regard to the parameter n, see my Elliptic Functions, Nos. 174 et seq., may be pre
sented under a very elegant form, by writing therein
sin2 <£ = x = sn2 u, sin <£ cos <£ A = y = sn u en u dn u,
and thus connecting the formula with the cubic curve
y2 = a; (1 - a;) (1 - k*x).
The parameter must, of course, be put under a corresponding form, say n = -- ,
a
where a = sn2 0, 6 = sn 0 en 0 dn 6, and therefore (a, b) are the coordinates of the point
corresponding to the argument 6. The steps of the substitution may be effected
without difficulty, but it will be convenient to give at once the final result and
then verify it directly. The result is
d_ b d y _ , 2 (
7/1 ? — K~ (a ~~ x),
do a—x du x — a
We, in fact, have
du
dx
-j- = 2 sn u en u dn u = 2y,
769] ON A FORMULA RELATING TO ELLIPTIC INTEGRALS OF THE THIRD KIND. 341
and thence
that is,
y -¥- = 2x [1 - (1 + k2) x
y du L
T? = cn2 u dn2 u — sn2 w dn2 u — & sn2 u en2
cm
Also
and hence
d y 1 (. .dy
dux — a (a — x)2 ( du '
= -, -, {- x - a + 2 (1 + A;2) ax
(a—sc)2 l
Interchanging the letters, we have
d b 1 , Lmn^iA ^u
-j-,. = j— x — a + 2 (1 + «r) ax + k-a?
\AJ\J \AJ ^~ 3C I Cv "~~ W/ )
and hence, subtracting,
dd a — x du x — a (a — .
I
(a - a?)
= &2 (a - x),
which is the required result.
k2 (a - x)3
342
770.
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
[From the American Journal of Mathematics, vol. iv. (1881), pp. 1—15.]
I HAVE, (by aid of Gundelfinger's formulae, afterwards referred to), calculated, and
I give in the present paper, the expressions of the 34 concomitants of the canonical
ternary cubic ax3 + bys + cz3 + <olxyz, or, what is the same thing, the 34 covariants of
this cubic and the adjoint linear function tfx + rjy+Zz: this is the chief object of
the paper. I prefix a list of memoirs, with short remarks upon some of them;
and, after a few observations, proceed to the expressions for the 34 concomitants;
and, in conclusion, exhibit the process of calculation of these concomitants other
than such of them as are taken to be known forms. I insert a supplemental table
of 6 derived forms.
The list of .memoirs (not by any means a complete one) is as follows :
HESSE, Ueber die Elimination tier Yariabeln aus drei algebraischen Gleichungen
vom zweiten Grade mit zwei Variabeln : Crelle, t. xxvm. (1844), pp. 68—96. Although
purporting to relate to a different subject, this is in fact the earliest, and a very
important, memoir in regard to the general ternary cubic; and in it is established
the canonical form, as Hesse writes it, yi3 + y23 + y/ + Q-rry^^.
ARONHOLD, Zur Theorie der homogenen Functionen dritten Grades von drei
Variabeln: Crelle, t. xxxix. (1850), pp. 140 — 159.
CAYLEY, A Third Memoir on Quantics : Phil. Trans., t. CXLVI. (1856), pp. 627—647 ;
[144].
ARONHOLD, Theorie der homogenen Functionen dritten Grades von drei Variabeln:
Crelle, t. LV. (1858), pp. 97—191.
SALMON, Lessons Introductory to the Modern Higher Algebra: 8°, Dublin, 1859.
CAYLEY, A Seventh Memoir on Quantics : Phil. Trans., t. CLI. (1861), pp. 277—292 ;
[269].
BRIOSCHI, Sur la thdorie des formes cubiques a trois indetermine'es : Comptes
Rendus, t. LVI. (1863), pp. 304—307.
770] ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. 343
HERMITE, Extrait d'une lettre a M. Brioschi: Crelle, t. LXIII. (1864), pp. 30 — 32,
followed by a note by Brioschi, pp. 32 — 33.
The skew covariant of the ninth order, which is y3 — z3 . z3 — x3 . a? — y3 for the canoni
cal form a? -f if + z3 + Qlxyz, and the corresponding contravariant if — £3 . £3 — £3 . f 3 — rf,
alluded to p. 116 of Salmon's Lessons, were obtained, the covariant by Brioschi and
the contravariant by Hermite, in the last-mentioned papers.
CLEBSCH and GORDAN, Ueber die Theorie der ternaren cubischen Formen : Math.
Annalen, t. I. (1869), pp. 56—89.
The establishment of the complete system of the 34 covariants, contravariants
and Zwischenformen, or, as I have here called them, the 34 concomitants, was first
effected by Gordan in the next following memoir:
GORDAN, Ueber die ternaren Formen dritten Grades : Math. Annalen, t. I. (1869),
pp. 90—128.
And the theory is further considered :
GUNDELFINGER, Zur Theorie der ternaren cubischen Formen: Math. Annalen, t. vi.
(1871), pp. 144 — 163. The author speaks of the 34 forms as being "theils mit den
von Gordan gewahlten identisch, theils moglichst einfache Combinationen derselben."
They are, in fact, the 34 forms given in the present paper for the canonical form
of the cubic, and the meaning of the adopted combinations of Gordan's forms will
presently clearly appear.
There is an advantage in using the form ax3 + by3 + cz3 + Qlxyz rather than the
Hessian form x3 + y3 + z3 + Qlxyz, employed in my Third and Seventh Memoirs on
Quantics : for the form ax3 + by3 + cz3 + Qlxyz is what the general cubic
(a, b, c, f, g, h, i, j, k, 1) (x, y, z}3
becomes by no other change than the reduction to zero of certain of its coefficients ;
and thus any concomitant of the canonical form consists of terms which are leading
terms of the same concomitant of the general form.
The concomitants are functions of the coefficients (a, b, ..., 1), of (f, 77, £), and of
(x, y, z) : the dimensions in regard to the three sets respectively may be distinguished
as the degree, class, and order; and we have thus to consider the deg-class-order of
a concomitant.
Two or more concomitants of the same deg-class-order may be linearly combined
together : viz., the linear combination is the sum of the concomitants each multiplied
by a mere number. The question thus arises as to the selection of a representative
concomitant. As already mentioned, I follow Gundelfinger, viz., my 34 concomitants
of the canonical form correspond each to each (with only the difference of a
numerical factor of the entire concomitant) to his 34 concomitants of the general
form. The principle underlying the selection would, in regard to the general form,
have to be explained altogether differently; but this principle exhibits itself in a
very remarkable manner in regard to the canonical form ax3 + by3 + cz3 + Qlxyz.
344 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
Each concomitant of the general form is an indecomposable function, not breaking
up into rational factors; but this is not of necessity the case in regard to a canonical
form : only a concomitant which does break up must be regarded as indecomposable,
no factor of such concomitant being rejected, or separated. So far from it, there is,
in regard to the canonical form in question, a frequent occurrence of abc + 8l3 or a
power thereof, either as a factor of a unique concomitant, or when there are two
or more concomitants of the same deg-class-order, then as a factor of a properly
selected linear combination of such concomitants: and the principle referred to is, in
fact, that of the selection of such combination for the representative concomitant ; or
(in other words) the representative concomitant is taken so as to contain as a factor
the highest power that may be of abc + 8l3. As to the signification of this expression
abc + 8l3, I call to mind that the discriminant of the form is abc (abc + 8l3)3.
As to numerical factors: my principle has been, and is, to throw out any common
numerical divisor of all the terms : thus I write S = — abcl + 14, instead of Aronhold's
S = — 4<abcl + 41*. There is also the question of nomenclature : I retain that of my
Seventh Memoir on Quantics, except that I use single letters H, P, &c., instead of
the same letters with V, thus HU, PU, &c. ; in particular, I use V, H, P, Q
instead of Aronhold's /, A, Sf, Tf. It is thus at all events necessary to make some
change in Gundelfinger's letters ; and there is moreover a laxity in his use of accented
letters; his B, B', B", B'", and so in other cases E, E', E" , &c., are used to denote
functions derived in a determinate manner each from the preceding one (by the
S-process explained further on) ; whereas his L, L' ; M, M' ; N, N' are functions
having to each other an altogether different relation ; also three of his functions are
not denoted by any letters at all. Under the circumstances, I retain only a few of
his letters ; use the accent where it denotes the S-process ; and introduce barred
letters J, K, &c., to denote a different correspondence with the unbarred letters /,
K, &c. But I attach also to each concomitant a numerical symbol showing its
deg-class-order, thus: 541 (degree = 5, class = 4, order = 1) or 1290, (there is no
ambiguity in the two-digit numbers 10, 11, 12 which present themselves in the system
of the 34 symbols); and it seems to me very desirable that the significations of
these deg-class-order symbols should be considered as permanent and unalterable.
Thus, in writing S = 400 = - abcl + 1\ I wish the 400 to be regarded as denoting its
expressed value — abcl + Z4 : if the same letter S is to be used in Aronhold's sense
to denote — 4>abcl + 4*1*, this would be completely expressed by the new definition
$ = 4.400, the meaning of the symbol 400 being explained by reference to the present
memoir, or by the actual quotation 400 = — abcl + 1*.
I proceed at once to the table : for shortness, I omit, in general, terms which
can be derived from an expressed term by mere cyclical interchanges of the letters
(a, 6, c), (£, 77, 0, (x, y, z).
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. 345
Table of the 34 Covariants of the Canonical Cubic ax? + by3 + cz3 + Qlxyz and
the linear form
First Part, 10 Forms. Class = Order.
Current No.
2 T = 600 = a262c2 - 20abcl3 - 8l6.
3 A = 011= %a; + iiy + &.
4 © = 222 = x"- [- Z2£2 -
5 0' = 422 = x2\l (abc + 2l3) % 2 + a (abc - 4Z3)
+ yz [6bcl2£2 - 21 (abc + 2l3
6 0" = 622 = a? [- (a&c + 2Z3)2 f + 12aZ2 (abc +
+ y* [366cZ4£2 + 2 (abc + 2l3)2
7 B = 333 = 0s [a2 (cr;3 - &£»)].. .
+ y*z [(abc + 8l3) if
+ yz2 [- (abc + 8l3)
8 B' = 533 = a? [3a2^2 (erf - 6£3)] . . .
+ y*z [~ V (abc + 8Z3) rft + 4<bl (- abc + I3) £2f - be (abc - 10F) f 77]. . .
+ yz* [I2 (abc + 81s) r)? + bc (abc - 10Z3) £f 2 - 4cZ (- abc + Is) ^2] ....
9 B" =733= a?[9aW(crf-b?y\...
+ y2z [I (abc + 8l3) (2abc + I3) ift,
+ b (abc + 21s) (abc - Wl3) £2£ + Qbcl2 (- abc + I3) f 17].
- 6bcl2 (- abc + I3) g- - c (abc + 2l3) (abc - Wl3)
10 B'" = 933 = a? [27a2Z6 (erf - 6£3)]. . .
+ fz [- (abc + 81s) (abc - I3)2 7?2£+ Qbl2 (abc + 213)Z £2%
-
+ yz2 [(abc + 81s) (abc - 13)2 ^- + 27bcl* (abc + 2l
Second Part, (4 + 4 =) 8 forms. Class = 0, and Order = 0.
Class = 0.
11 U= 103 = aa? + by3 + czs + Qlxyz.
12 H = 303 = I- (ax3 + by3 + cz3) -(abc + 2l3) xyz.
13 ^ = 806 = (ale + 8l3)2 {a*a? + b2y« + c2z* - 10 (bcysz3 + caz3^ + abxsy%
14 n = 1209 = (abc + 8l3)3 {by3 - cz3 . cz3 - ax3 . ax3 - by3}.
c. XL 44
346 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
Current No. Order = 0.
15 P= 330 = -l(bc?+car)3+abZ3) + (-abc + 4<l3)Zr)Z.
16 Q = 530 = (abc - W) (bc%3 + car,3 + ab£3) - Ql2 (oabc + 4,1s) &£
17 F = 460 = b2c2? + c2aV + a262£6 - 2 (abc + 161s) (ar,3^3 + b£3? + c?r,3)
- 24,1* (be? + car)3 + ab?) £77? - 24,1 (abc + 21s) ?r,2?.
18 n = 1290 = (abc + 8l3)3 {cr,3 - b%3 . a£3 - c? . b? - ar,3}.
Third Part, (8 + 8 =) 16 forms. Class less or greater than Order.
Class less than Order.
19 J= 414 =
20 K = 514 = (abc + 8l3) {£ [alx* - 2blxy3 - 2clxz3 + 3bcy2z2] ...}.
21 K' = 714 = (abc + 8l3) {£ [(abc + 2l3) (ax4 - 2bxy3 - Zcxz3) - I8bd2y2z*] . . .}.
22 E = 625 = (abc + 81s) {? (by3 - cz3) [21W + bcyz] . . .
+ rtW - czs) [4ala? + 2l2yz]...}.
23 E' = 825 = (abc + 8l3) {? (by3 - czs) [I (abc + 2l3) x* - Sbcl'-yz], . .
+ r)£ (by3 - cz3) [a (abc - 4>l3) x2 + 1 (abc + 21s) yz] ...}.
24 E" = 1025 = (abc + 8l3) {? (by3 - cz3) [(abc + 2Z3)2 #2 + ISbcVyz] . . .
+ rt (by3 - cz3) [- I2al* (abc + 21s) x2 + (abc + 2l3)* yz]... }.
25 M = 91 7 = (abc + 8l3)2 {(• (by3 - cz3) [5alx* - blxy3 - clxz3 - 3bcy2z2]. . .}.
26 M' = 1 1 1 7 = (abc + 81s)2 {f (by3 - cz3) [(abc + 2l3) (oax4 - bxy3 - cxz*)
+ I8bcl2y2zz]...}.
Order less than Class.
27 J = 841 = (abc + 8l3)2 [x%a (erf - b?) + yr)b (a£3 - cf ) + z& (b? - ar,3)}.
28 K= 541 = (abc + 8l3) [x [be? - 2ca^3 - 2ab%? - Qalr)2£2]. . .}.
29 K = 741 = (abc + 81s) [x [I2 (be? - 2cafr3 - 2ab^3) + a (abc + 21s) rf ?]...}.
30 E= 652 = (abc + 8l3) {x2 (cy3 - b£3) [2al? + a2^].. .
+ yz (cri3 -
31 W' = 852 = (abc + 8l3) {x2 (cr)3 - b£3) [a (abc - 4,l3) ? - 6aH2r,t;]. . .
+ yz (cr)3 - b£3) [4,1 (abc + 2l3) ?+a (abc - 4,l3
32 E* = 1052 = (abc + 8l3) {x2 (cr)3 - b£3) [- 3al2 (abc + 2l3) ? + 9a*l*r)S]. . .
+ yz (cr,3 - b£3) [(abc + 2l3)2 ? - Sal2 (abc - 4>l3
33 M = 771 = (abc + 8l3) {x (cr)3 - b?) [(abc - 8l3) ? - a2c£r)3 - a2
34 M' = 971 = (abc -f 8^3) {x (erf - b?) [I2 (lobe + 81s) ? - 3a2cl2fr3 - SaW^3
+ ±al (abc - I3) ?r)£+ a2 (abc - 10Z3) 17^ aj. ..}.
770] ON THE 34 CONCOMITANTS OP THE TERNARY CUBIC. 347
To this may be joined the following Supplemental Table of certain Derived
Forms :
Current No.
35 R = 1 200 = 64,8s -T*=- abc (abc + 8ls)s.
36 C = 703 = - TU + 24£ff = (abc + 81*) {(- abc + 4>l3) (aa? + by3 + cz3)
+ ISabclxyz}.
37 D = 903 = 8S2 U - 3TH = (abc + 8l3) {I2 (5abc + 4l3) (am? + by3 + cz3)
+ 3abc(abc-Wl3)a;yz}.
38 F = 930 = 3TP - 4SQ = (abc + 8Z3)2 {I (be? + cay* + ab?) - Sdbc&S}.
39 Z = 1130 = - 48S2P + TQ = (abc + 8Z3)2 {(abc + 21s) (be? + caif + ab£3)
+
40 3> = 1660 = 12 (abc + 81s)3 F- 288STP* + 768S2PQ - 8TQ*
= (abc + 81s)* {6
viz. these are derived forms characterized by having a power of abc + 8l3 as a factor :
R is the discriminant ; C, D, Y, Z occur in Aronhold, and in my Seventh memoir
on Quantics [269] : <J> in Clebsch and Gordan's memoir of 1869.
I regard as known forms A, U, H, P, Q, S, T, F, that is, the eight forms
3, 11, 12, 15, 16, 1, 2, 17; the remaining 26 forms are expressed in terms of these
by formulae involving notations which will be explained, viz. we have
13 ^ = 3(bc/ + b/c-2ff',...,gh' + g/h-af-a/f,...][X, F, Z\X' , Y', Z')
14 n = Jg Jac(Dr, H, ¥).
18 n =-^[Jac] (P, Q, F).
40= (bc-f2,...,gh-af,...$£ 77, £)2-
5 ©' = £80.
60"= | S2©.
7 B =-£Jac(tf, 0, A).
8 R = %SB.
9 B" = Jj8lB.
10 £"'= J-g&B.
19 / =-£Jac(J7, H, A).
27 /= HJac](P, Q. A>-
20 K = - f {^09^ + 3,09^ + 9f03,If } - fif CTA.
21 ^' = -(8)^.
28 K = 3 {9»0afP + ay08,P + 8,©a^P} + QA.
29 ^'= ^(8)^.
22 E =-^J&c(K, U, A).
23 E' =-l($)E.
44—2
348 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
24 E"=
30 E =-|Jac(^, U, A).
31 E' =-%($) R
32 E" = -$(&)E.
25 M = ^Jac(f7, V, A).
26 M' = -(B)M.
33 F = - £ [Jac] (P, P, A).
34 M' =
In explanation of the notations, observe that
U = ax3 + by3 + cz3 + Glxyz,
H = P (ax3 + by3 + cz3) - (abc + 2l3) xyz.
Hence, writing
QH = a'x3 + b'y3 + c'z3 + Ql'xyz,
we have
a', b', c', I' = 6aP, Qbl2, 6cZ2, - (abc + 2l3).
And this being so, we write
X, Y, Z = ax2 + 2lyz, by2 + 2lzx, cz2 + 2lxy,
a, b, c, f, g, h = ax, by, cz, Ix, ly, Iz,
for £ of the first differential coefficients, and £ of the second differential coefficients
of U; and in like manner
X', Y', Z' = a'x2 + 2l'yz, b'y2 + 21' zx, c'z2 + 2l'xy,
a', b', c', f, g', h' = a'x, b'y, c'z, I'x, I'y, I'z,
for £ of the first differential coefficients, and £ of the second differential coefficients
of 6H.
Jac is written to denote the Jacobian, viz. :
dxU, dyU,
Jac(tf, H, ^) = '
and in like manner [Jac] to denote the Jacobian, when the differentiations are in
regard to (f, 97, £) instead of (x, y, z): 8 is the symbol of the S-process, or sub
stitution of the coefficients (a', b', c', I') in place of (a, b, c, I); in fact,
B, S2, &c., each operate directly on a function of (a, b, c, 1), the (a', b', c', I') of the
symbol 8 being in the first instance regarded as constants, and being replaced ultimately
by their values ; for instance,
Sabc = a'bc + ab'c + abc', &abc = 2 (ab'c' + a'bc' + a'b'c), 83abc = Ga'b'c'.
770] ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. 349
In several of the formulae, instead of 8 or 8-} the symbol used is (8) or (82) ;
in these cases, the function operated upon contains the factor (abc + 8l3) or (abc + 813)'2,
and is of the form (abc+ 8l3) (aU+bV+cW) or (abc + 8l3)2 (a*-U+abV + &c.) : the
meaning is, that the 8 or 82 is supposed to operate through the (abc + 8l3) a, or
(abc + 8l3)2 a2 , &c., as if this were a constant, upon the U, V, &c., only; thus:
(8).(abc+8l3)(aU+bV+cW) is used to denote (abc + 8l3)(a8U + b8V+ c8W). As to
this, observe that, operating with 8 instead of (8), there would be the additional
terms U8 (abc + 81s) a + &c. ; we have in this case
8 (abc + 8l3) a, = a (2a'bc + ab'c + abc + 24W) + 8l*a',
= 24>a2bcl2 - 24al2 (abc + 21s) + 4,8al5, = 0 ;
or the rejected terms in fact vanish. For (S2) . (abc + 8l3) (a U + b V + c W ), operating
with 82, we should have, in like manner, terms U82 (abc + 8l3) a, &c. ; here
82 (abc + 8l3) a = a! -be + 2aba'c' + 2aca'b' + aW + 24Z2aT + 24aW'2,
which is found to be = - 24a (abc + 81s) (- abcl + Z4), that is, = - 24$ (abc + 8l3) a ; and
the terms in question are thus = — 24:8 (abc + 8l3)(aU+bV + cW), viz.
(abc + 8l3)(aU+bV+cW)
being a co variant, this is also a co variant ; that is, in using (82) instead of 82, we
in fact reject certain covariant terms ; or say, for instance, 82E being a co variant,
then (S2) E is also a covariant, but a different covariant. The calculation with (8)
or (S2) is more simple than it would have been with 8 or S2. See post, the calcula
tions of K, K', &c.
I give for each of the 26 covariants a calculation showing how at least a single
term of the final result is arrived at, and, in the several cases for which there is
a power of abc + 8l3 as a factor, showing how this factor presents itself.
Calculations for the 26 Covariants.
13. ¥ = 3 (be' + Vc - 2ff, . . . , gh' + g'h - af - a'f, . . .%X, T, Z\X\ Y', Z') + TU2,
= 3 ((be' + b'c) yz - 211' 'x2 , . . . , Zll'yz -(al' + a 1) a?, ... $a#2 + Zlyz, . . . $> 'a? + 21'yz, . . . )
The whole coefficient of x6 is
- Gll'aa' + Ta2, = 36a2l3 (abc + 2l3) + Ta2,
viz. the coefficient of aV is
= 36Z3 (abc + 2l3) + a2b2c2 - 20abcl3 - 81s
= a2b2c2 + IGabcl3 + 64Z6
= (abc + 8l3)2.
X, X', ^
14. n = T^Jac(?7, H, ^), =£ F, Y', ^3^
^, Z', ^8^
350 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
Here
YZ' - Y'Z = (by2 + 2lzx) (c'z2 + 21'xy) - (cz2 + *2lxy) (b'z2 + 21'xy)
= (be' - b'c) y2z2 + (2blf - b'l) xy3 - 2 (cl' - c'l) xz3
= - 2 (abc + 8l3) x (by3 - cz3) ;
\ • i "dx& — \ (a2^ — 5abx2y3 — oacx^z3).
Hence the whole is
- _ (abc + 8l3) [a2x* (by3 - cz3} + b2y6 (cz3 - ax3) + c2z6 (ax3 - by3)},
= (abc + 8Z3) (by3 - cz3) (cz3 - ax3) (ax3 - by3).
18.
viz. if, in this calculation, we write
£ i.e. a , b , c , 1 = — 6lbc, — Qlca, — 6lab, — abc + 4Z3,
I'^C, „ a', b', c', 1' = (abc - Wl3) (be, ca, ab), -l-(5abc + 4l3
then
n—
Here
or since
= (be' - b'c) rf? + 2 (bl' - b'l) ^ - 2 (cl' - c'
bc'-b'c = 0,
bl' - b'l = - Glca . - I- (oabc + 4Z3) - (abc - Wl3) ca (- abc +
= ca {61s (babe + 4,l3) + (abc - 4,1s) (abc - IQl3)}
= ca (abc + 8l3)*,
and the like for cl' — c'l, the expression is
= 2 (abc + 8l3)2 (car)3 - ab?) % ;
and the whole is thus
= - £ (abc + 813Y {(car)3 - ab?) % . £ d^F + . . .}
= - \ (abc + 8l3)"- {(car;3 - ab£3) [62c2f - (abc +
+ (a&£3 - 6cf ) [c-a^ - (abc + Wl3) (
+ (bc%3 - car}3) [a*b2? - (abc + 16Z3) (a
Here the coefficient of %*rf, inside the {}, is
a62c3 + be2 (abc + Wl3), = 26c2 (abc + 8l3),
) + &c.]
^3) + &c.]
c.]}.
770] ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC,
and consequently the whole is
4.
351
= (abc + SI3)3 {(cy3 - 6£3) (a? - cf ) (6f -
0 = (bc-fV..,gh-af,...$£, 77, O
= (bcyz - I2x-) % 2 + . . . + 2 (l*yz - o
which are the terms of the final result
0 = a? [- l*j? - 2al>r)Z] + yz [be? + 2^].
5 and 6. The ^-process applied to the terms of © just written down gives
©' = | S0 = x> [- II'? - (aV + a'l) vQ + yz [J (be + b'c) ?
0" = $ §2© = x«. j-_ pp _ 2a'l'^] + yz [b'c'? + 2l'2<n£] ;
substituting for a', b', c', I' their values, we have the corresponding terms of ©' and 0"
respectively.
X, 3*®,
7.
7, 0, A), =-
Y, 8,0,
Z, 8,0,
A term is X (rjdz® — £dy®), and if, in this calculation, we write
®=(A, B, C, F, G, H~$x, y, z)-, i.e. A = -l-^-
then the term is
Here
2 (Gr, - H& = 77 (car + m) - C (ab? + P
and hence the whole term in a? is = a?oP (cy3 — b£3).
8, 9, 10. The coefficient of a?r)3 in B is a2c, and hence in 8B, 82B, 83B the coefficients
of this term are 2a'ac + a2c', 2a'2c H- 4aa'c', 6a/2c', whence in
B', B", B'" =
the coefficients are
respectively,
£ (aV + Zaa'c), •& (a'2c + 2oa'c'), £ aV,
= 3Z2a2c, 9^4a2c, 27^6a2c respectively.
X, X', £
19 J=-£Jac([T, H, A) = -£ Y, Y, rj
9 at ^
6, 6 , <s
a term is — ^ ( YZ' — Y'Z) %, where, as in a previous calculation,
YZ' -Y'Z = -:L (abc + 81s) x (by3 - cz3).
352 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
Hence, the whole is
= (abc + 8l3) [%x (by3 - a?) + r^y (czs - ax3) + & (ax3 - by3)}.
27. J=HJac](P, Q, A) =
, as
y
if, as in a previous calculation
6P = af + b??3 + c£3 + Glfitf, Q = a'|3 + by + c'£3 + 61'£/£
Here, as before,
(V + 21#) (c'£2 + 21'fT/) - (bV + 21'(g) (c£2 + 21&) = 2 (a&c + 8Z3)2 (car}3
Hence, the whole is
= (abc + 8l3)* {x%a (C7?3 - b?) + yrjb (a? - cf ) + z& (6f - ay3)}.
20. K=-\ (ds®dxH+ dfldyH + d^dzH) - SUA,
Avhich, H being
= £ (aV + b'y3 + c'z3 + §l'xyz\
and putting
® = (A, B, C, F, G, #$£ 77, ^)2, ^l=-^ + 6c^, ..., F= -
is
- (- abcl + I*) U (
+ (c'z*
The whole coefficient of £ is thus
= _ | {(av + 21'yz) A + (Vyn- + Wzx) H + (c'z* + 2l'xy) G} - (- abcl + 1*) Ux
= - f {(aV + 21' yz) (- Va? + bcyz) + (b'y2 + 21' zx) (- clz2 + I2xy)
+ (c'z* + 21' xy) (- bly* + Pzx)} - (- abcl + 1*) {ax* + bxy3 + cxz3 + Qltfyz],
and herein the coefficient of a? is
= f o!lz - al (- abc + I3), = 9al* - al (- abc + I3), = (abc + 81s) al ;
viz. we have thus the term (abc + 81s) £ . alx* of the final result.
21. K'=-(S)K, where K is of the form (abc + 81s) (aU+bV+ cW ); operating
with (B), we obtain (abc + 8l3) (aSU + bSV + cSW ). Taking for instance the term of
K, (abc + 8l3) f [alas1 - 2blxy3 - Zclxz3 + 3bcy2z2], then, in operating with (8), the term be
may be considered indifferently as belonging to bV or cW, and the resulting term
of K' is
K' = -(S)K=- (abc + 8l3) % [al'x? - Wxy3 - 2cl'xz3 + Sbc'fz*],
= (abc + 8l3) % [(abc + 2l3) (ax4 - 2bxy3 - 2cxz3) -
770] ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. 353
28. K = 3 {dxSdf P + av®a,P + d&dfP] + QA ; viz. writing
® = (A, B, C, F, G, H^x, y, z}\ A
then this is
= 3 {[- Sbcl? + (- abc + 4Z3) tf] 2 (Ax + Hy + Gz)
+ [- Scalrj2 + (- abc + 4>l3) (gj 2 (Hx + By + Fz)
+ [- 3aWf8 + (- a&c + 4£3) £77] 2 (&» + Fy + Cz)}
+ {(abc - W) (be? + cm?3 + a&(?) - 6
The Avhole coefficient of x is thus
= 3 {[- Sbcl? + (- abc + 4,1s) ^] (-
+ {(abc - Wl3) (be? + cafr3 + abg?) - 61* (oabc + 4>l3) ^} ;
herein the coefficient of f4 is ISbcl3 + (abc — IQl3) be, = (abc + 8l3) be, giving, in the final
result, the term (abc + 8l3) f . bcx4.
29. K'=$(S)K.
Here K is of the form (abc + 8l3) (a U + b V + c W), and we have
K' = $ (abc + 8l3)(aBU+bSV+ cS W).
A term of aU + bV + cW is x [6c|4 — 2caf rf — 2a6f£3 — 6a^2^2], where 6cf4 may be con
sidered as belonging indifferently to bV or cW; and so for the other terms. The
resulting term in $(a&U+ bSV+cSW ) is thus
£ 0 [fec'f 4 - 2ca'
which is
= as [I2 (be? - 2caf7f - 2a6f f8) + a (a6c
and we have thus a term of K'.
22. E = -Js3w(K, U, A):
K contains the factor a&c + 8£3, and if, omitting this factor, the value of K is called
A £ + Brj + C£, then we have
E = -
and the term herein in f2 is - ^ ? (ZdyA - YdzA), where A is
= ate* - Zblxy3 - Zclzz3 + SbcyW ;
c. XL 45
354
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
[770
viz. the coefficient of f2 is
= - £ {(cz2 + 2lxy) (- Qblxy* + Gbcyz2) - (by2 + 2lzx) (- Qclxz2 + Qbcy2z)}
= b2cy*z - bc2yz* + 2blWy3 - 2c£W
= (2l2x2 + bcyz) (by3 - cz3).
Hence, restoring the omitted factor (abc + 8l3), we have in E the term
(abc + 8l3) Is (by3 - cz3) [2l2a? + bcyz].
23, 24. E' = -\ (8) E, E" = £(S2) E :
E is of the form (abc + 8l3)(aU + bV + cW), and, as before, in a term such as
(abc + 8l3) £2 (by3 - cz3) (21^ + bcyz),
we operate with 8 or B2 only on the factor 2l2a? + bcyz ; and in E' and E" respec
tively, operating upon this factor, we obtain
}, and ± {4ZV + 2b'c'yz],
viz. we thus obtain in E' the term
(abc + 81s) f 2 (by3 - cz3) [I (abc + 2l3) x2 - Sbcfiyz],
(abc + 81s) ? (by3 - cz3) [(abc + 213
, X,
and in E" the term
30. E = - £ Jac (K, U, A), = - %
r, z, r
and, if omitting in K the factor abc + 8l3, we write K = Ax + By + Cz, where
A, X,
A = 6c|4 - 2ca£7?3 -
which contains the term
, this is = -
, F, 77
, s, r
fy), = i (a«2 + Zlyz) {{(car)* -
- rj (ab? - 2
= (ax2 + 2lyz) (cy3 - b?) (2l%2 + atf).
Hence, restoring the factor abc + 8l3, we have the terms
E = (abc + 8l3) [x2 (crj3 - b?) [2al? + a2^] + yz (cy3 - b?) [4l2? +
31 and 32. E' = -$(S)E, E" = -% (S2) E :
E is of the form (abc + 8l3) (aU + bV + cW), and we operate with & and S2 on the factors
277£ &c.; viz.
B (2al? + a^O = 2 (al' + a'l) ?- + Zaa'tf,
770] ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
and we thus obtain in E' the term
(abc + 8l3) x2 (crj3 - b%3) [a (abc - 4/3) £2 - Qa2l2^],
and in E" the term
(abc + 8l3) x2 (crj3 - b£3) [- Sal2 (abc + 2l3) f + 9a2^£].
25. M=^JsLC(U, V, A): this, omitting the factor (abc + 8l3)2 of ^, is
ax2 + 2lyz, ax2 (ax3 — 5by3 — 5cz3), %
by2 + 2lzx, by2 (by3 — ocz3 — oax3), rj ;
cz* + 2lxy, cz2 (cz3 — oax3 —
355
the coefficient of £ herein is
= \ {(bcyzz2 + 2clxz3) (cz3 - 5aa? - 5by3) - (bcy2z- + Zblxy*) (by3 - ocz3 - oaoc3)},
= % [bcyW (- Qby3 + Qcz3} + 2lx [- %6 + c2^6 + 5a^ (by3 - cz3)]},
= (by3 - czs) [5a^4 - Uxy3 - clxz3 - 3bcy*z2].
Hence, restoring the factor (a&c + 8£3)2, we have the term
(abc + 8l3)2 . | (by3 - cz3) [5alo? - Uxy3 - clxz3 - 3bcy2z2].
26. M' = -($)M. Here M is of the form (abc + 8l3)2 (a- U + &c.) ; and the B operates
through the (abc + 8l3)2 a2, &c. ; we, in fact, have in M' the term
- (abc + 8Z3)2 . £ (by3 - cz3) [Zal'tf - bl'xy3 - cl'xz3 - Sbc'fz2],
which is
= (abc + 8l3)2 . % (by3 - cz3) [(abc + 2l3) (5ax* - bxy3 - cxz3} + I8bcl2y2z2].
- Slbc? + (- abc + 4Z3) r)£, d^F, x
J?--|[Jac](P, F, A), =-
- Slcarj2 + (- abc + 4Z3) §f ,
y
- Slab? + (- abc + 4,1s) fa d^F, z
and the whole coefficient of x is thus
2 + (abc - 4Z3) ff] ^F - [Mob? + (abc - 4^3) fr] V},
or substituting for ^d^F, ^d^F their values, this is
= {Wcav2 + (abc - U3) &} [azb2? - (abc + IQl3) (b?? + a£V)
- 4^2 (bcpq 4- cafr* + 4a6^^3) - 81 (abc +
- {Slab? + (abc - U3) &} [a2cy - (abc + IQl3) (arf? + c^f )
- 4Z2 (beg* + 4cafr3Z+ ab^) - 81 (abc + 2
45—2
356 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
Collecting, first, the terms independent of abc — 4<l3, and, next, those which contain
abc—4il3, each set contains the factor cr)3 — b%3, and the whole is =crf — b£3 multiplied by
- 3la3bcr)2£2 - 3aH (abc + 8l3) r)2^ - 121s (ale? + a2cfr3 + a2b^3) - 24aJ2 (abc + 21s)
+ (abc - 4Z3) {a*cfrs + a2b^3 - (abc + 16/3) £4
and here collecting the terms in £4, £ (erf 4- &£3)> f2?7£> and rfZ*, each of these contains
the factor a&c+8£3, and, finally, the term of M is
= (abc + 8l3) (cr)3 - b?) [(abc - 81s) £4 - a*c£rf - a2b^3 - I2al2^ - QaHrj^2] x.
34. M'
Here M is of the form (abc +8l3)(aU+bV+ cW) ; and, operating with 8 through the
(abc + 8l3) a, &c., we obtain in M' the term
£ (abc + 8l3) x (cy3 - b?) [(a'bc + ab'c + abc' - 24^') ? + &c.],
where
a'bc + ab'c + abc' - 2 W = I8abcl2 + 24^2 (abc + 21s), = Ql2 (7abc + 81s),
and the term thus is
= (abc + 81s) x (erf - 60 [(7 abc + 81s) I2? + ...].
This concludes the series of calculations.
Cambridge, England, 17 May, 1881.
771] 357
771.
SPECIMEN OF A LITEEAL TABLE FOR BINARY QUANTICS,
OTHERWISE A PARTITION TABLE.
[From the American Journal of Mathematics, vol. iv. (1881), pp. 248 — 255.]
THE Table, commencing 1 ; b; c, b2 ; d, be, b3 ; . . . , is in fact a Partition Table,
viz. considering the letters b, c, d, ... as denoting 1, 2, 3, ... respectively, it is 1°;
1; 2, 11 ; 3, 12, 111; ... a table of the partitions of the numbers 0, 1, 2, 3, ...,
expressed however in the literal form, in order to its giving the literal terms which
enter into the coefficients of any covariant of a binary quantic. The table ought to
have been made and published many years ago, before the calculation of the covariants
of the quintic ; and the present publication of it is, in some measure, an anachronism :
but I in fact felt the need of it in some calculations in regard to the sextic; and
I think the table may be found useful on other occasions. I have contented myself
with calculating the table up to 5=18, that is, so as to include in it all the partitions
of 18 : it would, I think, be desirable to extend it further, say to z = 26 ; or even
beyond this point, but perhaps without introducing any new letters, (that is, so as
to give for the higher numbers only the partitions with a largest part not exceeding
26): the question of the space which such a table would occupy will be considered
presently.
As to the employment of the table, observe that, in applying it to the case of
a quantic (a, b, c, dfyx, y)3, the terms containing the letters e, f, etc., posterior to
the last coefficient d of the quantic are to be disregarded ; and that the terms are
to be rendered homogeneous by the introduction of the proper power of the first
coefficient a, rejecting any term for which the exponent of a would be negative (or
what is the same thing, any term of too high a degree in the coefficients b, c, d);
358
SPECIMEN OF A LITERAL TABLE FOR BINARY QUANTICS,
[771
thus, for the cubicovariant, where the coefficients are of the degree 3, and of the
weights 3, 4, 5, 6 respectively, from the portion of the table
d e f g
be bd be bf
b3 c- cd ce
b2c b2d d"
64 be2 b2e
bsc bed
we at once copy out the terms
etc.
a2d
abc
b3
abd
acd
b2d
' be2
ad2
bed
c3
which compose the coefficients in question.
As regards the formation of the table, this is at once effected, and the successive
terms are obtained currente calamo, by Arbogast's rule of the last and the last but
one : observing that each term is to be regarded as containing implicitly a power
of a, so that operating on any term such as 64, the operation on the last letter gives
b3c, and that on the last but one letter gives b5. There is little risk of error except
in the accidental omission of a term ; but of course any one omission would occasion
the omission of all the subsequent terms derivable from the omitted term, and would
so be fatal : to remove this source of error, observe that for the successive numbers
0, 1, 2, 3, etc., the number of partitions should be
0123456 7 8 9 10 11 12 13 14 15 16 17 18 ...
1 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 ...
and we can thus, for each partible number successively, verify that the right number
of partitions has been obtained.
But as the number of partitions becomes large, a further control is convenient,
and even necessary — say we have the 176 partitions of 15, we have by the rule to
derive thence the 231 partitions of 16, and it is not until the whole of this derivation
is gone through, that we could by counting the number of the new terms ascertain
that the right number of 231 terms has been obtained. To break up the verification,
it is convenient to know that for the partitions of 16 into 1 part, 2 parts, 3 parts?
4 parts, etc., the numbers of partitions are 1, 8, 21, 34, etc., respectively: we can
then as soon as the derivations giving the partitions into 1 part, 2 parts, 3 parts,
etc., respectively, have been performed, verify that the right numbers 1, 8, 21, 34, etc.,
of terms have been obtained. The numbers are contained in the following table, each
column of which is calculated from the preceding columns according to a rule which
771]
OTHERWISE A PARTITION TABLE.
359
is easily obtained, and which is itself verified by the condition that the sums of
the numbers in the several columns give the before mentioned series of numbers 1,
1, 2, 3, 5, 7, etc.
No. of
Parts.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
PARTIBLE NUMBER.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
1
1
2
3
4
5
7
8
10
12
14
16
19
21
24
27
1
1
2
3
5
6
Q
11
15
18
23
27
34
39
47
1 1
2
3
5
7
10
13
18
23
30
37
47
57
1
1
2
3
5
7
11
14
20
26
35
44
58
1
1
2
3
5
7
11
15
21
28
38
49
1
1
2
3
5
7
11
15
22
29
40
1
1
2
3
5
7
11
15
22
30
1
1
2
3
5
7
11
15
22
1
1
2
3
5
7
11
15
1
1
2
3
5
7
11
1
1
2
3
5
7
1
1
2
3
5
1 1
2
3
1
1
2
1
1
1
1 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385
The practical rule for the construction of the table thus is: — On a sheet of
paper ruled in squares, and which is read as a continuous column from the bottom
of one column to the top of the next column, form the terms by Arbogast's method
as already explained; writing down in pencil a batch of terms, and counting them
360
SPECIMEN OF A LITERAL TABLE FOR BINARY QUANTICS,
[771
to see that the right number has been obtained, then, at the same time verifying
the derivations, mark these over in ink; and so on with another batch of terms,
until the whole number of the partitions of any particular number is obtained.
The foregoing series 1,1, 2, 3, ..., 385, for the number of the partitions of the
successive numbers 0, 1, 2, 3, ..., 18 is carried by Euler up to the number of partitions
of 59, = 831820, see the paper " De Partitione Numerorum," Op. Arith. Coll. I.,
bottom line of the table pp. 97 — 101 : the continuation from the number 385 and
for the partible numbers 19 to 30 is as follows :
19 20 21 22
23
24
25
26
27
28
29 30
490 627 792 1002 1255 1575 1958 2436 3010 3718 4565 5604'
the whole number of terms 1, 1, ..., 5604 amounts to 28629, which at the rate of
500 to a page would occupy somewhat under 60 pages; or, at the rate here employed
of 369 to a page, somewhat under 78 pages.
THE PARTITION TABLE, 0 TO 18.
0.3
4.5
6.7
7.8
8. 9
9
9. 10
10
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0.11
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771]
OTHERWISE A PARTITION TABLE.
THE PARTITION TABLE, 0 TO 18 (continued}.
361
11
11.12
12
12.13
13
13.14
14
14
14
di
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610e
C. XL
46
362 SPECIMEN OF A LITERAL TABLE FOR BINARY QUANTICS,
THE PARTITION TABLE, 0 TO 18 (continued).
14.15
15
15
15
15.16
16
16
16
16
b9cd
Vgh
bcdef
bT
6Vrf
cdl
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15
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176
bdfg
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65c5
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771]
OTHERWISE A PARTITION TABLE.
THE PARTITION TABLE, 0 TO 18 (continued).
363
16
16.17
17
17
17
17
17
17
17. 18
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b2cd3e
b7dh
bsd3
bfm
bl°dz
#
Mg
Wh
bc*def
Vd*
Peg
6Ve
bgl
46—2
364
SPECIMEN OF A LITERAL TABLE FOR BINARY QUANTICS.
THE PARTITION TABLE, 0 TO 18 (continued}.
[771
18
18
18
18
18
18
18
18
18
bhk
befi
b2dej
d3ef
bc2deg
b3cdf2
w%
b5cd2f
b»e2g
bij
begh
Vdfi
d*e3
bc2df2
b'ce-f
b*c2dh
b*cde2
b«cdf
c2o ' >
bf2h
b\lgh
b*n
bc2e2f
b3d3g
b*c*eg
b5d3e
b*ce2
cdn
w
5W
b*cm
bcd3g
Vd2ef
64c2/2 64cV
b*d2e
cent
c3m
Vefh
Vdl
bcdzef
b3de3
b*cd2g V<?df
Vff
cfl
c2dl
6V
Vek
bcde3
bWi
b*cdef bW
V<?de
cgk
c2ek
bT9
Wj
bd*f
bVdh
b4ce* 6W2e
Vcd3
chj
tfj
b<?l
bty
bd3e2
b2c3eg
Vd3/ b*cd*
Wc*e
ci2
c*gi
bc2dk
ft4/*2
C5i
b2c3/2
b*d2e2 ' 6V/
bWd2
d2m
c2h2
bc^ej
b3<rl
C4dh
b2c2d2g
bWi
b3c*de
b5C5d
del
cd2k
bc2fi
b3cdk
c*eg
b2c2def
b3c3dg
bWd3
We7
dfk
cdej
bc2gh
b3cej
cT
bW
V<?ef
b*cee
bllh
dgj
cdfi
bcd2j
Vcfi
<?d2g
b2cd*f
b3C2d2f
b2C5d2
bwcg
dhi
cdgh
bcdei
b3cgh
c3def
b2cdV
b3c2de2 \ bc7d
bwdf
#h
ceH
bcdfh
b3d2j
c3e3
bWe
b3cd3e
c9
610e2
efj
cefh
bcdg"2
bsdei
cWf
b<fh
b3d5
b*j
We2/
egi
ceg2
bce2h
bsdfh
c2d2e2
bc*dg
Wc*g
Wei
Wcde
eh?
4*9
beefy
Wdg*
cd*e
bclef
b2c4df
Wdh
Wd3
f*i
d3j
bcf3
b3e*h
ds
bcWf
bW
Weg
W<?e
fgn
d2ei
bd3i
b3efg
b6m
bc3de2
62c3c?2e
bT
bsc2d2
93
djh
bd*eh
b3/3
b5cl
bc2d3e
62c2^4
b7C2h
b7c*d
b3p
dY
bdffi
6V4
b5dk
bed5
bcsf
b7cdg
b*c6
Wco
de2h
bde^g
Wfdj
b*ej
c6g
bc5de
b7cef
b12ff
Wdn
defg
bdef*
bVei
b*fi
<?df
bc*d3
b7d2f
b»cf
Wem
df3
b<?f
6V/7i
Vgh
c5e2
c7e
b7de*
bude
byi
*9
c*k
62cy
b*C2k
c4d2e
csd2
bVg
bwc2e
Wgk
eT
c3dj
b*cd2i
b4cdj
cW
bsk
Wc2df
bwcd2
Whj
Wo
(?ei
b2cdeh
b*cei
VI
b7cj
bW
bVd
Wi2
Wen
c3fh
tfcdfg
Vcfh
b6ck
Vdi
Wcdze
We5
bc2n
Wdm
cY
bWg
&v
b'dj
b7eh
Wd*
V3f
bcd?n
Wei
c2dH
tfcef2
b*d2i
b6ei
Vf9
v<y
b12ce
bcel
Vfk
c2deh
b*d3h
b*deh
b6fh
b«c*i
Wc*de
bl*d*
bcfk
*>3ffj
<?dfg
b*d2eg
Vdfff
»v
b«cdh
bWd3
bnc2d
bcgj
Whi
Mg
tfd2/2
bWg
b*c2j
b6ceg
b4c*e
bwc4
bchi
62c2m
cV2
b2de2f
Vef*
b^cdi
b«cf*
bWd*
bl4e
bdH
Wcdl
cd3h
b2e4
b3C3j
bsceh
b6d2g
bsc6d
bl3cd
bdek
b2cek
cd2eg
bc4j
b3C2di
b*cfg
bedef
6V
bl2c3
bdfj
Wj
cdj*
bc?di
b3c2eh
b5d2h
b6e3
b10i
bl5d
bdgi
b2cgi
cde*f
bc3eh
bVfg
bsdeg
V<?k
b»ch
bl4C2
bdh2
Wch*
ce4
bc3fg
b3cd2h
b5df2
b*c2dg
Wdg
bisc
b^j
b*d*k
rfV
bcWh
bsodeg
b*ey
b5c2ef
b»ef
bis
772] 365
772.
ON THE ANALYTICAL FORMS CALLED TREES.
[From the American Journal of Mathematics, vol. iv. (1881), pp. 266 — 268.]
IN a tree of N knots, selecting any knot at pleasure as a root, the tree may
be regarded as springing from this root, and it is then called a root-tree. The same
tree thus presents itself in various forms as a root-tree ; and if we consider the
different root-trees with N knots, these are not all of them distinct trees. We have
thus the two questions, to find the number of root-trees with N knots; and, to find
the number of distinct trees with N knots.
I have in my paper "On the Theory of the Analytical Forms called Trees,"
Phil. Mag., t. xm. (1857), pp. 172—176, [203] given the solution of the first question;
viz. if (})N denotes the number of the root-trees with N knots, then the successive
numbers <f>1} <£2, <f>3} etc., are given by the formula
viz. we thus find
suffix of </> 1 2 3 4 5 6 7 8 9 10 11 12 13
<£=1 1 2 4 9 20 48 115 286 719 1842 4766 12486.
And I have, in the paper "On the analytical forms called Trees, with application
to the theory of chemical combinations," Brit. Assoc. Report, 1875, pp. 257 — 305, [610]
also shown how by the consideration of the centre or bicentre " of length " we can
obtain formulae for the number of central and bicentral trees, that is, for the number
366 ON THE ANALYTICAL FORMS CALLED TREES. [772
of distinct trees, with N knots : the numerical result obtained for the total number
of distinct trees with N knots is given as follows :
No. of Knots 1 2 3 4 5 6 7 8 9 10 11 12 13
No. of Central Trees 101123 7 12 27 55 127 284 682
„ Bicentral „ 010113 4 11 20 51 108 267 619
Total 1 I I 236 11 23 47 106 235 551 1301.
But a more simple solution is obtained by the consideration of the centre or
bicentre "of number." A tree of an odd number N of knots has a centre of number,
and a tree of an even number N of knots has a centre or else a bicentre of number.
To explain this notion (due to M. Camille Jordan) we consider the branches which
proceed from any knot, and (excluding always this knot itself) we count the number
of the knots upon the several branches; say these numbers are a, ft, 7, 8, e, etc.,
where of course a + ft + j + 8 + e + etc. = N — 1. If N is even we may have, say
a = ^N; and then /3 + y + 8 + e + etc. = |-Ar— 1, viz. a is larger by unity than the sum
of the remaining numbers: the branch with a knots, or the number a, is said to
be "merely dominant." If N be odd, we cannot of course have a = ^N, but we may
have a>^N; here a exceeds by 2 at least the sum of the other numbers; and the
branch with a knots, or the number a, is said to be "predominant." In every other
case, viz. in the case where each number a is less than %N, (and where consequently
the largest number a does not exceed the sum of the remaining numbers), the several
branches, or the numbers a, ft, 7, etc., are said to be subequal. And we have the
theorem. First, when N is odd, there is always one knot (and only one knot) for
which the branches are subequal : such knot is called the centre of number. Secondly,
when N is even, either there is one knot (and only one knot) for which the branches
are subequal : and such knot is then called the centre of number ; or else there is
no such knot, but there are two adjacent knots (and no other knot) each having a
merely-dominant branch : such two knots are called the bicentre of number, and each
of them separately is a half-centre.
Considering now the trees with N knots as springing from a centre or a
bicentre of number, and writing tyN for the whole number of distinct trees with N
knots, we readily obtain these in terms of the foregoing numbers fa, fa, fa, etc., viz.
we have
s = coeff. xz in (1 — #)~*»,
4 = %fa (fa + 1) + coeff. a? in (1 - #)-*»,
5 = coeff. x* in (1 — #)-*> (1 —
6 = $fa (fa + 1) + coeff. a? in (1 - #)-*' (1 -
7 = coeff. x6 in 1 — #"*» 1 —
772]
ON THE ANALYTICAL FORMS CALLED TREES.
367
and so on, the law being obvious. And the formulae are at once seen to be true.
Thus for N=6, the formula is
+ lie fc (fc + 1) (fc + 2) (fa + 3) (fc + 4).
We have fa root-trees with 3 knots, and by simply joining together any two of
them, treating the two roots as a bicentre, we have all the bicentral trees with
6 knots: this accounts for the term ^fa(fa + I). Again, we have fa root-trees with
1 knot, fa root-trees with 2 knots ; and with a given knot as centre, and the
partitions (2, 2, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1) successively, we build up the central
trees of 6 knots, viz. 1° we take as branches any two fa's and any one fa; 2° any
one fa and any three fa's ; 3° any five fa^s ; the partitions in question being all
the partitions of 5 with no part greater than 2, that is, all the partitions with sub-
equal parts. We easily obtain
suffix of
1 2 3 4 5 6 7 8 9 10 11 12
13
= 1 1 1 2 3 6 11 23 47 106 235 551 1301
agreeing with the results obtained by the much more complicated formulae of the
paper of 1875.
368
[773
773.
ON THE 8-SQUAEE IMAGINARIES.
[From the American Journal of Mathematics, vol. iv. (1881), pp. 293 — 296.]
I WRITE throughout 0 to denote positive unity, and uniting with it the seven
imaginaries 1, ... ,7, form an octavic system 0, 1, 2, 3, 4, 5, 6, 7, the laws of com
bination being
02 = 0, 12 = 22 = 32 = 42 = 52 = 62 = 72 = -0,
123 = £l, 145 = €s, 167 = 63,
246 = e4, 257 = e5)
347 = c6, 356 = e7,
where e = ±, viz. each e has a determinate value + or — as the case may be ; and
where the formula, 123 = e1} denotes the six equations
23= €ll, 31= e,2, 12= ei3,
32 = -e1l, 13 = -e12J 21 = -ei3,
and so for the other formulae. The multiplication table of the eight symbols thus is
0 1234567
0
0
1
2
3
4
5
6
7
1
1
0
«i3
-Cl2
€25
-c24
<37
-€36
2
2
-Cl3
0
ill
€46
%7
-€44
-<55
3
3
«i2
-«il
0
^67
e76
-€75
-^4
4
4
-c25
-^6
-*67
0
€,l
e42
€63
5
5
*24
-%7
-676
-%1
0
*73
%a
6
6
-€37
^4
675
-e42
-c73
0
%i
7
7
e36
€55
%4
-eG3
-*82
-%1
0
773]
ON THE 8 -SQUARE IM AGIN ARIES.
369
(04' + 0'4 + 6,51 + e462 + e673
Hence if 0, 1, 2, 3, 4, 5, 6, 7 and 0', 1', 2', 3', 4', 5', 6', 7' denote ordinary
algebraical magnitudes, and we form the product
(00 + 11 + 22 + 33 + 44 + 55 + 66 + 77) (O'O + 11 + 2'2 + 3'3 + 4'4 + 5'5 + 6'6 + 77),
this is at once found to be =
(00' - 11' - 22' - 33' - 44' - 55' - 66' - 77') 0
+ (01' + 0'l+e123 + e245 + e367 )1
+ (02' + 0'2 + e131 + e446 + es57 )2
)3
) 4
)5
+ (06' + 0'6 + e37l+e424 + e735 )6
+ (07' + 07 + e316 + e525 + e«34 ) 7,
\
where 12 is written to denote 12'— 1'2, and so in other cases.
The sum of the squares of the eight coefficients of 0, 1, 2, 3, 4, 5, 6, 7 respectively
will, if certain terms destroy each other, be
= (O2 + 12 + 2- + 3- + 42 + 52 + 6s + 72) (O'2 + I'2 + 2'- + 3'2 + 4'2 + 5'2 + 6'2 + 7/2) ;
viz. the sum of the squares contains the several terms
eie223.45, eie323.67, eie431.46, fje.,31.57, elf612.47, eie712.56, e2e345 . 67,
e4e724.35, e4e662.73, e,e714.63, e26651 . 73, e2e514.72, e2e451.62, e4e546 . 57,
e56625.34, e5e772.63, e3eB16.34, e;)e77l.35, e:je471.24, e3e516.25, 66e747 . 56,
and observing that 21 = — 12, etc., and that we have identically
23 . 45 + 24 . 53 + 25 . 34 = zero, etc.,
then the three terms of each column will vanish, provided a proper relation exists
between the e's : viz. the conditions which we thus obtain are
= 62 64 =
We may without loss of generality assume ej = e2 = e3 =
become
+ = -e4e6= e5e7,
; the equations then
C. XI.
47
370
ON THE 8 -SQUARE IMAGINARIES.
[773
and writing 6 = ± at pleasure, these are all satisfied if — e4 = e5 = e6 = e7 = 6. The terms
written down all disappear, and the sum of the squares of the eight coefficients thus
becomes equal to the product of two sums each of them of eight squares, viz. this
is the case if ex = e2 = e3 = +, — e4 = e5 = e6 = e7 = 6, d being = + at pleasure : the resulting
system of imaginaries may be said to be an 8-square system.
We may inquire whether the system is associative ; for this purpose, supposing
in the first instance that the e's remain arbitrary, we form the complete system of
the values of the triplets 12 . 3, 1 . 23, etc., (read the top line 12 . 3 = — e2 0,
1 . 23 = — ejO, the next line 12 . 4 = e, e67, 1 . 24 = e.e47, and so in other cases) :
12.3 =
1 . 23 =--
_ Cl
_ £j
0
12.4 =
1 . 24 =
Mi
Ml
7
12.5 =
1.25 =
ei€7
~ «3«6
6
12.6 =
1.26 =
- Cl«7
- e,£4
5
12.7 =
1.27 =
~ el€6
£2e5
4
13.4 =
1.34 =
- «1«4
~ ^3^6
6
13.5 =
1.35 =
- «1«5
€;5£7
7
13.6 =
1.36 =
*1*4
€2€7
4
13. 7 =
1.37 =
«!««
_€2c6
5
14.5 =
1.45 =
- «a
- « 2
0
14.6 =
1.46 =
£._,£7
Cl£4
3
14.7 =
1.47 =
Ml
- «1«6
2
15.6 =
1.56 =
- V«
- el€7
2
15.7 =
1.57 =
- «2<6
€lc5
3
16.7 =
1 . 67 =
- «3
- «3
0
23.4 =
2.34 =
•A
_C5£6
5
23.5 =
2.35 =
~ <1*2
- «4«7
4
23.6 =
2.36 =
•A
- «S«7
7
23.7 =
2.37 =
^1^3
- *4€6
6
24.5 =
2.45 =
^4^7
- «1«2
3
24.6 =
2.46 =
- £4
- £,
0
24.7 =
2.47 =
«3«4
•,«,
25.6 =
2.56 =
~ *3«5
M?
1
25.7 =
2.57 =
- *5
-£5
0
26.7 =
2.67 =
- VB
- *iC3
3
34.5 =
3.45 =
- «S«6
£j£.,
2
34.6 =
3.46 =
- «1«4
1
34.7 =
3.47 =
- «6
- *6
0
35.6 =
3. 56 =
-£7
-%
0
35.7 =
3.57 =
€3£7
~«l%
1
36.7 =
3.67 =
- «S<7
€l£;j
2
45.6 =
4. 56 =
>A
- «B«7
7
45.7 =
4.57 =
- f2£3
- «4£5
6
46.7 =
4.67 =
-C4£5
- €2€;j
5
56.7 =
5. 67 =
-^7
Mi
4.
773]
ON THE 8 -SQUARE IM AGIN ARIES.
371
Write as before ej = e2 = ea = + ; then, disregarding the lines (such as the first line)
which contain the symbol 0, and writing down only the signs as given in the third
and fourth columns, these are
«4«7
«4
~~ €6€7
We hence see at once that the pairs of signs in the two columns respectively cannot
be made identical: to make them so, we should have eg = e4, e7 = — es, e7=e4, that is,
e4= e6 = e7 = — e5, which is inconsistent with the last equation of the system — e6e7 = +.
Hence the imaginaries 1, 2, 3, 4, 5, 6, 7, as denned by the original conditions, are
not in any case associative.
If we have e1 = e.2 = e3=+ and also — e4 = e5= e6 = e7 = 6, that is, if the imaginaries
belong to the 8-square formula, then it is at once seen that each pair consists of
two opposite signs; that is, for the several triads 123, 145, 167, 246, 257, 347, 356
used for the definition of the imaginaries, the associative property holds good,
12 . 3 = 1 . 23, etc. ; but for each of the remaining twenty-eight triads, the two terms
are equal but of opposite signs, viz. 12 . 4 = — 1 . 24, etc. ; so that the product 124 of
any such three symbols has no determinate meaning.
Baltimore, March oth, 1882.
47—2
372
[774
774.
TABLES FOR THE BINARY SEXTIC.
THE LEADING COEFFICIENTS OF THE FIRST 18 OF THE 26 COVARIANTS.
[From the American Journal of Mathematics, vol. iv. (1881), pp. 379 — 384.]
INCLUDING the sextic itself, the number of covariants of the binary sextic is = 26,
as shown in the table p. 296 of Clebsch's Theorie der binaren algebraischen Formen^
Leipzig, 1872 ; viz. this is
Order
Deg.
0
10
12
1
/
2
A
*
H
3
z
?
(f, i}
T
4
B
(/ 0,
(/, 0
(H,i)
5
(*> Oa
(». 0
(ff, I)
6
4m
(p. 0
((/ 1), 0.
7
(/, ^),
//* ^v
\»/ ' /3
8
(i, 1%
9
((/-•), P),
10
(/, 0.
(f, 1%
12
((/ <), ^)a
15
((/ *), *),
774]
TABLES FOR THE BINARY SEXTIC.
373
Or, using the capital letters A, B,..., Z to denote the 26 co variants in the same
order, the table is
024 6 8 10 12
7
8
9
10
12
15
E
I
E
M, = (C, Ef
A
D
F
R, = (G, EV
II
A is the sextic. P is Salmon's C, p. 204.
B is Salmon's A, p. 202. W ,, „ D, p. 207.
/ „ „ B, p. 203. Z „ „ E, p. 253.
The references are to Salmon's Higher Algebra, 2nd Ed., 1866.
In the present short paper I give the leading coefficients of the first 18 covariants,
A to R (some of these are of course known values, but it is convenient to include
them) : for the next four covariants S, T, U, V, the leading coefficients depend upon
the coefficients of .A, G, G and E-, viz. writing
A = (a, b, c, d, e, / #$>, y)6,
we have
€=(«', 1/8',
G =(^^",
S, Coeff. a? = ae — b& + cy — d/3 + ea,
T, „ a* = aS - 2by + 3c/8 - 4rfa,
U, „ x> = 2a'S - /3'7 + y'/3 - 28'a,
V, „ g* = 280a"e - 3o/S"S + 107"7 - 20S"/3 + 24e"a.
Similarly the invariant W and the leading coefficients of X, Y depend on the coefficients
of A, G and E3; and the invariant Z depends on the coefficients of G and E4.
374
TABLES FOR THE BINARY SEXTIC.
[774
But these two invariants W and Z have been already calculated; viz. as already
mentioned, W is Salmon's invariant D, and Z his invariant E, given each of them
in the second edition of his Higher Algebra (but not reproduced in the third
edition) : on account of the great length of these expressions, it has been thought
that it was not expedient to give them here.
For the reason appearing above, I have added the expressions for the remaining
coefficients of C, E, G.
A,
C, x*
F
G, a*
ff, x1'2
a + 1
ag + 1
a°i/- 6
ce + 15
a e +1
a°6d - 4
c2 + 3
a c 4- 1
a°62- 1
a c^ + 1 a ce +1
df - 3 d2 - 1
e* + 2 a°62e - 1
a be — 5
cd + 2
a\l + 1
a be - 3
a°63 + 2
tP-10
a°62# — 1 bed + 2
a°62d + 8
6c/ + 3
bde -1
c3 - 1
6c2 - 6
^>2g ^
cd2 + 2
7,
A", x6
L,
M,
, X*
0, Xs
aceg + 1
a2/2 + 1
a2c?y + 1
a V + 1 a V + 1
tfcfg 1
a?cdg 0
c/2 - 1
abef - 10
ef ~ 1
rtfe — 1 dfg - 6
</6f/ + 1
cef - 1
#9 -1
cdf + 4
a beg — 3
aby-i
eV + 8
df* + 3
rf2/ + 3
de/ +2
cea +16
W/ - 2
bee - 2
ef2 - 3
«2/ - 3
de2 - 2
e3 - 1
rf2e - 12
6e3 + 5
bd2 + 4
a 6V - 1
abyg + 1
a b-dg 0
a°6Vy - 1
a°62^ + 16
c2/ + 9
c-d - 1
bcfg + 6
bceg + 2
62e/ + 1
by2 + 1
6V + 9
cde — 17
a°bse + 3 bdeg - 34
bcf2 - 3
bc-g 0
6cd# + 2
&cy - 12
c?:< + 8
6W- 6 &c#'2 + 48
bd*y - 4
bcdf - 14
6ce/ - 2
bcde — 76
a°680 + 2
6c:i +3 be2/ - 18
6rfe/- 12
6ce- +11
id2/ - 2
W3 + 48
62c/ - 6
c2ey + 18
be3 +15
bd~e + 1
fccfe2 + 2
c3e + 48
62rfe + 2
c2/2 - 45
c'2c?y + 1
c3/ + 9
<?g -1
c2c/2 - 32
6c2e + 6
cd'2g + 4
c2e/ + 9
c'-We - 14
c2^/" + 2
6c^2 — 4
cdef + 78
crfy + 4
cd3 + 6
cV + 1
ce:t - 36
cde2 - 21
a°63cy 0
c^2e - 3
d*f - 48
d3e + 8
63^ + 8
cZ4 + 1
rfV + 28
a°6% - 3
6V - 9
a°62cey 0
tfcdg + 6
6V/- 6
62^V+ 64
62ce/+ 9
b*cde+ 16
6^-144
Vd*f+ 32
bzd3 - 8
6V + 81
62de2 - 39
6c3e - 3
btfdg- 96
bc'y - 3
6c<W2+ 2
6c2e/ + 108
6c-'^- 66
6cd2/+ 96
6cV + 18
bcde2 - 126
6cc^e + 76
6(/3e + 16
bd* - 32
cV + 36
cy +27
c3df - 72
cade — 45
cse2 - 27
c2d3 + 20
c2rf2e + 96
cc^4 -32
774]
TABLES FOR THE BINARY SEXTIC.
375
P, x°
Q, a*
R, x6
df*
a bcdg*
beefy
bcf3
bdft
bde'2g
bey
c'2e'2g
cd'2eg
cdy2
odtf
ce4
d*g
d3ef
rfW
a°bsdg2
b3efg
Wcef2
b'2de'2f
We*
bc2deg
bc'2df2
6cV/
bcd3g
bcd'2ef
bcde3
wy
+ 1
- 6
+ 4
+ 4
3
- 6
+ 18
- 12
+ 12
- 18
+ 6
+ 4
— 24
- 18
+ 30
+ 54
- 12
-42
+ 12
-20
+ 24
- 8
+ 4
- 12
+ 8
- 3
+ 30
-24
- 12
-24
+ 60
-27
+ 6
-42
+ 60
-30
+ 24
-84
+ 66
+ 24
- 24
+ 12
-27
- 8
+ 66
- 8
-24
-39
+ 36
Mf
- 1
tf(J
+ 9
f'A
- 8
a?bcg2
+ 3
bdfg
- 24
be'2g
- 45
bef2
+ 66
tfg
_ 2
+ 3
cdeg
+ 5
+ 48
cd/2
+ 6
- 12
cey
rj
- 51
d3g
- 3
- 16
d2ef - 3
+ 36
de3 + 4
- 8
ffl b3g2 0
_ 2
Vcfy + 4
+ 12
62<&# - 5
+ 192
tfdf2 - 6
- 48
b2e2f + 7 I - 144
bc2eg
- 5 - 159
be2/'2
6
+ 18
bcd^g
+ 7
48
bcdef
- 16
+ 24
bee3
+ 23
+ 279
bdy
+ 30
- 48
bd2e~
- 33
- 84
c'dg
- 1
+ 42
c3ef
+ 36
+ 153
C2d2f
- 37
- 36
cW
- 53
-399
cd3e
+ 79
+ 312
d5
- 24
- 64
vWfy
- 2
0
b3ceg
+ 5
0
b3cf2
+ 6
0
b3d2g
+ 2
- 224
b3def
+ 22
+ 144
bse3
27
+ 54
b2c2dg
- 8
+ 336
b2c2ef
- 39
- 108
b2cd2f
- 50
+ 384
b2cde2
+ 107
-684
b2d3e
- 22
+ 144
bc*g
+ 3
- 126
bc3'df
+ 84
- 648
bc*e2
- 21
+ 432
bc2d2e
- 102
+ 564
bed4
+ 44
- 288
C5f
27
+ 270
c4de
+ 45
- 450
C3d3
20
+ 200
376
TABLES FOR THE BINARY SEXTIC.
c
Remaining Coefficients of C, E, G.
E G
&
*
af +
2
be-
6
cd +
4
-y
ce -
9
d2 +
8
xy*
bg +
2
cf-
6
de +
4
y
eg +
1
e2 +
3
xy
ady +
1
aef -
1
beg -
1
bdf-
8
be- +
9
c2/ +
9
cde -
17
d* +
8
y1
aeg +
1
of2 -
1
bdg -
3
bef +
3
<?y +
2
cdf -
1
ce2 -
3
d2e +
2
Note.— In the tables on this page, a
has been treated like the other letters ;
on the preceding pages, the powers
of a have been suppressed except in
the first of every series of terms con
taining a common power of a.
*
«v +
1
abf +
2
ace -
19
ad2 +
8
b~f, -
6
bed +
44
c:i —
30
•V
aby +
i
acf -
14
ade -
14
b2f
0
bee —
21
bd2 +
112
C2d -
70
,¥
acy +
7
adf -
28
ae2 —
14
b2y +
14
bef -
42
bde +
168
c2e —
105
«y
ady
0
aef -
35
beg +
35
bdf
0
be2 +
105
#f -
105
**
aeg
7
of2
14
bdg
+ 28
bef
+ 42
&g
+ 14
cdf
- 168
ce-
+ 105
«y
Ofg
— 7
bey
+ 14
V*
0
cdg
+ 14
cef
+ 21
d2/
- 112
de2
+ 70
xy7
ay2
1
Vy
_ 2
ceg
+ 19
</2
+ 6
d*g
8
def
- 44
e*
+ 30
f
by2
1
C/y
+ 5
deg
- 2
df2
8
e*f
+ 6
The final result is that we have the values of the invariants B, I, P, W, Z
and the leading coefficients of the covariants A, C, D, E, F, G, H, J, K, L, M,
N, 0, Q, R: also the means of calculating the leading coefficients of the remaining
covariants 8, T, U, V, X, Y.
775] 377
775.
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
[Written in 1894: now first published.]
THE binary sextic has in all (including the sextic itself and the invariants) 26
covariants which I have represented by the capital letters A, B, C, ..., Z. The leading
coefficients of the covariants A to R (of course for an invariant this means the
invariant itself) are given in my paper " Tables for the binary sextic," Amer. Math. Jour.
vol. iv. (1881), pp. 379—384, [774]; the two invariants Z and W (Salmon's invariants
-D and E} had been already calculated. But I did not in my values of the leading
coefficients, nor did Salmon in his values of the two invariants, insert the literal
terms with zero coefficients : as remarked in my paper [143] " Tables of the covariants
M to W of the binary quiutic," it is very desirable to have in every case the
complete series of literal terms, and I have accordingly in the expressions of the
covariants A to R obtained for the leading coefficients, and in the expressions obtained
from Salmon for the invariants W and Z, inserted in each case the complete series of
literal terms.
I give a list of the 26 covariants nearly in the form of that given in the latter
paper [143] for the covariants of the quintic, only instead of a separate column of
deg-weights I insert these in the body of the symbol; thus
C = (3, 3, 4, 3, 3)2 4 to 8 (x, y}\
the 5 coefficients of the quartic function contain respectively 3, 3, 4, 3, 3 terms
(some of them it may be with zero coefficients), are of the degree 2, and of the
weights 4, 5, 6, 7, 8 respectively.
The list is as follows :
A = (1, 1, 1, 1, 1, 1, I)1 0 to 6 (x, y)6,
B = (4)2 6 (x, y)°, Invt.,
C = (3, 3, 4, 3, 3)2 4 to 8 (x, y)\
D = (2, 2, 3, 3, 4, 3, 3, 2, 2)2 2 to 10 (x, y}*,
C. XI. 48
378 TABLES OF COVARIANTS OF THE BINARY SEXTIC.
E = (8, 8, 8)3 8 to 10 (x, y)-}
JP=(7, 7, 8, 8, 8, 7, 7Y 6 to 12 (x, yY,
G = (5, 7, 7, 8, 8, 8, 7, 7, 5)3 5 to 13 (x, yY,
# = (3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3)3 3 to 15 (x, yY,
7 = (18)4 12 (x, yY, Invt,
J = (16, 16, 18, 16, 16)4 10 to 14 (x, yY,
#=(14, 16, 16, 18, 16, 16, 14)4 9 to 15 (x, yY,
£ = (10, 13, 14, 16, 16, 18, 16, 16, 14, 13, 10)4 7 to 17 (x, yY0,
J/=(32, 32, 32)5 14 to 16 (x, yY,
N=(30, 32, 32, 32, 30)B 13 to 17 (x, yY,
0 = (25, 29, 30, 32, 32, 32, 30, 29, 25)5 11 to 19 (x, yY,
P = (58)6 18 (x, yY, Invt,
Q = (5l, 55, 55, 58, 55, 55, 51)6 15 to 21 (x, yY,
R = (5l, 55, 55, 58, 55, 55, 5l)6 15 to 21 (x, yY,
S = (94, 94, 94)7 20 to 22 (x, yY,
T=(90, 94, 94, 94, 90)7 19 to 23 (x, yY,
[f=(147, 151, 147)8 23 to 25 (x, y}2,
V= (221, 227, 227, 227, 221)9 25 to 29 (x, yY,
TF=(338)10 30 (x, yY, Invt,
X = (332, 338, 332)10 29 to 31 (x, yY,
7 = (668, 676, 668)12 35 to 37 (x, yY,
£ = (1636)15 45 (x, v)°, Invt.
[77;
a + 1
b + 6
c + 15
d + 20
e + 15
/+ 6
</ + !
0, Invt.
ag +
I
¥ -
6
ce +
15
d2 —
10
16
775]
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
379
ae + I
af+2
ag + 1
bg+2
C# + 1
bd-4:
be-Q
¥ -
c/-6
rf/-4
c2 + 3
cd + 4
ce - 9
de + 4
e2 +3
d- + 8
+ 4
+ 6
+ 9
+ 6
+ 4
aty-
a:5?/3
xy7
ac + 1
a^ + 4
«e + 6
«/+ 4
a# + 1
fy + 4
cff + 6
rf^ + 4
eg + 1
&2 -1
6c -4
6^+4
be + 16
If + 14
c/ + 16
rf/+ 4
e/ -4
/2 -1
c2 - 10
erf -20
ce + 5
rfe - 20
e2 - 10
rf2 -20
+ 1
a;6
+ 4
10
20 + 20
E=( Jx,
20
acy +1
a dy + 1
a e<7 +1
df -3
«/ - 1
/2 -1
e2 + 2
a°bcg - 1
a°6% - 3
a°6V - 1
6c//- 8
bef+ 3
bcf+ 3
be2 + 9
cV + 2
6<fo-l
c2/ + 9
CC//-1
c2« -3
cde - 17
ce2 -3
crf2+ 2
d3 + 8
rf3 + 2
±3
+ 5
+ 1
+ 26
+ 1
+ 7
F=(
+ 1
a*g ...
a bg ...
a eg +1
a^ + 2 a^ +1
a^ ... i a.g-2 ...
abf ...
cf +2
rf/ +2
ef •- 2
/2 - 1
a°beg + 2
a°4#
ce + 1
de - 2
e2 - 3
a°6c^ - 2
a^bdg + 2
6/2-2
ceg + 1
d* - 1
a°b2f - 2
aWy - 1
&rf/+ 4
bef - 2
cdg - 2
c/2 - 1
a°62e - 1
bee + 2
Hf-2
6e2 -- 2
cV - 3
ce/ + 2
d*ff ~ 1
bcd+ 2
6c?2 + 2
6rfe + 4
c2/ — 2
cdf+ 4
rfy+ 2
rf6/+2
c3 -1
c2rf - 2
2yJ 1^ O
cote + 6
ce2 + 2
rfe2 -2
e3 -• 1
rf2 -3
ct ^
o?2e-3
±1 ±2 +3 +2 + 1 +6 +3
±2 +4 +6 +10 +8
48—2
380
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
[775
t»
I
°
OS CD 00
O
CO
I I
i— i <M O
00
O
iQ
O
I I
t
»o 10
CO CO
1O 1O
o o
1-H 1-H
I -H
00
o
fe
(M OS CO
»O <M CO
O
I
co
I
o
n
lO
+1
+ 1
+ 1
o
^H
+ 1
+1
+1
+1
,— c CO <M
ej
\ + \
°S
co ; »o <M
. 1— 1 i-H
>
i $ •? i
°8
CO GO O •* O
^
\ \ + \ \
0 fcs &S f VL
^ *s •£ ^ v
»«O OJ C C v
T-I -* O O O
<M CO CD 00
: «
»
1 1 + 1 +
i
rt
*b» jfe» I" %» ^
^ ™»
8 8
OS lO CD ^^ 1O
^H CD lO O
O *O
fc
\ \ \ + \
^ ^ V, "I* ^
+ 1
8 °e
-M ^* 00 (M O
I-H (M Tt^ OS OS
o o :
^
11 + 1 +
+ i
rt
g5 •" ^ ^ ^
1 -1 ^
8 "8
: co oo : ^H
o :
1-H
<M
«L
I + I
+
V
^ ^ Is ^ H
!t^ *8 •
N » O n,^
O O w
8 °S
C-l 00 O -+ <M
o o :
%
1-H
+ 1 - 1 f +
l +
"*
B5 "^ "^ ^ "§*
"§ e? %
8 8
OS iO O O >O
i— i lO CD O
O lO
1-H
-»
1 +
°^"
5SS X ^ SS S
s ?
8 "8
r- 1 •«* O O O
m co oo CD
:• 8
H
+ + 1 1 +
^**"S ^ ^i ^
13
t*O 'o
W8 e =e
CO 00 O "f O
sT"
"8
%* 1 " ?
co ; 10 <M
_?s
+ I +
H
^ »«O O f^
"e e 8
I-H CO (M
"8
4 ^
«ji__*
775]
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
381
/-<
ceg + 1
d2g — 1
def +2
e3 -1
62/2 + 1
bcdg + 2
foe/ - 2
fo?2/ - 2
fofe2 + 2
c3</ — 1
c2df + 2
cV + 1
crf2e - 3
a"4 +1
±3
+ 9
a;4
+ 142
Or?/2
a;?/3
a?eg
«!fr * 2
a2/ + 1
abff2 + 2
a eg2
S* + i
a fo# — 10
abfff - 6
'/</ - 10
dfg
a bdg
ft/2 - 8
cey — 6
C?6gf + 4
e*g
bef - 10
cdg + 4
c/2 + 6
^ + 16
*r-
c2g
ce/ + 26
d*ff + 4
«2/ 12
a°&y + i
cdf + 4
#/ - 8
rfe/ + 12
a"62/^ - 8
fo/y - 10
ce2 + 16
rfe2 - 8
<? - 12
bceg + 26
bdeg + 4
rf2e - 12
a°#% + 16
a?bzey + 6
bcf* + 24
&<^2+ 16
a°b2cg
b2ef + 24
ft2/2
6rf^ - 8
be2/ - 12
b2df + 16
bc2g - 12
6crfy+ 12
bdef - 64
c2^ + 16
62e2 + 9
forff- 64
beef + 18
6e3 + 36
c2/2 + 9
be2/ - 12
foe2 - 42
6cZ2/- 96
c?dg - 8
cd*g - 12
bcde - 76
foPe + 56
bde2 + 60
c2e/ - 42
ccfe/- 76
fo*3 + 48
c3/ + 36
c3^ - 12
cd2f + 56
ce3 + 48
c3e + 48
c2c?e + 4
<?df + 60
cde- + 4
rf-/ + 48
c2rf2 - 32
erf3 - 16
cV - 99
tZ3e - 16
rfV - 32
c^2e + 84
d4 - 32
+ 168
263
+ 168
142
x-y
K=(
xy5
a% + 1
a2egr + 2
a%
ay
a6/
a eg2 2
a d<f - 1
«/ - 1
/» _ 2
a beg + 10
a 6/jy
tfy - 10
rf/flr + 2
«/</ + 3
a fo<7 — 3
a bdg — 2
ft/2 - 10
ce/y
C?6(/ + 15
9 /»
tfy 4- O
/3 2
ft^/* — 2
fo/ + 2
cdg — 15
c/2 - 20
df + 10
e/2 - 6
a°bcg2 + 1
fo2 + 5
CV ~ ^
ce/ - 5
d\j
e0/ - 15
a°6y + 2
fo%+ 2
c2/ + 9
cc// + 28
c/2/ + 60
def + 60
aWfy + 10
6c/^ — 2
6e2(/ — 9
cde — 17
ce2 — 26
de2 - 40
e3 - 40
bceg + 5
fo?e</ — 28
fo/2+ 6
d3 + 8
d2e + 4
a°b2dg — 10
a°b2eg + 20
6c/2 - 30
fo?/'2 + 32
<?fff - 5
a°bsg + 2
a°b2cg + 6
62e/ + 30
ft2/2 ...
bd2g — 60
fo2/ - 6
cdeg+ 17
62c/- 6
b2df- 32
bc2g + 15
ftca7*/- 60
6c?e/ +110
c2eg + 26
cdf'2— 2
b2de+ 2
6V + 36
foe//'- 110
foe/
fo3 - 45
c2/2 - 36
ce2/ - 6
fo2e + 6
fo2/+ 6
foe2 + 15
bd2f
c2<&/ + 40
cd2g - 4
c% - 8
bed2- 4
bcde- 58
bd-e + 40
bde-+ 20
c2e/ - 15
crfe/ + 58
rf2e/+ 4
C3d
bd'A + 32
c3/ + 45
c3y + 40
cd2f - 40
ce3 30
rfe3 ...
c3e + 30
c2a*e — 25
c2a7- 20
cde2 + 25
dsf - 32
c2d2 - 20
cd3
cV
d3e
d-e? + 20
+ 33
+ 146
+ 215
+ 140
+ 215
+ 146
+ 33
382
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
[775
I I
*5 V v
jp i-o i-c
a
O 1C
CO 1-H
COCOCMOOOSCMCOfMOS-^-H-lC
— 4 r-H i— i •*!< CM 00 •**
Si "v. f
* £ >
i i-O hO
%
CM CO
l-H
I I
CM CM CO CM -^
I-H ^H TO CO
CM I-H CO CO CO CO
+1 I .+ I +
6 ^S
(M
I
(O
»o
J
-i
O CM (M
O
I
•^ CM •»*
CO •* 00
S
SC >-0
"b e
•"
o
L-
CM
I
I
•t
-O
•fe
CM CM CO
co
r—
I
s
CO
I
J>
eo
I
•^ OO CM
CO I-H ,— i
T*< -^ O O
CO CO CO ^*
"» j» 5s
%
00
I
CO OS CM OS
I + + I
00
I
V
co
I
o
CO
•I
X £
jC «3
II I
<o
I
*e
775]
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
383
M=(
ar2
xy
aV + 1
aW ~
2 «2ey2 + 1
dfg - 6
efg +
4/ U
8 /V - 1
e2y + 8
/' -
6 a bdy1 — 6
ef2 — 3
a 6c^2 +
8 befy + 6
«6y - i
Wy -
20 bf3
bcfy + 6
be\j -
24 cV + 8
bdeg- 34
bef* +
36 cdfy - 34
bdf* + 48
<?fy -
24 ce2^r + 18
6e2/ - 18 cdey +
76 ce/2
<?eg + 18 cdf* +
36
d2eg + 4
c2/2 - 45 ce2/ -
72
c/2/2 +64
cd2y +4 d*y —
32
de2f — 96
7 /• , /T Q 7 0 J?
8
e4 + 36
ce3 - 36
d<? +
24
a°b2C(f - 3
d3/ - 48
a°6V -
6 b*df(/+ 48
(Z2e2 + 28
62c/7 +
36 62eV- 45
a°bsfy
tfdeg+
36 62e/2 •••
Wcey
tfdf*
bc2fy — 18
62c/2
tfey-
54 bcdey+ 78
62d2</ + 64
b<?eg-
72 bcdf2- 144
62de/- 144
6c2/2-
54 6c«y+ 108
6V + 81
bcd^y —
8 bd3g - 48
bc2dy- 96
bcdef-
36
6^2e/+ 9G
6c2e/+ 108
bce* +
216
bde3 - 72
bcd2f+ 96
bdsf +
128
c3^^ — 36
r\/*f1 0" 1 v f\
C/vt-t-O i- -J w
bd*#-
192
c:/2 + 81
770 T /»
bdAe + lo
c3dg +
24
c2t/2y+ 28
cV + 36
c3e/ +
216 c-def- 126
csdf - 72
cWf-
192
cV - 27
c3e2 - 27
c*de2 -
378
cJ:/ + 16
c2c£2e + 96
cd3e +
464
ccfV+ 96
cd* - 32
d* -
128
d4e - 32
+ 9 ±
8 + 1
182
180 + 136
497
1120 + 551
± 688 ±
1308 ± 688
384
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
Xlf
ov
aV - 1
cPdg2'
aV + 1
*w ...
tfy - i
dfg + 4
efy + 3
/V ~ 1
a 6ei72 + 1
deg + 1
*g
/3 — 3 a 6rf<,2 — 4
&/V 1
rf/2 + 3
/»2 Q
a 6c<?2 — 3
6e/^ + 4
crf^2 1
e2/ - 3
afiy + 1
bdfg
6/3
ce/J/ - 2
a b*fg + 1
bcfg — 4
be-g - 15 cy
c/3 + 3
bcey + 2
6rfe</ — 16
fie/2 + 18 crf/^ + 16
rf2/, + 4
6c/2 - 3
6rf/2 ...
c2/, + 15
ceV - 22
rfeV - 1
6rf2^ - 4
7 9.^* 1 O
oej + lo
f*fl p/i
t/tct/U • • •
ee/2 + 6
rf./2 - 6
fide/ - 12
c% +22
erf/2 -36
rf2e_i7 + 8
g.y + 3
6e3 + 15
c2/2 + 3
ce2/ + 9
rf2/2 -32
a«6%2 - 3
c*dy + 1
crf'2<7 • 8
d*ff
rfe2/ + 36
b^efg + 3
cV + 9
cde/ -48
d-ef + 24
e4 - 12
62/3
erf'/ + 4
ce3 + 12
rfe3 - 12
a°6%2 + 3
6c2^2 + 3
crfe2 - 21
rf:/ + 32
a°6V2 + 3
Vdfff -
6crf/# + 12
rf3e + 8
rf2e2 — 12
Vtfy -is
*
fice2^ — 9
«°6V/ - 3
a°63/y
b deg +36
62e/2 ...
beef2 — 9
1>T
62ce^ — 6
62rf/2 ...
bc-fg — 18
bd2eg — 4
Wcdg + 6
6V2 ...
62e2/ - 27
6crfe# + 48
6rf2/2 - 32
62ce/ + 9
62rf2<7 +32
bc*eg - 9 6crf/2 ...
6rfe2/ + 66
6'V2/ + 32
62rf</ ...
6c2/2 + 27 6ce2/ - 18
6e4 - 27
62rfe2 - 39 6V - 27
ficrf2^ — 24
6rf3r/ - 32 c3// - 15
6c3</ — 3 6c2rf<7 — 36
bcdef
7 -J2 af _i_ QO
c2rfea +21
6c2rf/ - 66
be"ef +18
bee3 + 27
6rfe3 - 12
c2rf/2 + 39
ficV +18
6crf/ - 32
bd*f ...
c3eg - 12
cV/ - 18
6crf2e +76
6crfe2 +84
6rfV - 12
c3/2 +27
cd*g - 8
6rf4 - 32
6rf:!e — 32
<?dg +12
c2rfV +12
cd-ef - 76
c4/ + 27
cV + 12
cV -27
c2rfe/ — 84
crfe3 + 45
c-^rfe - 45
c?df +12
(Affif +12
c2e:i + 45
rf4/ + 32
c2rf3 + 20
c3e2 - 45
c2rfe2
crf:/ + 32
rf3e2 - 20
c^d'e + 20
crf3e
crfV - 20
erf4 ... rf3
rf4e
775]
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
385
xs
x7y
a?4?/4
xy1
1
«y ...
<,% ...
«v - i
»w ...
a2eg2 + 1
aW
ay
abg3
v
a*bfg ...
cfg - 3
o£/# + 4
efff + 5
/V i
a beg2 + 3
aW .-
c/9* .-
1g ...
ceg - 1
deg + 10
e3<7 + 7
/3 - 5
a bdg2 — 4
bf2g - 3
cegr2 + 1
deg2 ...
f*
c/2 - 2
df2 + 2
e/2 - 10
a 6c^2 — 5
befg + 18
cdg2 — 10
c/V - 1
^"V -
if — I
do ~h o
e2/ 9
a&y + 1
da/flr ...
6/3 • 14
cefg + 22
/72/-/2 ^
Cc- U • - O
e2/^ ...
y + 3
efe/ + 6
ab2fg + 3
bcfg - 18
be*g +10
2,j2 7
ef3 - 12
de/g' +14
ef3 ...
e2 - 2
e3 - 6
6ce^ - 22
6c?e^ — 16
6e/2 - 5
coj/^ + 16
cPfg + 19
df3 - 8
a°b2fg2
dg
a 62e(/ + 1
bcf2 - 2
bdf*+ 28
c2fg -10
ce?g - 1
de2g - 24
e\j - 9
beeg2+ 1
''ef + 1
62/2 + 2
bd2g- 19
^e2/ + 4
cde^
ce/2 - 1
rfe/2 - 4
e2/2 + 6
&c/V- 1
ty ...
6cc&/ — 14
bdef + 34
cV + 1
c^/*2 + 80
£^2e^ - 13
e3/ + 9
a»6V+ 2
bd2g2— 3
•df - 14
beef ...
6e3 + 3
c2/2 + 24
ce2/ -55
df2 + 38
a°b2dg2- 2
62/V- 2
bdefg+ 14
:e2 +11
bd2f - 18
<?dg + 24
cc?2<7 + 13
dV •-•
(l($f Q 2
b2efg + 2
bcdg2— 6
6o?/'3 — 8
I2e + 1
bde2 +26
c-2e/ + 4
cdef - 6
d2ef — 65
e4 + 30
bfs
&ce# ...
6e3y - 9
f + 9
cV + 9
cd2/- 58
ce3 - 37
de3 + 50
n!9 9 T /^
(TO eg + 10
bc2g2 + 9
T /»Q , /»
6c/ + b
6e2/2+ 6
de -14
c2df + 10
cde2 - 42
d3f - 52
a°&3<72 + 5
b2dfg— 28
bcdfg— 34
6^+ 18
c*dg2+ 2
I3 + 6
cV +13
d3e + 38
dV + 58
62c/^+ 5
6V^r — 24
bce2g — 4
bde2g— 10
c2efg-ll
eg
/•/72*» ^
Get 6 — OO
a°b3eg + 12
a?b3fg + 14
62c%- 80
52^-2 + 42
6ce/2+ 57
bdef2 -20
c2/3 + 9
df + 8
d4 + 24
&;/2
b2ceg + 1
62^/2 ...
^>c!/5r ~~ ^
bd*eg+ 58
besf +12
cd?fg- 1
s2 — 9
aWcfcr + 8
62c^+ 4
62c/2 _ 42
6V/ + 60
bcdeg •*• 6
bd2f2+ 16
«y + 6
cde2g+ 14
c2/- 6
b3ef - 6
5V/- 57
&2d2#- 38
&c2i?<7 + 55
6co/2
6efey- no
c2^- 26
cde/2- 16
cde + 16
b2c2g — 6
b2dy— 16
68<fe/ ...
bc2f2 — 60
See2/ — 18
6e4 + 45
22 T O
C & O — I O
ce3/ + 3
i3 - 8
&2cd/"+ 20
62cfe2 + 30
62e3 + 36
bcd2g+ 55
6o?3^ + 52
c3fg - 3
c2e/2 + 21
d3eg - 6
3e - 3
62ce2 -21
oc Cf 9
bc2dg+ 62
&ct&/ . . .
bd2ef— 66
c2deg+ 42
cd2eg+ 53
d3/2 + 8
"$ + 2
b2d2e- 2
bc2df+ 110
6cV+ 18
ftce8 -60
6*3 + 30
c2(//"2_ 30
cd2f2+ 2
d?e2f- 2
i ...
fee3/ - 12
6cV + 12
6c^'/+ 66
bd3f ...
c3e# + 37
cV/- 12
cde-f- 52
de* ...
bc2de +52
&cd2e- 87
6cdea-126
WV+60
c3/2 - 66
cd3^ — 38
ce4 + 15
6cd3 - 28
bd* + 16
6^3e + 24
c% -50
c2d2g— 58
cd2e/+ 87
d*g -24
c4e - 15
c4/ + 45
c4^ - 30
c3e/ + 60
c2de/+ 126
cde3 - 40
d3ef + 28
c3o?2 +10
c3rfe + 40
c3df - 30
c2d2f- 15
cV - 60
dy -- 16
d2e3 - 10
C2C^3 _ ^Q
c3e2 + 60
cW ...
cd*f - 24
d3e2 + 10
O 7O -I e;
G Ct r 1 *")
c^3e ...
cdV+ 15
cd4
d5
^ -
C. XL
49
386
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
[771
d g + l
defg - 6
df3 + 4
e3^ + 4
e2/2 - 3
a 52e<?2
a b cdg2 — 6
cefg +18
c/3 - 12
d2fg + 12
d&g - 18
e3/ + 6
6°cy + 4
c^dfg — 18
eVty - 24
c2e/2 + 30
cd2eg +54
erf2/2 - 12
crf2</ — 42
ce4 + 12
d'g - 20
dsef + 24
rf2e3 — 8
a^df
+ 4
e/9
- 12
f3
+ 8
b*cy
- 3
**9
+ 30
cef*
-24
d2eg
- 12
dT
- 24
dff
+ 60
e4
-27
bc3fg
+ 6
(?deg
- 42
C2rf/2
+ 60
cV/
- 30
cd3g
+ 24
erf2*/
- 84
crfe3
+ 66
vf
+ 24
rfV
-24
bWeg
+ 12
c4/2
-27
<?dzg
- 8
<?def
+ 66
cse3
- 8
C2rf3/
- 24
cW
-39
cd4e
+ 36
f
- 8
775]
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
387
51
55
55
58
55
55
51
10
20
30
#Hf ...
a3b°eg2
«w ...
«w ...
a26^3
aW ...
aW,3 ...
efg
«/ *7
f*g
a26 e^2
«2&//
b'cfg* ...
d/^2
*// ...
/3
Cub dg
/v ...
6°ce/ ...
c?egr2 + 5
e2gr2 + 2
/v ...
a?b eg2
ffg . . .
6°cd/ - 5
cf*ff -
dfff- 5
e/V - 4
a b eg3
dfg ...
f3
cefg +15
dy ...
e2/^ - 5
> + 2
dfg2 ...
e2g
a2b°c2g2 - 2
c/3 -10
defg + 20
e/3 + 5
a 6V3
ey + 2
e/2
cdfg + 6
d2fg + 20
c^/"3 - 20
«w
6c//2
e/V - 4
b°c2fg -• 2
2 , o
defy - 25
esg - 20
6ce/ - 15
deg'2 — 6
/4 + 2
cdeg + 5
ce/2 - 10
de/2 - 10
e/2 + 20
c/V + 15
d/~g + 6
b^/g2 ...
cd/2 + 6
TO 1 /\
« 6<7 — 1U
e3/ + 15
a b2eg2
dy - 20
e% + 6
cdeg2— 5
ce2/ - 7
d2/2 + 18
a b2dg2 + 5
f2g •••
de/^ + 90
e/3 - 6
cd/V+ 5
«fy - 3
de2/ - 22
efg - 15
b cdg2 - 20
7 /»o F\ A
ct/ — ou
ce2/^+ 5
TO /• O
a e/ — o
e4 + 12
/3 +10
cefg ...
e>g - 35
ce/3 - 5
de3 + 4
a b2cg2 + 4
&cy + 5
c/3 + 20
e/2 - 15
cdy+ 10
dy + 3
a&y
dfg — 6
cd/<? - 90
d*fg ...
bWdg2 + 25
cdefg+ 4
dV?- 7
b2cfg + 4
*g - 8
ceV + 40
c?e2gr + 40
c2e/^ — 40
cd/s - 24
W2Ai O
tty — ^
dej7 — 5
e/2 + 10
ce/2 + 40
de/2 - 20
c2/3 + 15
ce3^ — 8
deV + 1
d/"2 — 6
a 6 c2fg — 6
d% + 50
e3/ - 20
cd%- 50
ce2/2+ 18
de/2 + 8
e2/ + 7
cde<7 — 4
d2/2 + 10
0 C O "f* Ij\J
cde2g+ 45
dsfg - 22
4 /» o
6y — O
6 c2e<7 — 5
cd/"2 — 68
de2/ - 40
c2dfg— 40
cc/e/2+ 5
d2e2g + 14
a«>6y
c2/2 - 6
ce2/ + 76
e4 - 15
c2eV ...
ce3/* ...
d e/ + 42
b2c/y2
cdV + 7
d*g + 22
b°csfg +35
c2e/2 - 20
ct 6^ — 0
de3/ - 46
deg2 — 6
cde/ - 16
d2ef + 38
c2deg— 45
c^% ...
d/2 + 50
e5 + 12
d/V+ 6
ce3 + 23
de3 - 58
C2C//2 ...
cd2f2+ 60
d2e/— 65
«w
e2/g + 6
d3f + 30
b°c3eg + 8
cV/- 65
cde2/- 20
de4 + 20
6 a o 1 /\
ce^ + 10
e/3 • 6
dV - 33
c3/2 + 42
CC?3^r + 5
ce4 + 20
«°&V +10
c/2# - 10
7. 2 2 . T
o c eg + /
b°c3dg - 1
c2d2g — 14
cd2ef+ 65
dV ...
f-g -10
dy — is
c2/V- 7
<?ef + 36
c2*/- 82
cc^e3 -45
d3e/ — 20
b2cdg2 +10
defg + 68
cdy+ 3
c'd2/- 37
c2e3 - • 44
d*f - 20
d2e3 ...
ce/gr - 40
d/3 - 32
cdefg+ 16
cW - 53
cdsf + 12
c^e2 — 20
ft b dg + ^iO
c/3 + 30
e3^ 42
cd/3 - 22
49—2
388
TABLES OF COVAEIANTS OF THE BINARY SEXTIC.
[775
51
55
Q = ( *$x, ?/)6 (continued").
55 58 55
55
ab°cd3e + 79
ab°cd*<?+ 122
a°b3cg2 — 5
a°b3efg - 20
cPWd^fg — 10
a°b2e2f2 + 24
a°b ce3g
d5 24
d4e - 44
dfg + 50
/3
de2^
6 c2d<72 + 22
ce2/2
a'Ufg 2
a'fty - 2
e2^ — 15
62cy — 20
de/2 + 10
o _/» w />
c^e^^ _ 7 b
d3fff
63ce<7 + 5
&V0 + 6
e/2 - 30
cd/g' + 20
e3/ -.
c2/3 + 54
d*e2g
c/2 + 6
deg + 24
b*c?fff — 15
ceV + 20
b c3/ - 15
ctffg - 38
d2e/2
d^ + 2
d/2 + 32
cdeg — 5
ce/2 ...
c2d/y+ 40
cefe2gr + 82
de3/ -
de/ + 22
e2/ 54
cd/2 - 10
G?2e<yT — 60
c2e2^ + 65
cde/2 - 50
e5
e3 27
b^c2eg — 18
ce2/ + 60
d2/2 ...
c2e/2 + 60
ce3/ + 6
6°c%2 -
6W0r - 8
c2/2 - 24
d3£ -50
de2/ + 40
cdzeg- 65
d3e<7 — 12
<?*fg •
c2e/ - 39
cdV - 42
d2e/ - 10
e4
cdy2+ 10
d3/2 +64
cT -
cd2/ - 50
cde/ + 50
de3 + 30
&c% +20
cdeif+ 10
d2ey - 82
cWfg-
cde2 + 107
ce3 + 54
b c3eg
c2deg+ 20
ce4
de4 + 30
C2de2g-
d3e - 22
d3/ -• 64
c3/2 ...
c2d/2— 40
d'g + 20
6°cy 12
c2de/2-
b c4g + 3
d2e2 + 32
<?<Pg +65
cV/ ...
dV - 10
c3d/# + 58
cV/ -
c3d/ + 84
b csdg + 46
c2de/- 10
cd3^ + 20
d2e3 + 5
c3e2^ + 44
cd3e# -
c3e2 - 21
c3e/ - 6
c2e3 - 30
cd2e/ ...
b°c*fg +15
c3e/2 - 54
cd3/2 H
C2^2g_ 1Q2
c d j + 82
cd3f + 10
cde3 - 40
c3deg+ 45
ctd2eg- 122
cdV/H
cd4 + 44
c2de2- 112
cd2e2 — 75
d4/ ...
c3d/2- 30
c2dy2- 32
cde4 -
b°c5f 27
cdse - 34
d*e + 40
d3e2 +20
c3e2/ + 30
c2dey+ 112
dV H
c4de + 45
d5 + 32
b°c4dg — 20
b°c*eg - 20
<?d*g+ 20
cV 30
d4e/ -
c3d3 - 20
6°cV 12
c4e/ ...
c4/2 ...
c'd2^- 75
cd4^ + 44
dV -(
c4df - 30
c3^2/- 5
cUV •••
c2de3 — 50
TO /» »l ^
cd e/ + o4
c4e2 + 30
c3de2 + 50
csdef+ 40
cd4/ - 40
cd2e3 — 30
c3d2e + 30
c2d3e-25
cV
cd3e2 + 25
d5/ - 32
c2d4 - 20
cd5
c2d3f— 20
d5e
d4e2 + 20
d6
40
50
60
776] 389
776.
ON THE JACOBIAN SEXTIC EQUATION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvni. (1882),
pp. 52—65.]
THE Jacobian sextic equation has been discussed under the form
(z - a)6 -4a(z- a)5 + 106 (z - a)3 -4>c(z-d) + 562 - 4>ac = 0,
(see references at end of paper), but the connexion of this form with the general
sextic equation has not, so far as I am aware, been considered. And although this
is probably known, I do not find it to have been explicitly stated that the group
of the equation is the positive half-group, or group of the 60 positive substitutions
out of the 120 substitutions, which leave unaltered Serret's 6- valued function of six
letters.
Invariantive Property of the Jacobian Sextic.
Taking z — a as the variable, and comparing the equation with the general sextic
equation
(a, b, c, d, e, f, g$> - a, 1)6 = 0,
we have
a, b, c, d, e, f , g
= 1, - fa, 0, £6, 0, - f c, 562 - 4ac ;
the Jacobian equation is thus an equation
(a, b, c, d, e, f, g$«, y)« = 0,
for which c = 0, e = 0, ag + 9bf - 20d2 = 0 ; but of course any equation, which can be by
a linear transformation upon the variables brought into this form, may be regarded
as a Jacobian equation.
Hence, using henceforward the small italic in place of the small roman letters,
the Jacobian sextic may be regarded as an equation
(a, b, c, d, e, f, g$x, y)« = 0,
linearly transformable into the form
(a, b, 0, d, 0, /, g\x, yj = 0,
390
ON THE JACOBIAN SEXTIC EQUATION.
[776
where ag + 9bf— 2Qd2 = 0. It is to be shown, that this implies a single relation between
the four invariants A, B, C, and A of the sextic function.
I call to mind that the general sextic has five invariants A, B, C, D, E of the
orders 2, 4, 6, 10, 15 respectively; the last of them E is not independent, but its
square is equal to a rational and integral function of A, B, C, D; and instead of D,
we consider the discriminant A which is an invariant of the same order 10. The
values of A, B, C are given, Table Nos. 31, 34, and 35 of my Third Memoir on
Quantics, Phil Trans., vol. CXLVI. (1856), pp. 627—647, [144]; those of D, A, E were
obtained by Dr Salmon, see his Higher Algebra, second ed. 1866, where the values of
A, B, C, D, A, E are all given ; only those of A, B, C, A are reproduced in the third
edition, 1876.
It may be remarked, that for the general form we have A = ag — Qbf+ loce — 1(M2,
and that B is the determinant
a,
b,
c,
d,
b,
c,
d,
e,
c,
d,
e,
f,
d
f
C and A are complicated forms, the latter of them containing 246 terms. But writing
c = 0, e = 0, there is a great reduction ; we have
A =
*-
c=
^> =
ag + 1
ad^g — I
a^d^g2 + 1
ay + i
6/6
b2/2 + 1
„ df3 + 4
a*bfg* - 30
d2 - 10
bd2f-2
abd2fg + 12
„ dy 300
d* + I
„ d4g - 20
„ df^g 2500
aWdg"2 + 4
,,/6 3125
7.3/3 , Q
,, Oy + o
a"A^r */?•* 1 ^\
*/ c/ -^ ^
„ i2^/2- 24
„ 6(^2// 4800
„ 6e#y + 24
„ W/V - 7500
,,d* - 8
„ c?y + 30000
ii d3f3g + 50000
azb3dg4 — 2500
,,byy 410
„ b*dyy — 171 300
„ b\lf5 240000
„ bd4fg2 + 780000
„ W3/4 + 1200000
„ dy - 1000000
„ d*f* - 1600000
a frdfg3 - 7500
,,b'f4g 11520
„ b3dy + 50000
,,b3d~f3g + 83200
a°6y 3125
„ tfdfy - 240000
„ bsf5 331776
,,64d%2 + 1200000
„ 64rfy4 + 1843200
„ b3dy - 1600000
„ bWf3 - 2560000
776]
OX THK JArOlUAN SKXTH' KtjT ATloN.
It is clear that these are all functions of ag, If, d* and a?/9 + fry, say of a, /8, 8 and </>.
In fact, A and jB are functions of a, @, S; 0 contains two terms, coefficient 4, which
are = 4 V& . <f> ; A contains two terms
- 3125 (a*f* + by),
which are =- 3125 (<£a- 2ct3£3); and also several pairs of terms, each which pair contains
the factor <f>. We thus have
A =
B-
(7 =
A =
a + 1
aS - 1
a'S + 1
a8 + 1
ft - 6
ft2 + 1
aftS + 12
a*ft 30
8-10
ftS - 2
aS2 - 20
a'S 300
S3 + 1
ft3 +8
a2^3 + 5840 (=6250- 410)
ft?8 - 24
a3/33 15
ftS2 + 24
a3/?8 4800
83 - 8
a382 + 3000
<£ v/8 + 4
a-ft°~8 171300
a2/?82 + 780000
a283 - 1000000
aft4 11520
a/33S + 83200
ft5 331776
ft48 + 1843200
/8382 - 2560000
$ ^/S . a3 - 2500
„ aft - 7500
„ a8 + 50000
„ jS2 - 240000
„ y8S + 1200000
„ S2 - 1600000
<^>2. 3125
We have ante, the relation a + 9/3 — 20S = 0, and using this to eliminate a, we have
A, B, C, A as functions of & S, £ (that is, of &/, d? and a2/1 + 6y). Effecting the
substitution, we find the values of A, B, C without difficulty. As regards the value of
A, this is
= - 3125$2 + %<f>K VS + terms without <j>,
392
where
ON THE JACOBIAN SEXTIC EQUATION.
[776
2K = - 2500 ( 81/32 - 360/38 + 40082)
7500 (- 9/32 + 20/38 )
+ 50000 ( - 9/38 + 2082)
- 240000 ( /32 )
+ 1200000 ( /38 )
- 1600000 ( 82),
or, reducing and dividing by 2,
K = - 3125 (60/32 - 240/38 + 25682).
The calculation of the terms without <f> is much more laborious, but they come out
= - 3125 (60/32 - 240/38 + 25682)2 8.
Hence the value of A is
A = - 3125 { </>2
+ 20 (60/32 - 240/38 + 25 682) V8
+ (60/32 - 240/38 + 25682)2 8},
A = - 3125/12,
h = <f> + (60/32 - 240/38 + 256S2) V8,
= a2/3 + 63#2 + (6062d/2 - 2406d3/-
The values of A, B, C, and the foregoing value of h then are
say this is
where
that is,
A =
B =
C =
4-
(3-15
yS2 + 1
/33 + 8
/32 VS + 60
8+10
138+7
/328+ 51
/3S Vs - 24°
S2 - 19
/332 + 84
82 V3 + 256
83 - 8
0 + 1
<£,ys + 4
We may, if we please, regard /3, 8, (f> as irrational invariants of the sextic, viz. A, B, C
being rational and integral functions of /3, 8, 0, we have conversely /3, 8, <f> irrational
functions of A, B, C ; and then the equation for h, say
~j. V(- A) = </> + V8 (60/32 - 240/38 + 25682)
is the invariantive relation which characterises the Jacobian sextic.
776]
ON THE JACOBIAN SEXTIC EQUATION.
393
The expression for A in terms of A, B, C, D is
A = A5 - M5A3B - 62oA"-C + 3125/),
and it was in the foregoing investigation proper to use A in place of D. But I annex
the value of D for the case in question 6 = 0, /= 0 ; and also its value in terms of
a, @, 8, (f>. These are
z>-
D =
a4/6
1
fh" — J.
a'bdf'ff
12
a2/33 + 2
„ dy
+ 5
ay84 12
aW2/2<
^2- 90
ft5 72
„ b*df5
48
8 a2/32 - 90
» bdW
+ 246
,,a/33 + 168
„ bd3/*
+ 480
„ /34 + 552
» dsg2
-258
82a3 + 5
» d5f3
-432
„ a2^ + 246
a b4dfg3
12
„ a^2 + 240
» by*g
12
,,y33 -976
„ b3dyst
1 + 168
83a2 — 258
„ bWf\
/ + 240
,,a/3 -168
„ bd6fg
- 168
„ ft2 + 336
,,dsg
-228
84a - 228
a°bsg*
1
» /? - 408
» bsdf*g
! - 48
85 - 240
„ b5/5
• 72
<£ ^/S a^g _ 12
» Vd'fff
+ 480
/3'2- 48
,, 64c/y3
+ 552
<^>8 ^8 (3 - 480
,, b3d5(j?
-432
i J2 / £ A Q O
0O V^ — TO-i
» bsd*f3
-976
„ 62c?6/2
+ 336
» ^8/
+ 408
,,^°
- 248
The Group of the Jacobian Sextic.
The solution of the Jacobian sextic equation depends upon that of a quintic ; in
fact, calling the roots zx, z0, zly z%, zs, z4, then there exists a quintic having the roots
VO*. -Zo.Zz-Zs-Zi- ZT),
V i^oo &i • &3 2*4' ZQ Z»)i
c. xi.
50
394 ON THE JACOBIAN SEXTIC EQUATION. [776
the coefficients of which are rational functions of the coefficients a, b, d, f, g, and of
the fourth root of the discriminant, i.e., \/h. But the meaning of this has not, so far
as I am aware, been noticed. Passing to the quintic whose roots are the squares of
the foregoing values, i.e., z!K—z0.z2 — z3.z4—z1, &c., the coefficients are here rational
functions of a, b, d, f, g and h ; that is, they are rational functions of a, b, d, f, g. The
symmetrical functions of these roots zm — ZQ . £2 — 2s • z± — z\> &c., are thus rational func
tions of the coefficients of the sextic ; each such rational function is a 12-valued function
of z^, z0, zl} 22, z3, zt) invariable by all the substitutions of a group of 60 substitutions;
and therefore also every like 12-valued function of the roots z^, z0, z1} z2, z3) z± is
invariable by the substitutions of this group of 60 ; or, in other words, this group of
60 is the group of the Jacobian sextic equation.
I write for convenience, in this section only,
£„, *o. *i, za> *3, z*=f, a, b, c, d, e;
and writing further ab for shortness instead of a — b, &c., (so that of course ba = — ab\
and putting B, C, D, E, F = — ab . cd. ef, — ac . bf. de, ad. be. ef, ae.bd. cf, of. be . cd, then
the five functions are B, C, D, E, F, and the group of 60 which leaves unaltered every
symmetrical function of these functions is made up of the substitutions
1
15
1.
ab . ce ,
ab . df,
ce . df,
ac . bf,
ac . de,
bf . de,
ad . be ,
ad . ef,
be . ef,
ae . bd,
ae . cf,
bd . cf,
af . be,
af . cd,
be . cd.
abode,
acebd,
adbec,
aedbc,
afbce,
abefc ,
acfeb ,
aecbf,
abdef,
adfbe,
aebfd,
afedb,
afced,
acdfe,
aefdc,
adecf,
afdbc,
adcfb,
abfcd,
acbdf,
bdcef,
bcfde ,
bedfc,
bfecd.
abc . dfe,
acb . def,
abd. cfe,
adb . cef,
abe . cfd,
aeb . cdf,
abf . ced,
afb. cde,
acd . bef,
adc . bfe,
ace . bfd,
aec . bdf,
acf . bed,
afc . bde,
ade . bfc,
aed . bcf,
adf. bee,
afd. bee,
aef . bed,
afe . bdc,
24
20
60
776]
ON THE JACOBIAN SEXTIC EQUATION.
395
where the symbols, ab, abode, abc, &c. denote cyclical substitutions. It is easy to verify
that each of these substitutions does in fact merely permute B, C, D, E, F; thus
B
C
D
abcde on — ab . ce . df, — ac . bf . de, ad .be . ef,
= — be . da . ef, — bd . cf . ea, be . cd . of,
E
ae . bd . cf,
ba . ce . df,
F
of .be . cd
bf . ca. de
= + ad . be . ef,
D
ae . bd . cf, af . be . cd, — ab . ce , df, — ac . bf . de
E
B
C,
which (expressed as a cyclical substitution) is = BDFCE, and so in other cases.
We may to the foregoing 60 substitutions join the 60 other substitutions:
30
cdef,
cfed,
bdfe,
befd,
beef,
bfce ,
bcdf,
bfdc,
bced,
bdec ,
aedf,
afde,
acef,
afec ,
acfd,
adfc,
adce,
aecd,
abfe,
aefb,
adbf,
afbd,
abed,
adeb,
abcf,
afcb,
acbe ,
aebc ,
abdc,
acdb.
ab . cd . ef,
ab . cf . de,
ac .bd . ef,
ac .be . df,
ad .be . cf,
ad. bf . ce,
ae .be . df,
ae . bf . cd,
af .be . de,
af . bd . ce.
abcefd,
adfecb,
abfdec,
acedfb,
abecdf,
afdceb,
abdfce ,
aecfdb,
acfbde,
aedbfc,
acbfed,
adefbc,
acdebf,
afbedc,
adbcfe,
aefcbd,
adcbef,
afebcd,
aebdcf,
afcdbe,
10
20
60
each of which changes B, C, D, E, F into a permutation of — B, — C, — D, — E, — F.
50—2
396
OX THE JACOBIAN SEXTIC EQUATION.
[776
The 60 and 60 substitutions form together a group of 120 substitutions, which
leave unaltered any even symmetrical function of B, C, D, E, F, or say any symmetrical
function of .B2, C2, D2, E'2, F2; such a function is thus a 6-valued function of a, b, c, d, e,f,
viz. it is Serret's 6-valued function of 6 letters.
Transformation of the Jacobian Sextic into the Resolvent Sextic of a special
quintic equation.
Starting from the Jacobian Sextic Equation
(a, 6, 0, d, O,/ g^z, 1)6 = 0,
ag + 9bf— 20d2 = 0, I effect upon it the Tschirnhausen transformation
X = -az3-6bz2-Wd;
which, it may be remarked, is a particular case of the Tschirnhausen-Hermite form
X (az + b) B + (az2 + 6bz + oc)C + (azs + 6bz2 + I5cz + Wd) D
+ (az* + Qbz3 + I5cz2 + 20dz + lOe)E + (az5 + Qbz* + locz3 + 20dz2 + loez + of) F.
Writing for convenience Y=X + 10d, Z = X — Wd, this is
az3 + 6bz2- . +F=0,
and we thence have
az* + 6bz3 . + Yz . = 0,
az5 + 6bz* . + Yz2 . . = 0,
- Zz3 . - 6fz - g = 0,
- Z* . - 6fz2 - gz . = 0,
-Zz5 . -6fz3- gz2 . . =0,
or, eliminating, the resulting equation is
a, 66,
a, 66, . F, ',
a, 66, \. Y, . .
Z, ., 6/, g
z> • 6/, g, .
Z, . 6/, g, . .
The developed form is most easily obtained by expanding the determinant in the form
= 0.
123 . 456 - 456 . 123, &c.,
where the terms 123, &c., belong to the matrix
a, 66, . F
a, 66, . F,
a, 66,
F,
776]
ON THE JACOBIAN SEXTIC EQUATION.
397
and those of 123, &c., to the matrix
z,
z,
The several terms are
123
. 456
+ -a3 .-(;'
-124
. 356
- - 6a26 . - 6/#2
+ 125
. 346
+ 0 . - 36/2
-126
.345
--a2F . tf-Z
+ 134
.256
+ - 36a62 . 0
- 135
. 246
- a2F --g-Z
+ 136
. 245
+ - Qab Y . - Qfgi
+ 145
. 236
+ 6a6 F . - Gfgi
-146
. 235
- 0 . - 36/
+ 156
. 234
+ - a F2 . - gZ*
-234
. 156
a2F-21663. g-Z
+ 235
. 146
+ 6a6 F . Qfgi
-236
. 145
--3662F . S6/2
-245
. 136
- 3662F . 0
+ 246
. 135
+ aF2 . gfr
-256
. 134
66 F2 . 6/Z-
+ 345
. 126
+ -aF2 .-gZ°-
-346
. 125
- 66 F2 . - 6/Z2
+ 356
. 124
+ 0 .0
-456
. 123
- - F3 Z\
Hence, collecting and reducing, the equation is
+ YZ
+ F
+ Z .- 2166V2
(Sag + 726/)
(Say + 36a#6/+ 129662/)
- 36ay&/,
398
ON THE JACOBIAN SEXTIC EQUATION.
[776
where Y, Z denote X+lOd, X — lOd respectively, and consequently YZ = X2 —
Hence, writing as before a, /3, 8, </> to denote ag, bf, d2 and a2/3 + b3g- respectively, the
result finally is
1
0
a + 3
0
a2 + 3
a2/3 + 216
^8 + 2160
(3+72
a/8+ 36
63<72 — 216
a3 + 1
8 -300
a8 - 600
a2/3 36
/32 + 1296
a28 - 30
/2S- 14400
a(38 - 360
82 +30000
a82 + 30000
P*8 + 12960
/?82 + 720000
83 - 1000000
where observe that the coefficient of the term in X is 21 6 (a2/3 — fcfy2), = 216 V($2 — 4«2/33).
We have as before 0*7 + 96/— 20c?2 = 0, that is, a + 9/? — 208 = 0 ; and using this equation
to eliminate a, also in the constant term writing its value for 0 in terms of h,
(- 60/32 + 240/88 - 256S2)
the new equation is
-5 x
5x
5x
1
0
ft - 9
0
(F - 243
-216VA
A Vs + 432
8+48
/2S-1872
/83 729
82 +3840
^8 + 4184
/?S2 - 11520
83 + 8292
where
A = {A + (- 60/32 + 240/38 - 256S2) V§}2 - 4 (- 9/3 + 20S)2/33
-4S)3 (9/3 -168)-.
-4
/32 -60
/38 + 240
82 -256
It is to be shown that this Tschirnhausen-transformation of the Jacobian sextic is,
in fact, the resolvent sextic of the quintic equation
where
(a, 0, c, 0, e, f$sf 1)5 = 0,
a =
776]
ON THE JACOBIAN SEXTIC EQUATION.
399
I consider the general quintic (a, b, c, d, e, /$#, I)5 = 0 ; taking the roots to be
xlt x», #3, #4, x5, and writing
<k = 12345 - 24135,
</>3 = 14235 - 43125,
04 = 21435 - 13245,
-14325,
06 = 41325-12435,
where 12345 is used to denote the function
) V(20),
(this numerical factor V(20) being inserted for greater convenience), then the equation
whose roots are 01} 02, 03, 04, 05, 06, which equation may be regarded as the resolvent
sextic of the given quintic equation, is
a6 x
- 5«4 x
5a2x
-^D)-^
+ 5
1
0
ae
-2a2df
+ 1
+ Ia3cf3
-±bd
+ 3aV
- 2a?def
+ 3c2
&c.
+ &c.
D = a4/4 + &c., the discriminant of the quintic : see p. 274* of my paper " On a new
auxiliary equation in the theory of equations of the fifth order," Phil. Trans, t. CLI.
(1861), pp. 263—276, [268].
I now write 6 = 0, d = 0, but, to avoid confusion again, write roman instead of
italic letters, viz. I consider the resolvent sextic of the quintic equation
(a, 0, c, 0, e, f$>, I)5.
Many of the terms thus vanish, and the equation assumes the form
a6 x
- 5a4
5a2
-aVD
+ 5
1
0
ae + 1
a2e2 + 3
-1- 1
a3cf2 + 1
c2 +3
ac2e- 2
a3e3 + 1
c4 + 15
aW-11
ac4e + 35
c6 -25
and then if, as before,
or say
a = 1, c = 2d, e = - 96/+ 36d2, f2 = 216A,
a = l, c = 2\/8, e = - 9/3 +368, f2=216A,
* [This Collection, vol. iv., p. 321.]
400 ON THE JACOBIAN SEXTIC EQUATION. [776
this becomes identical with the foregoing Tschirnhausen-transformation equation ; thus
ae + 3c2 = - 9/8 + 368 + 128, = £ - 9
S + 48;
and similarly
3a2e2 - 2ac2e + 15c4 = fi- + 243,
£8 - 1872,
82 + 3840.
So for the constant term, + Ia3cf2 gives the term 432/i «/$, and + Ia3e3, &c., give the
remaining terms - 729/33, &c. of the value in question.
It only remains to verify the equality of the coefficients of X,
216VA = VD or 46656A=D.
Here D, the discriminant of the quintic (a, 0, c, 0, e, f$#, I)5, from the general
form (see my Second Memoir on Quantics, [141], or Salmon's Higher Algebra, third
edition, p. 209) putting therein b = 0, d = 0, is
D = a4f4 + 1,
a3ce2f2 + 160,
a3e5 + 256,
a2c3ef2 - 1440,
a*cse4 -2560,
ac5f2 + 3456,
ac4e3 +6400,
and writing for a, c, e, f their values 1, 2^8, 9 (-/3 + 4S), 216A, the value becomes
D = (216)2 . h* .
+ 432A VS . 12960 (/3 - 4S)2
+ 345 60 (/3- 48)8
+ 55296 82
- 256 . 95 . (J3- 48)5
- 10240 . 94 . (/3 - 4S)4 S
- 102400. 93. (y3-4S)3S3.
The whole divides by (21 6)2, and we thus obtain
A = A2 + 2AVS. 60 (-£ + 4S)2. + (-/3-4S)3. 324 (/3 - 4S)2
- 240 (- /3 + 48) 8 + 1440 (0 - 4S) 8
+ 256 82 + 1600 82,
which is, in fact, equal to the foregoing value of A.
776] ON THE JACOBIAN SEXTIC EQUATION. 401
The conclusion is that, starting from the Jacobian sextic
(a, b, 0, d, 0, /, ff%z, 1)6=0,
where ag + Qbf— 2Qd? = 0, and effecting upon it the Tschirnhausen-transformation
X = - az3 - 6bz* - lOd,
so as to obtain from it a sextic equation in X, this sextic equation in X is the resolvent
sextic of the quintic equation
(1, 0, c, 0, e, f$>, 1)5 = 0,
where
c = 2d, e = -9bf+ 36d2, f =
and, A being the discriminant of the Jacobian sextic, then
h = ^r= V(- A), = a2/3 + &0*
O d
As to the subject of the present paper, see in particular Brioschi, "Ueber die
Auflosung der Gleichungen vom funften Grade," Math. Annalen, t. xni. (1878), pp. 109—
160, and the third Appendix to his translation of my Elliptic Functions, Milan, 1880,
each containing references to the earlier papers.
C. XI. 51
402 [777
777.
A SOLVABLE CASE OF THE QUINTIC EQUATION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvin. (1882),
pp. 154—157.]
THE roots of the general quintic equation
(a, b, c, d, e, /$>, I)5 = 0
may be taken to be
--+ B+ C+ D+ E
ct
- „ + w4 „ + <w3 „ + &)2 „ + o> „
- „ + &)3 „ + 03 „ + ft)4 „ + O)2 „
- „ + o)2 „ + &)4 „ + o) „ + ca3 „
- „ + (o „ + w2 „ + &>3 „ + o>4 „ ,
where <o is an imaginary fifth root of unity; and if one of the four functions B,
C, D, E is = 0, say if E — 0 (this implies of course a single relation between the
coefficients), then the equation is solvable.
Writing x — % , we have
(a, 6, c, d, *,/)($-$ lY=(a', 0, c', d' , e', /'$£ I)5,
\ ct /
where
a' = a,
ac' =ac — b2,
a?d' = a-d - 3a6c + 263,
a3e' = ase - 4,a-bd + 6a62c - 364,
a4/' = a4/ - 5asbe + Watfd - Wab2c + 465,
and the roots of the new equation
(a, 0, c', d', e',f^, 1)5 = 0
777] A SOLVABLE CASE OF THE QUINTIC EQUATION. 403
have the
difficulty
have the above-mentioned values, omitting therefrom the terms ; we find without
Cv
2 -, = -BE-CD,
a
2-, = - R-D - BC2 -CE2- D2E,
a
-, = - B3C - B?E2 + BCDE + BD3 + C3E + C2D2 - DE-,
a
= -B5 + 5B3DE - 5B°-C2E - 5B2CD2 + oBC3D + 5BCE3
a
- 5BD2E2 -C5 + 5CD3E - 5CD2E2 - D5 - E5,
and hence, when E = 0, we have
2-! = - CD,
a
2 + 5BCSD
a
or, as these may be written,
-'
a
equations which imply a single relation between the coefficients of, c', d', e', /'.
Supposing this satisfied, we may attend only to the first three equations; or, writing
for convenience,
S=-2, =- (a?d
S = - -,-^n, =-\{a*(
ci a a
the equations are
51—2
A SOLVABLE CASE OF THE QUINTIC EQUATION.
[777
fy
= , and substituting this value in the other two
The first equation gives
JL-'
equations, we have
B2D3 + Bj4 -BD- = Q,
B3y + BD4 + 0D = 0.
Eliminating B, the result is obtained in the form Det. = 0. where in the last
column of the determinant each term is divisible by D ; and omitting this factor,
the result is
D3, 72, -BD = 0.
7~\o o £ T\o
D\ 7-, - 8D2,
D3, 72, - BD2,
7, 0, - D4, 0
7, 0, - D\ 0D,
If, in order to develope the determinant, we consider it as a sum of products,
each first factor being a minor composed out of columns 1 and 2, and the second
factor being the complementary minor composed out of columns 3, 4, 5 (the several
products being of course taken each with its proper sign), the expansion presents
itself in the form
D3j (-
2 + B2D7),
+ BD9 - 02
3 . - BD2 (BD5 - 6<f)
(yn-BD5 - 08D5 - <y )
Hence, collecting, and changing the sign of the whole expression, we obtain
3D15 - (27S2 + rd + 02) D10 + (- yS + 37S0 + B3) 72D5 + 770 = 0,
a cubic equation for D5. We have then as above C = j), and B is given rationally
as the common root of the foregoing quadric and cubic equations satisfied by B.
Substituting for 7, B, 6 their values in terms of the original coefficients, the
equation for D5 becomes
2 (a?d - 3abc +
( a4 (ae - 4>bd + 3c2)2
+ < + a2 (ac - 62)2 (ae - 4bd + 3c2)
(-16 (ac - Z>2) (a*d - 3a6c + 26
( 28 (ac - 62)3 (a*d- 3a6c + 26s)
+ 4 (ac - b2)- <+l2a2 (ac - b2) (a2d - Sabc + 2b3) (ae - 4fod + 3c2)
U 8 (a2d - Babe + 263)3
- 128 (ac - b2)7 [a2 (ae - 4bd + 3c2) + (ac2 - 62)2} = 0,
and the solution of the given quintic equation thus ultimately depends upon that of
this cubic equation.
778] 405
778.
[ADDITION TO MR HUDSON'S PAPER "ON EQUAL ROOTS OF
EQUATIONS."]
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvm. (1882),
pp. 226—229.]
IT seems desirable to present in a more developed form some of the results of
the foregoing paper.
Thus, if the equation (a0, al} ..., a^x, l)n = 0 of the" order n has n— v equal
roots, where v is not > \n — 1, then we have ^r(r, v + 1, m) = 0, where m has any one
of the values 0, 1, ..., n — 2v — 2, and r any one of the values
2v + 2, 2v + 3, . . . , n — m.
The signification is
•^ (r, v + 1, m) = r 1 . r^js+i am ar+m
(r ^ p+i- i
~~ v ~ L) • i
+ (r-4)
+ (-)« (r - 25)
+ (-)"+1 (r-2v- 2).
Thus, when v = 0, the condition is
1 ' |> — 1]V
+ 1. v + 2 1
1.2
[v+l]s
= 0,
406
that is,
ADDITION TO MR HUDSON'S PAPER
[778
satisfied when the equation has all its roots equal.
The values of m are 0, 1, 2, ...,w — 2, and those of r are 2u + 2, 2v + 3, ..., n-m:
in particular, if ??i = 0, the values of r are 2, 3, ..., w, and the corresponding conditions
are
a0a2 — tti" = 0,
a0as — ajtta = 0,
and so for the different values of m up to the final value n — 2, for which r = 2,
and the condition is
„ n nl — C\ '
an—2ctn — a n—i — u ,
we have thus, it is clear, the whole series of conditions included in
= 0,
which are obviously satisfied in the case in question of the roots being all equal.
Again, when v = 1, the condition for n — 1 equal roots is
r .1.
r - 2) . 2 .
r . r — 1 . r — 2
1
r-l.r-2.r-3
1
that is,
• * • n — — o — — 7 u'jn+2"'H
r-2.r-3.r-4
2 __ ~
~
r_l.r_2 r-l.r-3r-2.r-3
or, what is the same thing,
(r - 3) amar+m - 2 (r - 2) am+lar+m-i + (r - 1) am+2ar+m_2 = 0,
where n — 4 at least, and m, r have the values
0, 1, 2, ...,?i-4
r =
4, 4,
5, 5
?i-l
thus, when n = 4, the only values are m = 0, r = 4, and the condition is
a0a4 — ^ava3 + 3a22 = 0.
778] "ON EQUAL ROOTS OF EQUATIONS."
Similarly, when v = 2, the condition for n — 2 equal roots is found to be
407
r— l.r — 2 . r — 3 r— 1 . r - 3 . r — 4 r — 2.?^ — 3.r— 5 r— 3.r — 4.r — 5
or, what is the same thing,
T — 4 . T — 5 . ttm Or+m
8 . f 2 . r 5. <zm+1ar^.m_1
+ 3.r-l.r-4. am+2a,.+m_3
. ^ 1 . T 2 . dwt+3(ir+Wt_3 = 0,
where n = 6 at least, and w, r have the values
m = ; 0, 1, ..., M — 6
= 0;
r=
6, 6,
7, 7
6
w-1
Observe that the sum of the coefficients is = 0, viz.
(r_4)(r_5)_3(r_2)(r-5) + 3(r-l)(r-4)-(r-l)(r-2) = 0,
this should obviously be the case, since the conditions for n — 2 equal roots must
be satisfied when the roots are all of them equal ; and the property serves as a
verification.
It is to be remarked that the equation -^ (r, v + 1, m) = 0 does not in all cases
give all the conditions for the existence of n — v equal roots in an equation of the
order n ; thus when n = 3 and v = l, we cannot by means of it obtain the condition that
a cubic equation may have 2 equal roots. The problem really considered is that of
the determination of those quadric functions of the coefficients which vanish in the
case of n — v equal roots ; and in the case in question (n = 3, v = 1) there is no
quadric function which vanishes, but the condition depends on a cubic function.
The question of the quadric functions which vanish in the case of n — v equal
roots, and to a small extent that of the cubic functions which thus vanish, is considered
in Dr Salmon's " Note on the conditions that an equation may have equal roots,"
Camb. and Dublin Math. Jour., t. v. (1850), pp. 159 — 165, and in particular the
equation there obtained p. 161 is the equation i/r(0, v + l, n) = Q.
408 [779
779.
[NOTE ON MR JEFFERY'S PAPER "ON CERTAIN QUARTIC
CURVES WHICH HAVE A CUSP AT INFINITY, WHEREAT
THE LINE AT INFINITY IS A TANGENT."]
[From the Proceedings of the London Mathematical Society, vol. xiv. (1883), p. 85.]
THE assumed form Ka3/3=u2, or, as this is afterwards written,
= aa? + 2bxy + cy* + 2ex + 2dy
is, I think, introduced without a proper explanation. Say, the form is a?y =• z* (*§x> V> z)z>
it ought to be shown how for a cuspidal quartic we arrive at this form ; viz. taking
the cusp to be at the point (x = 0, 2 = 0), z = Q for the tangent at the cusp, and # = 0
an arbitrary line through the cusp ; then the line z = 0 besides intersects the curve
in a single point, and, if y = 0 is taken as the tangent at that point, the equation
of the curve must, it can be seen, be of the form
(x3 + Qa?z) y=z* (a, b, c, f, g, h§x, y, zf.
The conic (a, b, c, /, g, h~$x, y, zf = 0 touches the quartic at each of the two inter
sections of the quartic with the arbitrary line x = 0 ; and we cannot, so long as the
line remains arbitrary, find a conic which shall osculate the quartic at the two points
in question; but, for the particular line cc + ^6z = Q, there exists such a conic, viz.
writing x instead of x + \Qz, the form is a?y = z-(a', b', c',f, g', h'Qx, y, z}-, and the
new conic (a', ...\x, y, z)z = 0 has the property in question. This is the adopted form,
and it thus appears that in it the line x = 0 is a determinate line, viz. the line
passing through the cusp and the two points of osculation of the osculating conic.
It thus appears that in the assumed form the lines x = 0, y = 0, z = Q are determinate
lines.
780]
409
780.
[ADDITION TO MR HAMMOND'S PAPER "NOTE ON AN EXCEP
TIONAL CASE IN WHICH THE FUNDAMENTAL POSTULATE
OF PROFESSOR SYLVESTER'S THEORY OF TAMISAGE
FAILS."]
[From the Proceedings of the London Mathematical Society, vol. xiv. (1883), pp. 88 — 91.
Read Dec. 14, 1882.]
THE extreme importance of Mr Hammond's result, as regards the entire subject
of Covariants, leads me to reproduce his investigation in the notation of my Memoirs
on Quantics, and with a somewhat different arrangement of the formula?. For the
binary seventhic
(a, 6, c, d, e, f, g, K$x, y}\
the four composite seminvariants of the deg- weight 5 . 11 (sources of covariants of the
deg-order 5 . 13) are
II.
1.7 4.6
1.0 4 . 11
a + 1
a?eh + 1
fg - i
a bdh - 4
beg - 2
bf* + 6
c*h + 3
cdg - 2
cef - 6
d»/ +10
de2 - 5
a°b'2ch 0
&V+67
6c2# - 15
6cd/-24
bee* - 30
bd?e - 10
c3/ +27
C ftfi — 4o
cd3 +20
2 . 10
3
.3
2 . 2
3
.9
ac + 1
ach
+ 2
62 — 1
dg
- 7
ef
+ 5
aWt
- 2
beg
+ 7
bdf
+ 22
be*
-25
cy
-27
cde
+ 45
Deg-order.
Deg-weight.
C. XI.
52
410
ADDITION TO MR HAMMONDS PAPER.
[780
m.
IV.
2 . 6
2 .4
3 .
3 .
7
1
ae +1
bd-4
c2 +3
abg
de
bee
bd-
+ 1
- 7
+ 9
5
+ 12
-30
+ 20
2 . 2
2 .6
3. 11
3 . 5
ag+ 1
6/- 6
ce +15
d2 - 10
a2/ +1
a be -5
cd +2
a°62d + 8
fee2 6
Deg-order.
Deg-weight.
and it is here at once obvious that there exists a syzygy of the form I. = III. — IV. ;
in fact, if in III. and IV. we write a = 0, then the values are each
= - 26 (4>bd - 3c2) (Qbf- 15ce + 10d2) ;
hence III. — IV. must divide by a, the quotient being a seminvariant of the deg-weight
4 . 11, which can only be a numerical multiple of the second factor of I., and is in
fact = this second factor, that is, we have the syzygy I. = III. — IV.
Working out the values of the four products, and joining to them the expression
for the irreducible seminvariant of the same deg-weight 5 . 11 (0, a? of my tables [774]
for the binary sextic), we have the table :
5 . 10
a?dh
eg
a?bch
bdg
bef
c*g
cdf
ce2
d?e
ab3h
Peg
tfdf
6V2
&c2/
bcde
bd3
c3e
b3cf
b3de
bc3d
c5
5 . 11
0
I.
III.
IV.
H.
a3eh
+ 1
+ 1
ft
- 1
+ 1
a?bdh
- 4
- 4
beg
- 2
- 7
- 5
ft/2
+ 6
- 6
c2h
+ 3
+ 3
+ 2
cdg
- 2
+ 2
- 7
cef
- 1
- 6
+ 9
+ 15
+ 5
d?f
+ 3
+ 10
- 10
de'2
- 2
- 5
- 5
atfch
- 4
Wdg
+ 20
+ 28
+ 8
+ 7
b'2ef
+ 1
+ 57
+ 12
- 45
- 5
bc2<7
-15
- 21
- 6
+ 7
bcdf
-14
-24
- 36
- 12
+ 22
bee2
+ 11
-30
- 30
-25
bd?e
+ 1
-10
+ 40
+ 50
c*f
+ 9
+ 27
+ 27
-27
c^de
-14
-45
- 15
+ 30
+ 45
cd? j + 6
+ 20
- 20
-20
a°b*h
- 2
b3cg
- 7
b3df
+ 8
- 48
- 48 ! -22
63e2
- 9
+ 25
62c2/
- 6
+ 36
+ 36 i +27
tfcde
+ 16
+ 120
+ 120 : -45
V2d3
- 8
- 80
- 80 +20
1)C^6
3
90
- 90 !
bcW
+ 2
+ 60
+ 60
c*d
!
I have prefixed to the table the literal terms of the deg-weight 5.10; for the deg-
weights 5 . 11 and 5 . 10, the numbers of terms are =30 and 26 respectively; and it is
the difference of these 30 — 26, = 4, which gives the number of asyzygetic semiuvariants
of the deg-weight 5 . 11.
781] 411
781.
ON THE AUTOMORPHIC TRANSFORMATION OF THE BINARY
CUBIC FUNCTION.
[From the Proceedings of the London Mathematical Society, vol. XIV. (1883),
pp. i 03— 108. Read Jan. 11, 1883.]
I CONSIDER the cubic equation (a, b, c, d$x, I)3 = 0. It is shown (Serret, Cours
d'Algebre sup&rieure, 4th ed., Paris, 1879, t. II. pp. 466 — 471) how, given one root of
the equation, the other two roots can be each of them expressed rationally in terms
of this root and of the square root of the discriminant ; viz. making the proper
changes of notation, and writing
A, B, C = ac- b\ ad - be, bd - c2, X = V^J,
H = B2 - 4>AC, = o?d? + 4ac3 + 4>bsd - 362c2 - Qabcd,
= + B +2(7 -^A - x yn - B
' ^~' ~
(values which give a + S = — 1, ccS — f3y = — 1, and therefore also
a2 + aS + S2 + /3y = 0,
which is the condition in order that the function <bx, = -- v > mav be periodic of the
<x + o
third order, <£3# = #), then, u being a root of the equation, say (a, b, c, d^u, I)3 = 0, the
other two roots are
, au + 13
and
+ ^(a + g) 8u-(3
where observe that, by the change of V^ in^o — \/fl, a, /3, j, 8 become 8, — /3, — 7, a ;
so that the last-mentioned value $~^u is, in fact, the value obtained from <f>u by the
mere change of sign of the radical.
52—2
412 ON THE AUTOMORPHIC TRANSFORMATION [781
It is to be observed that, if we have between two roots u, v of the equation
(a, b, c, d$as, 1)3 = 0,
a relation v = - — ^ , where a, (3, 7, 8 have given values, this implies in the first place
a relation between a, b, c, d (and the given values of a, ft, y, B), and it implies more
over that u, and consequently also v, are not any roots whatever, but two determinate
roots of the equation ; viz. u, v will be each of them expressible rationally in terms
of a, b, c, d and a, ft, y, 8. And if, in order that (a, b, c, d) may remain arbitrary,
we consider a, ft, 7, 8 as given quantities satisfying the relation which exists between
these quantities and (a, b, c, d), then in general we still have u, v determinate roots
of the cubic equation. But in the foregoing solution u is any root whatever of the
cubic equation.
To examine how this is, starting from the equations
au + ft
(a, 6, c, dftu, 1)3= 0, (a, 6, c, dtfv, ly — ^, v — » ,
7W + o
we have
a (it3 - v3) + 36 (w2 - v2) + 3c (w - v) = 0,
and therefore
a (u2 + wv + v2) + 36 (it + v) + 3c = 0,
that is,
av2 + (au + 36) v + aw2 + 36w + 3c = 0 ;
or, writing herein for v its value,
a (au + ft)2 + (au + ft)(yu + 8) (au + 36) + (yu + 8)2 (au2 + 3bu + 3) = 0 ;
that is,
a (au + ft)2 + ay (au3 + 36w2) + 72 (au* + 36 u3 + 3cu2)
+ (08 + fty) (au2 + 36w) + 278 (au3 + 36w2 + 3cw)
+ /38 (ait + 36) + 82 (aw2 + 36w + c) = 0 ;
or, reducing by the equation aw3 + 36u2 + 3cw + d = 0, this is
a (au + ft)2 + <*7 (— 3ci* — a7) + 72 (— du)
+ (a8 + fty) (au2 + Sbu) + 2y8 (- d)
+ ft8 (aw + 36) + 82 (au2 + 36w + 3c) = 0,
and, collecting the terms, this is
u*a (a2 + a8 + 82 + £7)
+ u [a (2a/3 + £8) + 36 (a8 + 8- + £7) - 3ca7 - dy2}
+ aft2 + 3bft8 + 3cS2 + d (- ay - 2y8) = 0.
We can, from this equation, and the equation au3 + 36w2 + 3cw + d = 0, eliminate u, thus
obtaining a relation between a, b, c, d, a, ft, y, 8; and, this relation being satisfied, the
two equations then determine u rationally in terms of these quantities.
781] OF THE BINARY CUBIC FUNCTION. 413
We may without loss of generality assume &8 — ft<y = 1 ; and, this being so, if we
then further assume a+8= — 1, then we have
a2 + a8 + S2 + j3y = 0,
which is, as above appearing, the condition for </>3# = 0. The foregoing equation in u
thus becomes
u [aft (a - 1) - 36a2 - 8007 - cfy2}
+ (a/32 + Zb/38 + 3cS2 - dy (8 - 1)} = 0 ;
a linear equation giving (in a simplified form) the like results to those given by the
quadric equation; viz. substituting in the cubic equation the value of u given by the
linear equation, we have a relation between a, b, c, d, a, ft, 7, 8 ; and, this relation
being satisfied, u has the determinate value given by the linear equation.
The only way in which u can cease to have this determinate value, and so be
capable of being any root whatever of the cubic equation, is when the linear equation
becomes 0=0; viz. if
a/3 (a— 1) — 36a2 — 3cay — d<f =0,
equations which are, in fact, satisfied by the foregoing values of a, ft, 7, 8, as may be
verified without difficulty.
/¥)/ | /O
It is to be remarked that if, instead of the root u and the equation v = - '
yu + 8 '
then, instead of a, ft,
have 8, —ft, —7, a, and the corresponding equations are
cs r\
we consider the root v and the equation u = — ; then, instead of a, ft, 7, 8, we
dy* =0,
a/32
equations which are also satisfied by the foregoing values of a, {3, 7, 8. And the four
equations, together with a8 — ft<y = l and a + 8 = 1, are more than sufficient to determine
the foregoing values of a, ft, 7, 8.
But we further verify without difficulty that the foregoing values of a, ft, 7, 8 give
identically
(a, b, c, ti$wc + fty, 7« + %)3 = (a, 6, c, d$>, y)3 ;
or the formulae lead to an automorphic transformation of the binary cubic (a, b, c, d$x, y}3.
And conversely, starting from the notion of the automorphic transformation of the binary
cubic, we ought to be able to obtain the foregoing formulae.
For greater convenience, I write the equation of transformation in the form
(a, b, c, dQax + fiy, yx + 8y)3 = - 6 (a, b, c, d^x, y)3;
ON THE AUTOMORPHIC TRANSFORMATION
[781
the equations to be satisfied by a, /3, 7, S, 0 then are
aa3 + b . 3a27 + c . 3a72 + c£y3 = - a0,
aa2/3 + b (o?S + 2a/37) + c (2a7S + /372) + dy*S = - 6(9,
aa/32 + b (2a/3S + /327) + c (aS2 + 2/37S) + dy&2 = - c0,
a/33 + b . 3/32S + c . 3/3S2 + dS3 =- d0.
Writing aS — {3y = V , and as before £1 for the discriminant, the theory of invariants
gives fiV6=n04. We are considering the case of the general cubic function (a, b, c, <T§x, y)3,
for which II is not = 0 ; and we have therefore V 6 — 04 = 0, or, what is the same thing,
we may write V = g2, 6 = q3, where q is arbitrary.
It is to be shown that a + 8 is = q or — 2q, the latter value giving the trivial
solution ax + j3y, yx + 8y = (x, y). The proper solution thus corresponds to V = q2, a + 8 = q,
that is,
(a + S)2 - (aS - /37) - 0, or a2 + 8- + aS + /37 = 0,
the condition for the periodic function (f>3x — ac = 0.
For this purpose, from the foregoing equations eliminating a, b, c, d, we have
3a27 ,
2a/37 + 0,
3a72
=0;
an equation which may be written
D + 6 (123 + 234 + 341 + 412) + 02 (12 + 23 + 34 + 41 + 13 + 42) + 0s (1 + 2 + 3 + 4) + 6* = 0,
where 123, &c., are the first diagonal minors, 12, &c., the second diagonal minors, 1, &c.,
the third diagonal minors, or diagonal terms of the foregoing determinant, writing
therein 6 = 0. We find without difficulty
1, 2, 3, 4 =a3, a2S + 2a/37, aS2 + 2/37S, S3,
12, 13, 14, 23, 24, 34 = (a4, a2 (aS + 3/37), (a2S2 + aS/37 + /3V),
(a2^ + aS/37 + /3272), S2 (aS + 3/37), S4} V ,
123, 124, 134, 234 = [a3, a (08 + 2/87), S(aS + 2/37); 83} V3,
D = V6,
and the equation thus is
+ 6s [a3 + a2S + aS2 + S3 + 2ay87 + 2/37S]
+ ^ V [a4 + a38 + ag3 + g4 + 3a2yg7 + 3
+ 0 V3 [a3 + S3 + (a + S) (08 + 2/37)]
+ V6 = 0.
781] OF THE BINARY CUBIC FUNCTION. 415
Putting herein a + B = m, a8 = n, /3y = n — V , it is found that n disappears altogether
from the equation; viz. the resulting form is
2V (m4- 3Vm2 + 2V2)+ <9V3ra(m2- 2V)+ V6 = 0,
or, what is the same thing,
ra4.02 V
Putting for 0, V, their values q3, q2, the equation divides by qs, and omitting this
factor it becomes
m* + 2m3q - 3m2q2 - 4>mq3 + ^q4 = 0 ;
viz. this is
or we have
m = q or — 2q ; that is, a + B = q, or a + S = — 2q.
Writing, as before, A, B, C=ac — b2, ad — be, bd — c2, we deduce from the foregoing
equations
62A = V2 [Aa? + B . «7 + Cy2],
= V2 [A . 2«/3 + 5 (aS + /3y) + C . 27S],
+5./3S +C82],
which are, in fact, the equations for the automorphic transformation of the Hessian
(A, B, G\x, y)z. And, writing herein 0, V = q3, qz, the equations become
A (a2 - q2) + Bay + C . y2 =0,
A 2a/3 +5 (aS + £7 - g2) + C . 278 = 0,
A/32 +B/3S
From the first and second of these we have
A :B : C=y2(V + q2) : - 27aV +278^2 : a2V - (a2 + aS + @y) q2 + q* ;
or, writing herein for V , q* the values q2, q2 (a.8 — /37), the three expressions divide by
27<22, and we have
A : B : C = y : B - a : -/3.
Combining these values in the first place with the equation a. + 8 = — 2q, we may write
a, & 7, B = -q-pB, - 2pC, 2pA, -q+pB,
where p is to be determined. Substituting in the last of the three equations, we have
A . 4>p2C2 - 2pBC (-q+ pB) + C (- 2pqB + p2B?) = 0,
that is,
p2C (4 A C - B2), = - p2 . <7H, = 0,
and the form (a, b, c, d~§x, yf being arbitrary, neither C nor fl is = 0 ; whence p = 0, and
the values are a, /3, 7, 8 = — q, 0, 0, — q, that is,
(a, b, c, d\-qx, - qy)3 = - q3 . (a, 6, c, d$x, yj,
a trivial result.
416 AUTOMORPHIC TRANSFORMATION OF THE BINARY CUBIC FUNCTION. [781
But, combining the same values with a + & = q, we have
a, /3, 7, S = ±q-pB, -2pC, 2pA, $q+pB;
and then, substituting in the third equation, we have
A . 4,p*C2 - 2pBC (fa + pB) + C (- f 32 + pqB + p*&) = 0,
that is,
C{(4AC-B*)p"--%q*} = 0,
or, omitting the factor C, and introducing the foregoing notation X2 = — 1, this is
4fl\y - (f = 0, or say p = ^—7^ q-
For the unimodular substitution 08— /9y— 1, we must have ^2=1: but, the transforma
tion being
(a, b, c, d~§ax + (3y, <yx + By)3 = — q3 . (a, b, c, d§x, y)3,
to make this strictly automorphic, we must have <f = — \, and the two conditions are
satisfied by q = — 1 ; we then have p = — _— — rp» > and the coefficients are
2(7 -
which are the values given at the beginning of the paper, and which belong to the
automorphic transformation
(a, b, c, dQctx + {3y, yx + Sy)3 = (a, b, c, d~$x, y)3.
The ou priori reason for the periodicity-equation <f>3x = x, is best seen by expressing
the cubic function as a product of factors
M(x — ay) (x — by) (x — cy).
The substitution must, it is clear, cyclically interchange these factors, and therefore,
when performed three times in succession on any one of these factors, or consequently
upon an arbitrary linear factor x — ft/, must leave the factor unaltered, and it must
thus be a periodic substitution <f>3x = x. But it was interesting to see how the condition
for this, a2 -f- 82 + aS + @y = 0, comes out from the consideration of the equation
(a, b, c, d$ax + Py, jac + 8y)3 = (a, b, c, d$ac, y)3.
782] 417
782.
ON MONGE'S "MEMOIRE SUR LA THEORIE DES DEBLAIS ET
DES REMBLAIS."
[From the Proceedings of the London Mathematical Society, vol. xiv. (1883),
pp. 139—142. Read March 8, 1883.]
THE Memoir referred to, published in the Memoires de I' Academic, 1781, pp. 666 —
704, is a very remarkable one, as well for the problem of earthwork there considered
as because the author was led by it to his capital discovery of the curves of curva
ture of a surface. The problem is, from a given area, called technically the Ddblai,
to transport the earth to a given equal area, called the Remblai, with the least
amount of carriage. Taking the earth to be of uniform infinitesimal thickness over
the whole of each area (and therefore of the same thickness for both areas), the
problem is a plane one; viz. stating it in a purely geometrical form, the problem is:
Given two equal areas, to transfer the elements of the first area to the second area
in such wise that the sum of the products of each element into the traversed
distance may be a minimum ; the route of each element is, of course, a straight line.
And we have the corresponding solid problem : Given two equal volumes, to transfer
the elements of the first volume to the second volume in such wise that the sum
of the products of each element into the traversed distance may be a minimum ; the
route of each element is, of course, a straight line. The Memoir is divided into two
parts : the first relating to the plane problem (and to some variations of it) : the
second part contains a theorem as to congruences, the general theory of the curvature
of surfaces, and finally a solution of the solid problem; in regard to this, I find a
difficulty which will be referred to further on.
I have said that Monge gives a theorem as to congruences. This is not stated
quite in the best form, — viz. instead of speaking of a singly infinite system of lines,
or even of the lines drawn according to a given law from the several points of a
surface, he speaks of the lines drawn according to a given law from the several points
c. XL 53
418 ON MONGE'S "MEMOIRE SUR LA [782
of a plane (but, of course, any congruence whatever of lines can be so represented) ;
and he establishes the theorem that each line of the system is intersected by each
of two consecutive lines, — viz. taking (#', y') as the coordinates of the point of
intersection of any line with the plane of xy, he obtains, as the condition of inter
section with the consecutive line a quadric equation in (dx, dy'}. He then considers
the nonnals of a surface, (which, as lines drawn according to a given law from any
point of a surface, require a slightly different analytical investigation), establishes for
them the like theorem, and shows moreover that the two directions of passage on
the surface to a consecutive point are at right angles to each other; or, what is the
same thing, that in the two sets of developable surfaces formed by the intersecting
normals, each surface of the one set intersects each surface of the other set in a
straight line, and at right angles. He speaks expressly of the lines of greatest and
least curvature, and generally establishes the whole theory of the curvature of surfaces
in a very complete and satisfactory manner; the particular case of surfaces of the
second order is not considered. It may be remarked that, although not explicitly
stating it, he must have seen that a congruence of lines is not, in general, a system
of normals of a surface (that is, the lines of a congruence cannot be, in general, cut
at right angles by any surface) ; he, in fact, assumes (quite correctly, but a proof
should have been given) that a congruence of lines for which the two sets of
developable surfaces intersect at right angles is a system of normals of a surface.
Reverting to the before-mentioned problem (plane or solid), I remark that this
is a problem of minimum sui generis. Considering the first area or volume as divided
in any manner into infinitesimal elements, we have to divide the second area or
volume into corresponding equal elements, in such wise that the sum of the products
of each element of the first area or volume into its distance from the corresponding
element of the second area or volume may be a minimum ; but, for doing this, we
have no means of forming the analytical expression of any function which is to be,
by the formulae of the differential calculus or the calculus of variations, made a
minimum.
For the plane problem, Monge obtains the solution by means of the very simple
consideration that the routes of two elements must not cross each other; in fact,
imagine an element A transferred to a, . and an equal element B transferred to b :
the lines Aat Bb must not cross each other, for if they did, drawing the two lines
S
Ab and Ba, the sum Aa + Bb would be greater than the sum Ab + Ba, contrary to
the condition of the minimum. Imagine the areas intersected by two consecutive lines
as shown in the figure : the filament between these two lines may be regarded as
782] THEORIE DBS DEBLAIS ET DBS REMBLAIS." 419
a right line ; and, assuming that some one element of the filament BD is transferred
to a point of bd (that is, so as to coincide with an element of the filament bd), it
follows that every other element of BD must be transferred so as to coincide with
some other element of bd; and this obviously implies that the filaments BD and bd
must be equal. Observe that, this being so, it is immaterial which element of BD is
transferred to wrhich element of bd ; in whatever way this is done, the sum of the
products will be the same*. The two lines may be regarded as the normals of a
curve ; and the problem thus is, to find a curve such that, drawing the normals
thereof to intersect the two areas, then that the filaments BD and bd, cut off by
consecutive normals on the two areas respectively, shall be equal. This leads to a
differential equation of the second order for the normal curve; one of the constants
of integration remains arbitrary, for the normal curve is any one of a system of
parallel curves. It is to be observed that the filaments are the increments of the
areas BCD and bed ; these increments are equal ; a position of the line must be the
common tangent Cc of the two areas (this, in fact, constitutes the condition for the
determination of one of the arbitrary constants), and for this position the areas are
each = 0. Hence, in general, the areas must be equal ; or the problem is, to find a
curve such that any normal thereof cuts off equal areas BCD and bed.
If, instead of the normal curve, we consider the curve which is the envelope of
the several lines, or, what is the same thing, the locus of the point N, then we
could, in like manner, obtain for this curve a differential equation of the first order:
the constant of integration would be determined by the condition that Cc is a
tangent. The curve in question is, of course, the evolute of the normal curve.
The several lines which intersect the two areas give rise to a finite arc IS of
this evolute, and, as remarked by Monge, it is only when this arc IS lies (as in the
figure) outside the two areas, that we have a true minimum.
Passing now to the solid problem, we may imagine a congruence of lines inter
secting the two volumes; each line of the congruence is intersected by two consecutive
lines, and the lines of the congruence thus form two sets of developable surfaces, each
surface of the one set intersecting each surface of the other set. And, considering
two consecutive surfaces of the one set, and two consecutive surfaces of the other set,
these include between them a filament; and, treating the filament as a right line, it
seems to follow (although it is more difficult to present the reasoning in a rigorous
form) that, if any one element of the filament BD be transferred to any one element
of the filament bd, then that every other element of the filament BD must be
* The most simple case is, take in the same straight line two equal segments AB, ab ; it is immaterial
how the elements of AB are transferred to ab, the sum of the products of each element into the traversed
distance will be in every case the same. Analytically, if dx = dx', then
I (x' -x) dx — I x'dx' - I x dx,
the equation dx' = dx meaning x' = x + & discontinuous constant. In the actual case of the filament, the formula
is, if rdr=r'drr, then
I (/ - r) r dr = I r'2 dr' - If2 dr.
53—2
420 ON MONGE'S "MEMOIRE SUE LA THEORIE DES DEBLAIS ET DBS REMBLAIS." [782
transferred to some other element of the filament bd; and, this being so, the two
filaments must be equal. But Monge goes on to argue that the condition of the
minimum further requires that the developable surfaces shall cut at right angles, and
I cannot say that I see this. He says (pp. 700, 701), "We know already that the
routes must be the intersections of two sets of developable surfaces such that each
surface of the first set intersects those of the second set in right lines; it remains
to be found under what angles these surfaces must cut each other to satisfy the
minimum. But it is evident that these angles must be right angles, for with these
angles the elementary spaces comprised between four developable surfaces will be greater,
and for equal distances the transported mass will be greater; therefore, in the case
of a minimum, the routes must be the intersections of two sets of developable surfaces
such that each surface of the one set cuts those of the second set in straight lines
and at right angles." And, this being so, he infers, and it in fact follows, that the
routes are the normals of a surface.
Admitting the conclusion, the problem becomes as follows: — Given two volumes,
it is required to find a surface such that, drawing the normals thereof to intersect the
two volumes, and considering the filament bounded by the developable surfaces which
belong to two consecutive curves of curvature of the one set and those belonging to
two consecutive curves of curvature of the other set, the portions cut off on the two
volumes respectively may be equal. And we are thus led to a partial differential
equation of the second order for determining the equation z=f(x, y} of the required
surface. As in the plane problem, it is immaterial how the elements of the one
filament are transferred to the other filament.
783] 421
783.
ON MR WILKINSON'S RECTANGULAR TRANSFORMATION.
[From the Proceedings of the London Mathematical Society, vol. xiv. (1883),
pp. 222—229. Bead May 10, 1883.]
CONSIDERING the three cones,
(p + X) X2 + (q + X) F2 + (r + X) Z2 = 0,
(p + rfX2 + (q + /i) F2 + (r + fi) Z2 = 0,
(p + v)X2 + (q + v) F2 + (r + v) Z* = 0,
where
p + q + r+\+ p+v = 0,
it is easy to see that these contain a singly infinite system of rectangular axes,
viz. we have in each cone one axis of a rectangular system, and for one of the
cones the axis may be any line at pleasure of the cone. In fact, taking for
the three axes (aj, y, z\ (x, y', z'\ (x", y", z") respectively, that is, for the first
axis X : Y : Z=x : y : z, and so for each of the other two axes, then (x, y, z)
being an arbitrary line on the first cone, we can find (x', y', z) and (x", y", z") such
that
(p + \)x2 +(q + \)y2 +(r-fX)22 =0,
(p + n}x"> +(q + fj,)y'2 +(r + riz'* =0,
(p + v) x"2 + (q+v) y"2 +(r+v) z"2 = 0,
x' x" + y' y" + z' z" = 0,
x"x +y"y +z"z =0,
x x' +y y' +z z' =0.
For, eliminating (x", y", z") from the third, fourth, and fifth equations, we have,
first,
x" : y" : z" = yz' — y z : zx — z'x : xy' — x'y,
422 ON MR WILKINSON'S RECTANGULAR TRANSFORMATION. [783
and consequently
v) (zxf - z'xf + (r+v)(xy'- x'yj = 0.
It is to be shown that this equation is implied in the remaining first, second, and
third equations; for, this being so, (x, y, z), (as', y', /) satisfy only these equations:
or (x, y, z) are any values whatever satisfying the first equation. The other two
equations then determine (x, y't z'\ and, these being known, (x", y", z") are then
determined as above.
In fact, attending to the sixth equation, the equation just obtained may be
written in the form
or, what is the same thing, in the form
for, comparing in the two forms, first the coefficients of oFx'*, these are
and -2
which are equal in virtue of p + q + r + \ + p + v = Q; and comparing next the
coefficients of y~z'~, these are
p+v and -(r+/i)-(2 + X),
which are equal in virtue of the same relation: and, similarly, the coefficients of the
other terms y^y'3, &c,, are equal in the two equations respectively.
Take now three arguments a,, fc,, c,, connected by the relation a, + \ + c, = 0,
and write a, a, A for the sn, en, and dn of a«: and similarly b, b, B and c, c, C
for those of bd and cv respectively: then we may write
£, -A),
£• ez)-
for, starting from the first set of values, we have the second set if only
We thence obtain
X_ o i _J b a _
^--
783] ON MB WILKINSON'S RECTANGULAR TRANSFORMATION. 423
and, in order to the identity of the two values of 0, we must have
.
that is,
(abc - b1) (ABC - A*) - (abc - aj) (ABC- #) = 0,
or, reducing,
(a* - b*) ABC - (A* - B1) abc + A-te - #aj = 0.
But
hence the whole equation divides by a1— 61, and, omitting this factor, it becomes
-ABC+l&bc + V* = 0,
which is a known relation between the elliptic functions of the arguments a*, &,,
connected by the equation a0 + bt + c, = 0. Similarly, for <£, we have
, c a C A
r-X-f- -*-~
and, comparing the two values of £, we have the same identical relation.
It thus appears that the three cones
(the coefficients whereof depend on the elliptic functions sn, en, and dn, of the
arguments a,, 6,, c, connected by the equation o, + ft, + c,= 0) contain a singly infinite
system of rectangular axes.
Considering an argument /„ and denoting its sn, en, dn by /, t, F respectively,
we have, for an arbitrary line on the first cone, the values
*, y, z =
In fact, substituting in the equation of the cone, we obtain the identity
and if we determine J/ by the condition that a? + if + 2? shall be =1, then we have
where the coefficient of if J is
= Jfc'^a + #4bc (1 -/*) -
424 ON MR WILKINSON'S RECTANGULAR TRANSFORMATION. [783
which is easily shown to be
so that the values of x, y, z are
= {V&'l4a, V&l4bc . f, */-a,BC . F} -H V)Wb'*6c (a2 - /2),
and, similarly taking the arguments g0} h0, and denoting their elliptic functions by
, \J, 11, 11, J.J.,
actively, the value
s
5111 ui luiuiinu
x, y, z =
^/k'-Ad.
vA^.Abc . f
V- a5(7 . J?'
x', y, z =
*Jk'2Bb
A/Z^2A?r*Q n
V A/ JLJ\s<M • I/
V- b(7^ . G
x", y", z" =
V&'2(7c
VAtfcb.A
^-cAB.H
a (62 - #2)
-=- Vjfe%'aa& (c2 - A2),
these values being such that a? + y2 + 22, a/2 + y'2 + z'2> x"z + 2/"2 + 2"'2 are eacn = *• Tne
radicals in the first line would be more correctly written, and may be understood as
meaning k' *JA */a, k */A \/b Vc> * Va V-S V^> and similarly as regards the second and
third lines respectively.
Taking now the arbitrary lines at right angles to each other, the condition for
the second and third lines is
which is satisfied if a0 = g0 — h0; similarly the condition for the third and first lines
is satisfied if b0=h0—f0; and we then have a0 + b0 = g0 — f0 ; that is, —C0=g0—f0 or
C<>=f0—g0) which is the condition for the first and second lines; hence the arguments
«o, b0, c0> f0, g0, h0 being such that
-/J0 -
go -fo • + c0 = 0,
- a0 - b0 - c0 . =0,
or, what is the same thing, a0, 60, C0, /„> #o, ^o being the differences of any four
arguments a, /3, 7, 8, the foregoing values of (x, y, z}, (x', y', z'), (x", y" , z") will
satisfy the equations
X2 +7/2 +Z°' =1,
x'2 +?//2 +z'2 =1,
x"x + y"y + z"z = 0,
x x' + y y' + z z' = 0,
for the transformation of a set of rectangular axes. These are, in fact, Mr Wilkinson's
expressions, the a0, 60, c0, /0, g0, h0 being his t-p, p-q, q-t, t, p, q respectively.
783]
ON MR WILKINSON'S RECTANGULAR TRANSFORMATION.
425
Returning to the three cones, it is to be remarked that, taking in the first of
them a line 1 at pleasure, then we have in the second of them two lines 2, 2'
each at right angles to the line 1, and such that the line 3 at right angles to
the plane 12, and the line 3' at right angles to the plane 12', lie each of them
in the third cone; or, what is the same thing, we have in the two cones respectively
the rectangular lines 1 and 2, and also the rectangular lines 1 and 2', such that the
planes 12 and 12' each of them envelope one and the same cone, the reciprocal of
the third cone ; where by the reciprocal cone of a given cone is meant the cone
generated by the lines through the vertex at right angles to the tangent planes of
the given cone. Introducing the notion of the absolute cone X2 + Y2 + Z2 = 0, a line
and plane through the vertex at right angles to each other are, in fact, reciprocal
polars in regard to this absolute cone ; and two lines at right angles to each other
are reciprocals (or harmonics) in regard to this absolute cone ; that is, the reciprocal
plane of either of them passes through the other. The two cones are cones inter
secting each other in four lines lying on the absolute cone ; and in virtue of this
relation they have the property in question, viz. taking in the first cone a line 1
at pleasure, then the reciprocal plane hereof in regard to the absolute cone meets
the second cone in a pair of lines 2 and 2' such that the planes 12 and 12' each
of them envelope one and the same cone ; the reciprocal of this cone is then the
third cone of the system, and as such it passes through the four lines on the absolute
cone.
In verification, observe that the coefficients p + \, q + \, &c. of the equations of
the three cones satisfy the equations
v,
1, q + \, q + fj,, q + v, (q + X) (q + /*) (q + v)
1, r+\ r + //,, r + v, (r + \) (r + /*) (r + v)
This is obviously the case for each equation such as
= 0.
1,
\,
= 0;
and any equation containing the fifth column is at once reducible to
that is,
p}
1, p, p3
or, dividing by 1, p, p2 , this is
the coefficients.
= 0,
1, p, ^1=0;
v = Q, the equation connecting
Hence, representing the three cones by
p X2 + q Y2 + r Z2 = 0,
and the absolute by
C. XL
54
426 ON MR WILKINSON'S RECTANGULAR TRANSFORMATION. [783
the coefficients p, q, &c., are connected by the equations
1, P, P', P", PPP"
'"
1, r, r', r", rr'r
= 0;
among these are of course included the equation | 1, p, p' \ = 0, which expresses that
the first and second cones intersect on the absolute; (p, q, r), (p', q, r') are any
quantities satisfying this relation, and, regarding them as given, we have then two
independent equations determining the ratios p" : q" : r". The theorem is that the
planes 12 and 12' envelope one and the same quadric cone
X* Y* *,
p" + q"+r"~
The equations | 1, p, p" = 0 and | 1, p, pp'p" | = 0 give
(q — r}p" + (r-p) q" + (p - q) r" = 0,
(q — r) pp'p" + (r—p) qq'q" + (p — q) rr'r" = 0,
and thence
(q — r)p" : (r —p) q" : (p — q) r" = qq' — rr' : rr' — pp : pp' — qq ;
or, observing that we have
q — r:r —p : p — q = qr — q'r : rp' — r'p : pq — p'q,
the equations may also be written
(qr' — q'r)p" : (rp' — r'p)q" : (pq' — p q) r" = qq' — rr' : rr —pp : pp' — qq'.
Starting with an arbitrary line (x, y, z) in the first cone, then the reciprocal
plane thereof (in regard to the absolute cone) is the plane Xx + Yy+Zz = 0, which
meets the second cone in two lines, say (2) and (2'), each of which is a line reciprocal
to the line (1); and we have thus two planes (12) and (12'), each of which
envelopes, as is to be shown, the same cone q"r"X2 + r"p"Y* +p"q"Z2 = 0.
Suppose, in general, that we have an arbitrary line (x, y, z) and an arbitrary
plane aX + /3Y+<yZ=Q, and that it is required to find the equation of the two
planes through the line (x, y, z), and the intersections of the plane aX+/3Y+<yZ = Q
with the cone .p'X* + q'Y* + rZ* = Q: the equation of the pair of planes is
(aX + flY+yZytjp'a? + q'y* + r'z2)
+ (CLX + /3y +jz )2 (p'X2 + q'Y2 + r'Z*)
-2(aX + pY + yZ)(cuK + j3y + <yz) (p'Xx + q'Yy + r'Zz) = 0.
In the present case, the plane aX + @Y+yZ = 0 is the plane xX+yY + zZ=0,
which is the reciprocal of the line (x, y, z) in regard to the absolute cone, and the
equation of the pair of planes is
+q'y2
xn- + y* + z-) (p'Xx + q'Yy + r'Zz) = 0,
783] ON MR WILKINSON'S RECTANGULAR TRANSFORMATION. 427
where the quantities (to, y, z), as belonging to a line on the first cone, satisfy the
condition px2 + qy2 + rz2 = 0. The equation may be written
(a, b, c,f, g, h^yZ-zY, zX-xZ, xY-yX)2 = 0,
where
«> b, c, f, g, h = q'z2 + r'y2, r'x2 + p'z2, p'y2 + q'x2, — p'yz, — q'zx, — r'xy,
and, as before, pa? + qy2 + rz2 = 0 ; viz. this is the equation of the pair of planes (12)
and (12').
The equation of the pair of tangent planes through the line (x, y, z) to the
cone q"r"X2 + r"p"Y2+p"q"Z2 = 0 is
(q"r"x2 + r"p"y2 + p"q"z2) (q"r"X2 + r"p"Y2 +p"q"Z2) - (q"r"xX + r"p"yY + p"q"zZ}2 = 0 ;
viz. omitting a factor p"q"r", this equation is
(p", q", r", 0, 0, Q^yZ-zY, zX-xZ, xY-yX)2 = 0.
And it is to be shown that this is equivalent to the former equation ; viz. writing
yZ — zYy zX — xZ, xY—yX = \ p, v, then that the two equations
, py + qx, —pyz, —qzx, —
p"K + q'p? + r"v* = 0,
are equivalent to each other.
We have identically \x + py + vz — 0, and thence also
(\x + fj,y + vz) [(p' -q- r') \x + (-p' + q - r') py + (-p -q' + r') vz} = 0,
where, on the left-hand side, the terms in pv, v\ and \/j, are
= — Zp'yz/jiv — 2q'zxv\ — 2r'xy\ft.
Hence the first equation may be written
[q'z2 + r'y- + (p' -q- r') x2} \2 + [r'x2 + p'z2 + (-p' + q'- r') y2] /j,2
+ [p'f + ?'«2 + (-p'-q'+ r') z2] v2 = 0,
and it is to be shown that this is equivalent to
viz. that we have p" : q" : r" =
q'z2 + r'y2 - ( p' - q' - r') x2
: r'x2 + p'z2 - (- p' + q' - r') y2
: p'y2 + q'x2 -(-p'-q' + r') z2,
where px2 + qy2 + rz~ = 0. Writing the equation in the form
p" : q" : r" = A : B : C,
54—2
428 ON MR WILKINSON'S RECTANGULAR TRANSFORMATION. [783
we have
A = q'z2 + r'y2 - p'a? + q'x2 + r'a?
= - p'a? + q' (a? + z2) + r' (of + y2)
= -pa? + (q' + r') (x* + f + z2) - q'y'2 - r'z2.
By what precedes, we have an identity of the form
a? + y- + z2 = a (pa? + q'y- + r'z'2) + fi (pa? + qy2 + rz2),
where, determining a from the equations 1 = q'ct + q/3, 1 = r'a + r{3, we find
a = (q - r) 4- (qrf - q'r) ;
but pa? + qy2 + rz- = 0, and the relation thus is
a? + y2 + z2 = a (pa? + q'y* + r'z2) ;
hence
A = {(q +r')a-l] (p'a? + q'y2 + r'z2),
or, substituting for a its value, this is
+ «0 (9-r)7 ir' + 9V|
qr — qr j ^
QQ' ~ rr'
= —, - - (p'a? + q'y2 + r'z2) ;
qr —qr^
and, forming the like values of B and C, the relations to be verified become
p» : q" : r" = ~rr> rr' ~ ' '-
qr — qr rp — rp pq — pq
which are, in fact, the values of the ratios p" : q" : r" obtained above; and the
theorem is thus seen to be true. It may be remarked that, if the first and second
cones, instead of intersecting in four lines on the absolute cone, had been arbitrary
cones; then, taking in the first cone a line (1) and in the second cone a line (2),
the reciprocal of (1) in regard to the absolute, the envelope of the plane (12) would
have been (instead of a quadric cone) a cone of the class 8.
784] 429
784.
PRESIDENTIAL ADDRESS TO THE BRITISH ASSOCIATION,
SEPTEMBER 1883.
[From the Report of the British Association for the Advancement of Science, (1883),
pp. 3—37.]
SINCE our l&st meeting we have been deprived of three of our most distinguished
members. The loss by the death of Professor Henry John Stephen Smith is a very
grievous one to those who knew and admired and loved him, to his University, and
to mathematical science, which he cultivated with such ardour and success. I need
hardly recall that the branch of mathematics to which he had specially devoted himself
was that most interesting and difficult one, the Theory of Numbers. The immense range
of this subject, connected with and ramifying into so many others, is nowhere so well
seen as in the series of reports on the progress thereof, brought up unfortunately
only to the year 1865, contributed by him to the Reports of the Association; but
it will still better appear when to these are united (as will be done in the collected
works in course of publication by the Clarendon Press) his other mathematical writings,
many of them containing his own further developments of theories referred to in the
reports. There have been recently or are being published many such collected
editions — Abel, Cauchy, Clifford, Gauss, Green, Jacobi, Lagrange, Maxwell, Riemann,
Steiner. Among these the works of Henry Smith will occupy a worthy position.
More recently, General Sir Edward Sabine, K.C.B., for twenty-one years general
secretary of the Association, and a trustee, President of the meeting at Belfast in
the year 1852, and for many years treasurer and afterwards President of the Royal
Society, has been taken from us, at an age exceeding the ordinary age of man. Born
October 1788, he entered the Royal Artillery in 1803, and commanded batteries at the
siege of Fort Erie in 1814; made magnetic and other observations in Ross and
Parry's North Polar exploration in 1818-19, and in a series of other voyages. He
430 PRESIDENTIAL ADDRESS TO THE [784
contributed to the Association reports on Magnetic Forces in 1836-7-8, and about
forty papers to the Philosophical Transactions; originated the system of Magnetic
Observatories, and otherwise signally promoted the science of Terrestrial Magnetism.
There is yet a very great loss : another late President and trustee of the
Association, one who has done for it so much, and has so often attended the meetings,
whose presence among us at this meeting we might have hoped for — the President
of the Royal Society, William Spottiswoode. It is unnecessary to say anything of his
various merits : the place of his burial, the crowd of sorrowing friends who were
present in the Abbey, bear witness to the esteem in which he was held.
I take the opportunity of mentioning the completion of a work promoted by the
Association : the determination by Mr James Glaisher of the least factors of the missing
three out of the first nine million numbers: the volume containing the sixth million
is now published.
I wish to speak to you to-night upon Mathematics. I am quite aware of the
difficulty arising from the abstract nature of my subject ; and if, as I fear, many or
some of you, recalling the Presidential Addresses at former meetings — for instance, the
resume and survey which we had at York of the progress, during the half century
of the lifetime of the Association, of a whole circle of sciences — Biology, Palaeontology,
Geology, Astronomy, Chemistry — so much more familiar to you, and in which there
was so much to tell of the fairy-tales of science; or at Southampton, the discourse
of my friend who has in such kind terms introduced me to you, on the wondrous
practical applications of science to electric lighting, telegraphy, the St Gothard Tunnel
and the Suez Canal, gun-cotton, and a host of other purposes, and with the grand
concluding speculation on the conservation of solar energy: if, I say, recalling these
or any earlier Addresses, you should wish that you were now about to have, from a
different President, a discourse on a different subject, I can very well sympathise with
you in the feeling.
But be this as it may, I think it is more respectful to you that I should speak
to you upon and do my best to interest you in the subject which has occupied me,
and in which I am myself most interested. And in another point of view, I think
it is right that the Address of a President should be on his own subject, and that
different subjects should be thus brought in turn before the meetings. So much the
worse, it may be, for a particular meeting; but the meeting is the individual, which
on evolution principles must be sacrificed for the development of the race.
Mathematics connect themselves on the one side with common life and the
physical sciences ; on the other side with philosophy, in regard to our notions of space
and time, and in the questions which have arisen as to the universality and necessity
of the truths of mathematics, and the foundation of our knowledge of them. I would
remark here that the connexion (if it exists) of arithmetic and algebra with the notion
of time is far less obvious than that of geometry with the notion of space.
As to the former side, I am not making before you a defence of mathematics,
but if I were I should desire to do it — in such manner as in the Eepublic Socrates
was required to defend justice, quite irrespectively of the worldly advantages which
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 431
may accompany a life of virtue and justice, and to show that, independently of all
these, justice was a thing desirable in itself and for its own sake — not by speaking
to you of the utility of mathematics in any of the questions of common life or of
physical science. Still less would I speak of this utility before, I trust, a friendly
audience, interested or willing to appreciate an interest in mathematics in itself and
for its own sake. I would, on the contrary, rather consider the obligations of
mathematics to these different subjects as the sources of mathematical theories now
as remote from them, and in as different a region of thought — for instance, geometry
from the measurement of land, or the Theory of Numbers from arithmetic — as a
river at its mouth is from its mountain source.
On the other side, the general opinion has been and is that it is indeed by
experience that we arrive at the truths of mathematics, but that experience is not
their proper foundation: the mind itself contributes something. This is involved in
the Platonic theory of reminiscence ; looking at two things, trees or stones or anything
else, which seem to us more or less equal, we arrive at the idea of equality: but
we must have had this idea of equality before the time when first seeing the two
things we were led to regard them as coming up more or less perfectly to this idea
of equality ; and the like as regards our idea of the beautiful, and in other cases.
The same view is expressed in the answer of Leibnitz, the nisi intellectus ipse,
to the scholastic dictum, nihil in intellectu quod non prius in sensu : there is nothing in
the intellect which was not first in sensation, except (said Leibnitz) the intellect
itself. And so again in the Critick of Pure Reason, Kant's view is that while there is
no doubt but that all our cognition begins with experience, we are nevertheless in
possession of cognitions a priori, independent, not of this or that experience, but
absolutely so of all experience, and in particular that the axioms of mathematics
furnish an example of such cognitions a priori. Kant holds further that space is no
empirical conception which has been derived from external experiences, but that in
order that sensations may be referred to something external, the representation of
space must already lie at the foundation ; and that the external experience is itself
first only possible by this representation of space. And in like manner time is no
empirical conception which can be deduced from an experience, but it is a necessary
representation lying at the foundation of all intuitions.
And so in regard to mathematics, Sir W. R. Hamilton, in an Introductory Lecture
on Astronomy (1836), observes : " These purely mathematical sciences of algebra and
geometry are sciences of the pure reason, deriving no weight and no assistance from
experiment, and isolated or at least isolable from all outward and accidental phenomena.
The idea of order with its subordinate ideas of number and figure, we must not indeed
call innate ideas, if that phrase be defined to imply that all men must possess them
with equal clearness and fulness : they are, however, ideas which seem to be so far born
with us that the possession of them in any conceivable degree is only the development
of our original powers, the unfolding of our proper humanity."
The general question of the ideas of space and time, the axioms and definitions of
geometry, the axioms relating to number, and the nature of mathematical reasoning, are
432 PRESIDENTIAL ADDRESS TO THE [784
fully and ably discussed in Whewell's Philosophy of the Inductive Sciences (1840), which
may be regarded as containing an exposition of the whole theory.
But it is maintained by John Stuart Mill that the truths of mathematics, in
particular those of geometry, rest on experience ; and as regards geometry, the same
view is on very different grounds maintained by the mathematician Riemann.
It is not so easy as at first sight it appears to make out how far the views
taken by Mill in his System of Logic Ratiocinative and Inductive (9th ed. 1879) are
absolutely contradictory to those which have been spoken of ; they profess to be so ; there
are most definite assertions (supported by argument), for instance, p. 263 : — " It remains
to enquire what is the ground of our belief in axioms, what is the evidence on which
they rest. I answer, they are experimental truths, generalisations from experience.
The proposition ' Two straight lines cannot enclose a space,' or, in other words, two
straight lines which have once met cannot meet again, is an induction from the
evidence of our senses." But I cannot help considering a previous argument (p. 259)
as very materially modifying this absolute contradiction. After enquiring " Why are
mathematics by almost all philosophers . . . considered to be independent of the
evidence of experience and observation, and characterised as systems of necessary
truth ? " Mill proceeds (I quote the whole passage) as follows : — " The answer I conceive
to be that this character of necessity ascribed to the truths of mathematics, and even
(with some reservations to be hereafter made) the peculiar certainty ascribed to them,
is a delusion, in order to sustain which it is necessary to suppose that those truths
relate to and express the properties of purely imaginary objects. It is acknowledged
that the conclusions of geometry are derived partly at least from the so-called
definitions, and that these definitions are assumed to be correct representations, as far
as they go, of the objects with which geometry is conversant. Now, we have pointed
out that, from a definition as such, no proposition unless it be one concerning the
meaning of a word can ever follow, and that what apparently follows from a definition,
follows in reality from an implied assumption that there exists a real thing conformable
thereto. This assumption in the case of the definitions of geometry is not strictly true :
there exist no real things exactly conformable to the definitions. There exist no real
points without magnitude, no lines without breadth, nor perfectly straight, no circles
with all their radii exactly equal, nor squares with all their angles perfectly right. It
will be said that the assumption does not extend to the actual but only to the
possible existence of such things. I answer that according to every test we have of
possibility they are not even possible. Their existence, so far as we can form any
judgment, would seem to be inconsistent with the physical constitution of our planet
at least, if not of the universal [sic]. To get rid of this difficulty and at the same
time to save the credit of the supposed system of necessary truth, it is customary to
say that the points, lines, circles and squares which are the subjects of geometry exist
in our conceptions merely and are parts of our minds ; which minds by working on
their own materials construct an a priori science, the evidence of which is purely
mental and has nothing to do with outward experience. By howsoever high authority
this doctrine has been sanctioned, it appears to me psychologically incorrect. The points,
lines and squares which anyone has in his mind are (as I apprehend) simply copies
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 433
of the points, lines and squares which he has known in his experience. Our idea of a
point I apprehend to be simply our idea of the minimum visibile, the small portion of
surface which we can see. We can reason about a line as if it had no breadth, because
we have a power which we can exercise over the operations of our minds : the power,
when a perception is present to our senses or a conception to our intellects, of
attending to a part only of that perception or conception instead of the whole. But
we cannot conceive a line without breadth : we can form no mental picture of such a
line ; all the lines which we have in our mind are lines possessing breadth. If anyone
doubt this, we may refer him to his own experience. I much question if anyone who
fancies that he can conceive of a mathematical line thinks so from the evidence of his
own consciousness. I suspect it is rather because he supposes that, unless such a
perception be possible, mathematics could not exist as a science : a supposition which
there will be no difficulty in showing to be groundless."
I think it may be at once conceded that the truths of geometry are truths
precisely because they relate to and express the properties of what Mill calls "purely
imaginary objects " ; that these objects do not exist in Mill's sense, that they do not
exist in nature, may also be granted ; that they are " not even possible," if this means
not possible in an existing nature, may also be granted. That we cannot " conceive "
them depends on the meaning which we attach to the word conceive. I would myself
say that the purely imaginary objects are the only realities, the 6W<»? ovra, in regard to
which the corresponding physical objects are as the shadows in the cave ; and it is only
by means of them that we are able to deny the existence of a corresponding physical
object ; if there is no conception of straightness, then it is meaningless to deny the
existence of a perfectly straight line.
But at any rate the objects of geometrical truth are the so-called imaginary
objects of Mill, and the truths of geometry are only true, and a fortiori are only
necessarily true, in regard to these so-called imaginary objects ; and these objects,
points, lines, circles, &c., in the mathematical sense of the terms, have a likeness to and
are represented more or less imperfectly, and from a geometer's point of view no matter
how imperfectly, by corresponding physical points, lines, circles, &c. I shall have to
return to geometry, and will then speak of Riemann, but I will first refer to another
passage of the Logic.
Speaking of the truths of arithmetic, Mill says (p. 297) that even here there is one
hypothetical element : " In all propositions concerning numbers a condition is implied with
out which none of them would be true, and that condition is an assumption which may be
false. The condition is that 1 = 1: that all the numbers are numbers of the same or of
equal units." Here at least the assumption may be absolutely true ; one shilling = one
shilling in purchasing power, although they may not be absolutely of the same weight
and fineness : but it is hardly necessary ; one coin + one coin = two coins, even if the one
be a shilling and the other a half-crown. In fact, whatever difficulty be raisable as to
geometry, it seems to me that no similar difficulty applies to arithmetic ; mathematician
or not, we have each of us, in its most abstract form, the idea of a number ; we can
each of us appreciate the truth of a proposition in regard to numbers ; and we cannot
but see that a truth in regard to numbers is something different in kind from an
C. xi. 55
434 PRESIDENTIAL ADDRESS TO THE [784
experimental truth generalised from experience. Compare, for instance, the proposition
that the sun, having already risen so many times, will rise to-morrow, and the next day,
and the day after that, and so on ; and the proposition that even and odd numbers
succeed each other alternately ad infinitum : the latter at least seems to have the
characters of universality and necessity. Or again, suppose a proposition observed to
hold good for a long series of numbers, one thousand numbers, two thousand numbers,
as the case may be : this is not only no proof, but it is absolutely no evidence, that
the proposition is a true proposition, holding good for all numbers whatever ; there are
in the Theory of Numbers very remarkable instances of propositions observed to hold
good for very long series of numbers and which are nevertheless untrue.
I pass in review certain mathematical theories.
In arithmetic and algebra, or say in analysis, the numbers or magnitudes which we
represent by symbols are in the first instance ordinary (that is, positive) numbers or
magnitudes. We have also in analysis and in analytical geometry negative magnitudes;
there has been in regard to these plenty of philosophical discussion, and I might refer
to Kant's paper, Ueber die negativen Grossen in die Weltweisheit (1763), but the notion
of a negative magnitude has become quite a familiar one, and has extended itself into
common phraseology. I may remark that it is used in a very refined manner in
bookkeeping by double entry.
But it is far otherwise with the notion which is really the fundamental one (and
I cannot too strongly emphasise the assertion) underlying and pervading the whole
of modern analysis and geometry, that of imaginary magnitude in analysis and of
imaginary space (or space as a locus in quo of imaginary points and figures) in
geometry : I use in each case the word imaginary as including real. This has not
been, so far as I am aware, a subject of philosophical discussion or enquiry. As
regards the older metaphysical writers this would be quite accounted for by saying
that they knew nothing, and were not bound to know anything, about it; but at
present, and, considering the prominent position which the notion occupies — say even
that the conclusion were that the notion belongs to mere technical mathematics, or
has reference to nonentities in regard to which no science is possible, still it seems to
me that (as a subject of philosophical discussion) the notion ought not to be thus
ignored ; it should at least be shown that there is a right to ignore it.
Although in logical order I should perhaps now speak of the notion just referred
to, it will be convenient to speak first of some other quasi-geometrical notions ; those
of more-than-three-dimensional space, and of non-Euclidian two- and three-dimensional
space, and also of the generalised notion of distance. It is in connexion with these
that Riemann considered that our notion of space is founded on experience, or rather
that it is only by experience that we know that our space is Euclidian space.
It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has
been considered as needing demonstration ; and that Lobatschewsky constructed a
perfectly consistent theory, wherein this axiom was assumed not to hold good, or say
a system of non-Euclidian plane geometry. There is a like system of non-Euclidian
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 435
solid geometry. My own view is that Euclid's twelfth axiom in Playfair's form of it
does not need demonstration, but is part of our notion of space, of the physical space
of our experience — the space, that is, which we become acquainted with by experience,
but which is the representation lying at the foundation of all external experience.
Riemann's view before referred to may I think be said to be that, having in intellectu
a more general notion of space (in fact a notion of non-Euclidian space), we learn
by experience that space (the physical space of our experience) is, if not exactly, at
least to the highest degree of approximation, Euclidian space.
But suppose the physical space of our experience to be thus only approximately
Euclidian space, what is the consequence which follows ? Not that the propositions of
geometry are only approximately true, but that they remain absolutely true in regard
to that Euclidian space which has been so long regarded as being the physical space
of our experience.
It is interesting to consider two different ways in which, without any modification
at all of our notion of space, we can arrive at a system of non-Euclidian (plane or
two-dimensional) geometry; and the doing so will, I think, throw some light on the
whole question.
First, imagine the earth a perfectly smooth sphere ; understand by a plane the
surface of the earth, and by a line the apparently straight line (in fact, an arc of
great circle) drawn on the surface ; what experience would in the first instance teach
would be Euclidian geometry ; there would be intersecting lines which produced a few
miles or so would seem to go on diverging : and apparently parallel lines which would
exhibit no tendency to approach each other; and the inhabitants might very well
conceive that they had by experience established the axiom that two straight lines
cannot enclose a space, and the axiom as to parallel lines. A more extended experience
and more accurate measurements would teach them that the axioms were each of them
false ; and that any two lines if produced far enough each way, would meet in two
points : they would in fact arrive at a spherical geometry, accurately representing the
properties of the two-dimensional space of their experience. But their original Euclidian
geometry would not the less be a true system : only it would apply to an ideal space,
not the space of their experience.
Secondly consider an ordinary, indefinitely extended plane ; and let us modify only
the notion of distance. We measure distance, say, by a yard measure or a foot rule,
anything which is short enough to make the fractions of it of no consequence (in
mathematical language, by an infinitesimal element of length) ; imagine, then, the length
of this rule constantly changing (as it might do by an alteration of temperature), but
under the condition that its actual length shall depend only on its situation on the
plane and on its direction : viz. if for a given situation and direction it has a certain
length, then whenever it comes back to the same situation and direction it must have
the same length. The distance along a given straight or curved line between any two
points could then be measured in the ordinary manner with this rule, and would have
a perfectly determinate value: it could be measured over and over again, and would
always be the same ; but of course it would be the distance, not in the ordinary
55—2
436 PRESIDENTIAL ADDRESS TO THE [784
acceptation of the term, but in quite a different acceptation. Or in a somewhat different
way : if the rate of progress from a given point in a given direction be conceived as
depending only on the configuration of the ground, and the distance along a given path
between any two points thereof be measured by the time required for traversing it, then
in this way also the distance would have a perfectly determinate value ; but it would be
a distance, not in the ordinary acceptation of the term, but in quite a different
acceptation. And corresponding to the new notion of distance we should have a new
non-Euclidian system of plane geometry; all theorems involving the notion of distance
would be altered.
We may proceed further. Suppose that as the rule moves away from a fixed
central point of the plane it becomes shorter and shorter ; if this shortening takes
place with sufficient rapidity, it may very well be that a distance which in the ordinary
sense of the word is finite will in the new sense be infinite ; no number of repetitions
of the length of the ever-shortening rule will be sufficient to cover it. There will be
surrounding the central point a certain finite area such that (in the new acceptation
of the term distance) each point of the boundary thereof will be at an infinite distance
from the central point ; the points outside this area you cannot by any means arrive at
with your rule ; they will form a terra incognita, or rather an unknowable land : in
mathematical language, an imaginary or impossible space : and the plane space of the
theory will be that within the finite area — that is, it will be finite instead of
infinite.
We thus with a proper law of shortening arrive at a system of non-Euclidian
geometry which is essentially that of Lobatschewsky. But in so obtaining it we put
out of sight its relation to spherical geometry: the three geometries (spherical, Euclidian,
and Lobatschewsky's) should be regarded as members of a system : viz. they are the
geometries of a plane (two-dimensional) space of constant positive curvature, zero
curvature, and constant negative curvature respectively ; or again, they are the plane
geometries corresponding to three different notions of distance ; in this point of view
they are Klein's elliptic, parabolic, and hyperbolic geometries respectively.
Next as regards solid geometry : we can by a modification of the notion of distance
(such as has just been explained in regard to Lobatschewsky's system) pass from our
present system to a non-Euclidian system ; for the other mode of passing to a non-
Euclidian system, it would be necessary to regard our space as a flat three-dimensional
space existing in a space of four dimensions (i.e., as the analogue of a plane existing in
ordinary space) ; and to substitute for such flat three-dimensional space a curved three-
dimensional space, say of constant positive or negative curvature. In regarding the
physical space of our experience as possibly non-Euclidian, Riemann's idea seems to be
that of modifying the notion of distance, not that of treating it as a locus in four-
dimensional space.
I have just come to speak of four-dimensional space. What meaning do we attach
to it ? Or can we attach to it any meaning ? It may be at once admitted that we
cannot conceive of a fourth dimension of space ; that space as we conceive of it, and
the physical space of our experience, are alike three-dimensional; but we can, I think,
conceive of space as being two- or even one-dimensional ; we can imagine rational
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 437
beings living in a one-dimensional space (a line) or in a two-dimensional space (a
surface), and conceiving of space accordingly, and to whom, therefore, a two-dimensional
space, or (as the case may be) a three-dimensional space would be as inconceivable
as a four-dimensional space is to us. And very curious speculative questions arise.
Suppose the one-dimensional space a right line, arid that it afterwards becomes a
curved line : would there be any indication of the change ? Or, if originally a curved
line, would there be anything to suggest to them that it was not a right line ?
Probably not, for a one-dimensional geometry hardly exists. But let the space be
two-dimensional, and imagine it originally a plane, and afterwards bent or converted
into a curved surface (converted, that is, into some form of developable surface) :
or imagine it originally a developable or curved surface. In the former case there
should be an indication of the change, for the geometry originally applicable to the
space of their experience (our own Euclidian geometry) would cease to be applicable ;
but the change could not be apprehended by them as a bending or deformation of
the plane, for this would imply the notion of a three-dimensional space in which
this bending or deformation could take place. In the latter case their geometry
would be that appropriate to the developable or curved surface which is their space :
viz. this would be their Euclidian geometry : would they ever have arrived at our
own more simple system ? But take the case where the two-dimensional space is a
plane, and imagine the beings of such a space familiar with our own Euclidian plane
geometry; if, a third dimension being still inconceivable by them, they were by their
geometry or otherwise led to the notion of it, there would be nothing to prevent
them from forming a science such as our own science of three-dimensional geometry.
Evidently all the foregoing questions present themselves in regard to ourselves,
and to three-dimensional space as we conceive of it, and as the physical space of
our experience. And I need hardly say that the first step is the difficulty, and that
granting a fourth dimension we may assume as many more dimensions as we please.
But whatever answer be given to them, we have, as a branch of mathematics,
potentially, if not actually, an analytical geometry of w-dimensional space. I shall have
to speak again upon this.
Coming now to the fundamental notion already referred to, that of imaginary
magnitude in analysis and imaginary space in geometry : I connect this with two
great discoveries in mathematics made in the first half of the seventeenth century,
Harriot's representation of an equation in the form /(#) = 0, and the consequent
notion of the roots of an equation as derived from the linear factors of f ' (x),
(Harriot, 1560 — 1621 : his Algebra, published after his death, has the date 1631), and
Descartes' method of coordinates, as given in the Geom&rie, forming a short supplement
to his Traite de la Methode, etc., (Leyden, 1637).
Taking the coefficients of an equation to be real magnitudes, it at once follows
from Harriot's form of an equation that an equation of the order n ought to have
n roots. But it is by no means true that there are always n real roots. In particular,
an equation of the second order, or quadric equation, may have no real root ; but
if we assume the existence of a root i of the quadric equation s? + 1 = 0, then the
438 PRESIDENTIAL ADDRESS TO THE [784
other root is = — i; and it is easily seen that every quadric equation (with real
coefficients as before) has two roots, a ± bi, where a and b are real magnitudes. We
are thus led to the conception of an imaginary magnitude, a + bi, where a and b are
real magnitudes, each susceptible of any positive or negative value, zero included. The
general theorem is that, taking the coefficients of the equation to be imaginary magni
tudes, then an equation of the order n has always n roots, each of them an imaginary
magnitude, and it thus appears that the foregoing form a + bi of imaginary magnitude
is the only one that presents itself. Such imaginary magnitudes may be added or
multiplied together or dealt with in any manner: the result is always a like imaginary
magnitude. They are thus the magnitudes which are considered in analysis, and
analysis is the science of such magnitudes. Observe the leading character that the
imaginary magnitude a + bi is a magnitude composed of the two real magnitudes a and
b (in the case 6 = 0 it is the real magnitude a, and in the case a = 0 it is the pure
imaginary magnitude bi). The idea is that of considering, in place of real magnitudes,
these imaginary or complex magnitudes a + bi.
In the Cartesian geometry a curve is determined by means of the equation
existing between the coordinates (so, y) of any point thereof. In the case of a right
line, this equation is linear; in the case of a circle, or more generally of a conic, the
equation is of the second order; and generally, when the equation is of the order n,
the curve which it represents is said to be a curve of the order n. In the case of
two given curves, there are thus two equations satisfied by the coordinates (x, y) of the
several points of intersection, and these give rise to an equation of a certain order for
the coordinate x or y of a point of intersection. In the case of a straight line and a
circle, this is a quadric equation; it has two roots, real or imaginary. There are thus
two values, say of x, and to each of these corresponds a single value of y. There are
therefore two points of intersection — viz. a straight line and a circle intersect always
in two points, real or imaginary. It is in this way that we are led analytically to the
notion of imaginary points in geometry. The conclusion as to the two points of
intersection cannot be contradicted by experience : take a sheet of paper and draw
on it the straight line and circle, and try. But you might say, or at least be strongly
tempted to say, that it is meaningless. The question of course arises, What is the
meaning of an imaginary point ? and further, In what manner can the notion be
arrived at geometrically ?
There is a well-known construction in perspective for drawing lines through the
intersection of two lines, which are so nearly parallel as not to meet within the limits
of the sheet of paper. You have two given lines which do not meet, and you draw
a third line, which, when the lines are all of them produced, is found to pass through
the intersection of the given lines. If instead of lines we have two circular arcs not
meeting each other, then we can, by means of these arcs, construct a line ; and if
on completing the circles it is found that the circles intersect each other in two real
points, then it will be found that the line passes through these two points : if the
circles appear not to intersect, then the line will appear not to intersect either of the
circles. But the geometrical construction being in each case the same, we say that
in the second case also the line passes through the two intersections of the circles.
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 439
Of course it may be said in reply that the conclusion is a very natural one,
provided we assume the existence of imaginary points ; and that, this assumption not
being made, then, if the circles do not intersect, it is meaningless to assert that the
line passes through their points of intersection. The difficulty is not got over by
the analytical method before referred to, for this introduces difficulties of its own : is
there in a plane a point the coordinates of which have given imaginary values ? As
a matter of fact, we do consider in plane geometry imaginary points introduced into
the theory analytically or geometrically as above.
The like considerations apply to solid geometry, and we thus arrive at the notion
of imaginary space as a locus in quo of imaginary points and figures.
I have used the word imaginary rather than complex, and I repeat that the
word has been used as including real. But, this once understood, the word becomes
in many cases superfluous, and the use of it would even be misleading. Thus, " a
problem has so many solutions " : this means, so many imaginary (including real)
solutions. But if it were said that the problem had "so many imaginary solutions,"
the word " imaginary " would here be understood to be used in opposition to real. I
give this explanation the better to point out how wide the application of the notion
of the imaginary is — viz. (unless expressly or by implication excluded), it is a notion
implied and presupposed in all the conclusions of modern analysis and geometry. It
is, as I have said, the fundamental notion underlying and pervading the whole of
these branches of mathematical science.
I shall speak later on of the great extension which is thereby given to geometry,
but I wish now to consider the effect as regards the theory of a function. In the
original point of view, and for the original purposes, a function, algebraic or transcen
dental, such as \lx, since, or log#, was considered as known, when the value was known
for every real value (positive or negative) of the argument ; or if for any such values
the value of the function became imaginary, then it was enough to know that for
such values of the argument there was no real value of the function. But now this
is not enough, and to know the function means to know its value — of course, in
general, an imaginary value X + iY, — for every imaginary value x + iy whatever of the
argument.
And this leads naturally to the question of the geometrical representation of an
imaginary variable. We represent the imaginary variable x + iy by means of a point
in a plane, the coordinates of which are (x, y). This idea, due to Gauss, dates from
about the year 1831. We thus picture to ourselves the succession of values of the
imaginary variable x + iy by means of the motion of the representative point: for
instance, the succession of values corresponding to the motion of the point along a
closed curve to its original position. The value X + iY of the function can of course
be represented by means of a point (taken for greater convenience in a different
plane), the coordinates of which are X, Y.
We may consider in general two points, moving each in its own plane, so that
the position of one of them determines the position of the other, and consequently
440 PRESIDENTIAL ADDRESS TO THE [784
the motion of the one determines the motion of the other: for instance, the two points
may be the tracing-point and the pencil of a pentagraph. You may with the first
point draw any figure you please, there will be a corresponding figure drawn by the
second point : for a good pentagraph, a copy on a different scale (it may be) ; for a
badly-adjusted pentagraph, a distorted copy: but the one figure will always be a sort
of copy of the first, so that to each point of the one figure there will correspond a
point of the other figure.
In the case above referred to, where one point represents the value x + iy of the
imaginary variable and the other the value X + iY of some function $(x + iy) of that
variable, there is a remarkable relation between the two figures : this is the relation of
orthomorphic projection, the same which presents itself between a portion of the earth's
surface, and the representation thereof by a map on the stereographic projection or on
Mercator's projection — viz. any indefinitely small area of the one figure is represented in
the other figure by an indefinitely small area of the same shape. There will possibly be
for different parts of the figure great variations of scale, but the shape will be unaltered;
if for the one area the boundary is a circle, then for the other area the boundary will
be a circle; if for one it is an equilateral triangle, then for the other it will be an
equilateral triangle.
I have for simplicity assumed that to each point of either figure there corresponds
one, and only one, point of the other figure ; but the general case is that to each point
of either figure there corresponds a determinate number of points in the other figure ;
and we have thence arising new and very complicated relations which I must just refer
to. Suppose that to each point of the first figure there correspond in the second figure
two points : say one of them is a red point, the other a blue point ; so that, speaking
roughly, the second figure consists of two copies of the first figure, a red copy and a
blue copy, the one superimposed on the other. But the difficulty is that the two copies
cannot be kept distinct from each other. If we consider in the first figure a closed
curve of any kind — say, for shortness, an oval — this will be in the second figure
represented in some cases by a red oval and a blue oval, but in other cases by an oval
half red and half blue ; or, what comes to the same thing, if in the first figure we
consider a point which moves continuously in any manner, at last returning to its
original position, and attempt to follow the corresponding points in the second figure,
then it may very well happen that, for the corresponding point of either colour, there
will be abrupt changes of position, or say jumps, from one position to another; so
that, to obtain in the second figure a continuous path, we must at intervals allow
the point to change from red to blue, or from blue to red. There are in the first
figure certain critical points called branch-points (Verzweigungspunkte), and a system
of lines connecting these, by means of which the colours in the second figure are
determined ; but it is not possible for me to go further into the theory at present.
The notion of colour has of course been introduced only for facility of expression ; it
may be proper to add that in speaking of the two figures I have been following Briot
and Bouquet rather than Riemann, whose representation of the function of an
imaginary variable is a different one.
I have been speaking of an imaginary variable (x + iy), and of a function
(j) (a + iy) =* X + iY of that variable, but the theory may equally well be stated in
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 441
regard to a plane curve: in fact, the x + iy and the X + iY are two imaginary
variables connected by an equation ; say their values are u and v, connected by an
equation F (u, v) = 0 ; then, regarding u, v as the coordinates of a point in piano, this
will be a point on the curve represented by the equation. The curve, in the widest
sense of the expression, is the whole series of points, real or imaginary, the coordinates
of which satisfy the equation, and these are exhibited by the foregoing corresponding
figures in two planes ; but in the ordinary sense the curve is the series of real points,
with coordinates u, v, which satisfy the equation.
In geometry it is the curve, whether denned by means of its equation, or in any
other manner, which is the subject for contemplation and study. But we also use the
curve as a representation of its equation — that is, of the relation existing between two
magnitudes x, y, which are taken as the coordinates of a point on the curve. Such
employment of a curve for all sorts of purposes — the fluctuations of the barometer, the
Cambridge boat races, or the Funds — is familiar to most of you. It is in like manner
convenient in analysis, for exhibiting the relations between any three magnitudes x, y, z,
to regard them as the coordinates of a point in space ; and, on the like ground, we
should at least wish to regard any four or more magnitudes as the coordinates of a
point in space of a corresponding number of dimensions. Starting with the hypothesis
of such a space, and of points therein each determined by means of its coordinates, it is
found possible to establish a system of n-ditnensional geometry analogous in every respect
to our two- and three-dimensional geometries, and to a very considerable extent serving
to exhibit the relations of the variables. To quote from my memoir " On Abstract
Geometry" (1869), [413]: "The science presents itself in two ways: as a legitimate
extension of the ordinary two- and three-dimensional geometries, and as a need in these
geometries and in analysis generally. In fact, whenever we are concerned with quantities
connected in any manner, and which are considered as variable or determinable, then the
nature of the connexion between the quantities is frequently rendered more intelligible by
regarding them (if two or three in number) as the coordinates of a point in a plane or
in space. For more than three quantities there is, from the greater complexity of the
case, the greater need of such a representation ; but this can only be obtained by means
of the notion of a space of the proper dimensionality ; and to use such representation we
require a corresponding geometry. An important instance in plane geometry has already
presented itself in the question of the number of curves which satisfy given conditions ;
the conditions imply relations between the coefficients in the equation of the curve ; and
for the better understanding of these relations it was expedient to consider the coefficients
as the coordinates of a point in a space of the proper dimensionality."
It is to be borne in mind that the space, whatever its dimensionality may be, must
always be regarded as an imaginary or complex space such as the two- or three-dimen
sional space of ordinary geometry ; the advantages of the representation would otherwise
altogether fail to be obtained.
I have spoken throughout of Cartesian coordinates ; instead of these, it is in plane
geometry not unusual to employ trilinear coordinates, and these may be regarded as
absolutely undetermined in their magnitude — viz. we may take x, y, z to be, not equal,
c. xi. 56
442 PRESIDENTIAL ADDRESS TO THE [784
but only proportional to the distances of a point from three given lines; the ratios of
the coordinates (x, y, z) determine the point ; and so in one-dimensional geometry, we
may have a point determined by the ratio of its two coordinates x, y, these coordinates
being proportional to the distances of the point from two fixed points; and generally in
n-dimensional geometry a point will be determined by the ratios of the (n+l) coordinates
(x, y, z, ...). The corresponding analytical change is in the expression of the original
magnitudes as fractions with a common denominator ; we thus, in place of rational and
integral non-homogeneous functions of the original variables, introduce rational and
integral homogeneous functions (quantics) of the next succeeding number of variables —
viz. we have binary quantics corresponding to one-dimensional geometry, ternary to two-
dimensional geometry, and so on.
It is a digression, but I wish to speak of the representation of points or figures in
space upon a plane. In perspective, we represent a point in space by means of the
intersection with the plane of the picture (suppose a pane of glass) of the line drawn
from the point to the eye, and doing this for each point of the object we obtain a
representation or picture of the object. But such representation is an imperfect one, as
not determining the object : we cannot by means of the picture alone find out the form
of the object ; in fact, for a given point of the picture the corresponding point of the
object is not a determinate point, but it is a point anywhere in the line joining the eye
with the point of the picture. To determine the object we need two pictures, such as
we have in a plan and elevation, or, what is the same thing, in a representation on the
system of Monge's descriptive geometry. But it is theoretically more simple to consider
two projections on the same plane, with different positions of the eye : the point in space
is here represented on the plane by means of two points which are such that the line
joining them passes through a fixed point of the plane (this point is in fact the
intersection with the plane of the picture of the line joining the two positions of the
eye) ; the figure in space is thus represented on the plane by two figures, which are
such that the lines joining corresponding points of the two figures pass always through
the fixed point. And such two figures completely replace the figure in space ; we can by
means of them perform on the plane any constructions which could be performed on the
figure in space, and employ them in the demonstration of properties relating to such
figure. A curious extension has recently been made : two figures in space such that the
lines joining corresponding points pass through a fixed point have been regarded by the
Italian geometer Veronese as representations of a figure in four-dimensional space, and
have been used for the demonstration of properties of such figure.
I referred to the connexion of Mathematics with the notions of space and time, but
I have hardly spoken of time. It is, I believe, usually considered that the notion of
number is derived from that of time ; thus Whewell in the work referred to, p. xx, says
number is a modification of the conception of repetition, which belongs to that of time.
I cannot recognise that this is so: it seems to me that we have (independently, I
should say, of space or time, and in any case not more depending on time than on space)
the notion of plurality ; we think of, say, the letters a, b, c, &c., and thence in the case
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 443
of a finite set — for instance a, b, c, d, e — we arrive at the notion of number; coordinating
them one by one with any other set of things, or, suppose, with the words first, second,
&c., we find that the last of them goes with the word fifth, and we say that the number
of things is = five : the notion of cardinal number would thus appear to be derived from
that of ordinal number.
Questions of combination and arrangement present themselves, and it might be
possible from the mere notion of plurality to develope a branch of mathematical
science: this, however, would apparently be of a very limited extent, and it is difficult
not to introduce into it the notion of number; in fact, in the case of a finite set of
things, to avoid asking the question, How many ? If we do this, we have a large
enough subject, including the partition of numbers, which Sylvester has called Tactic.
From the notion thus arrived at of an integer number, we pass to that of a
fractional number, and we see how by means of these the ratio of any two concrete
magnitudes of the same kind can be expressed, not with absolute accuracy, but with
any degree of accuracy we please : for instance, a length is so many feet, tenths of a
foot, hundredths, thousandths, &c. ; subdivide as you please, non constat that the length
can be expressed accurately, we have in fact incommensurables ; as to the part which
these play in the Theory of Numbers, I shall have to speak presently : for the moment
I am only concerned with them in so far as they show that we cannot from the notion
of number pass to that which is required in analysis, the notion of an abstract (real and
positive) magnitude susceptible of continuous variation. The difficulty is got over by a
Postulate. We consider an abstract (real and positive) magnitude, and regard it as
susceptible of continuous variation, without in anywise concerning ourselves about the
actual expression of the magnitude by a numerical fraction or otherwise.
There is an interesting paper by Sir W. R. Hamilton, " Theory of Conjugate
Functions, or Algebraical Couples : with a preliminary and elementary Essay on Algebra
as the Science of Pure Time," 1833 — 35 (Trans. R. I. Acad. t. xvil.), in which, as
appears by the title, he purposes to show that algebra is the science of pure time.
He states there, in the General Introductory Remarks, his conclusions : first, that the
notion of time is connected with existing algebra ; second, that this notion or intuition
of time may be unfolded into an independent pure science ; and, third, that the science
of pure time thus unfolded is coextensive and identical with algebra, so far as algebra
itself is a science ; and to sustain his first conclusion he remarks that " the history
of algebraic science shows that the most remarkable discoveries in it have been made
either expressly through the notion of time, or through the closely connected (and in
some sort coincident) notion of continuous progression. It is the genius of algebra to
consider what it reasons upon as flowing, as it was the genius of geometry to consider
what it reasoned on as fixed. . . . And generally the revolution which Newton made in
the higher parts of both pure and applied algebra was founded mainly on the notion of
fluxion, which involves the notion of time." Hamilton uses the term algebra in a very
wide sense, but whatever else he includes under it, he includes all that in contra
distinction to the Differential Calculus would be called algebra. Using the word in this
restricted sense, I cannot myself recognise the connexion of algebra with the notion of
time : granting that the notion of continuous progression presents itself, and is of
56—2
444 PRESIDENTIAL ADDRESS TO THE [784
importance, I do not see that it is in anywise the fundamental notion of the science.
And still less can I appreciate the manner in which the author connects with the
notion of time his algebraical couple, or imaginary magnitude a + bi (a + b V — 1, as
written in the memoir).
I would go further: the notion of continuous variation is a very fundamental one,
made a foundation in the Calculus of Fluxions (if not always so in the Differential
Calculus) and presenting itself or implied throughout in mathematics : and it may be
said that a change of any kind takes place only in time ; it seems to me, however, that
the changes which we consider in mathematics are for the most part considered quite
irrespectively of time.
It appears to me that we do not have in Mathematics the notion of time until
we bring it there : and that even in kinematics (the science of motion) we have very
little to do with it ; the motion is a hypothetical one ; if the system be regarded as
actually moving, the rate of motion is altogether undetermined and immaterial. The
relative rates of motion of the different points of the system are nothing else than the
ratios of purely geometrical quantities, the indefinitely short distances simultaneously
described, or which might be simultaneously described, by these points respectively.
But whether the notion of time does or does not sooner enter into mathematics, we at
any rate have the notion in Mechanics, and along with it several other new notions.
Regarding Mechanics as divided into Statics and Dynamics, we have in dynamics
the notion of time, and in connexion with it that of velocity : we have in statics and
dynamics the notion of force ; and also a notion which in its most general form I
would call that of corpus : viz. this may be, the material point or particle, the flexible
inextensible string or surface, or the rigid body, of ordinary mechanics ; the incompressible
perfect fluid of hydrostatics and hydrodynamics ; the ether of any undulatory theory ; or
any other imaginable corpus ; for instance, one really deserving of consideration in any
general treatise of mechanics is a developable or skew surface with absolutely rigid
generating lines, but which can be bent about these generating lines, so that the element
of surface between two consecutive lines rotates as a whole about one of them. We have
besides, in dynamics necessarily, the notion of mass or inertia.
We seem to be thus passing out of pure mathematics into physical science ; but it
is difficult to draw the line of separation, or to say of large portions of the Principia,
and the Mecanique celeste, or of the whole of the Mecanique analytique, that they are
not pure mathematics. It may be contended that we first come to physics when we
attempt to make out the character of the corpus as it exists in nature. I do not at
present speak of any physical theories which cannot be brought under the foregoing
conception of mechanics.
I must return to the Theory of Numbers ; the fundamental idea is here integer
number: in the first instance positive integer number, but which may be extended to
include negative integer number and zero. We have the notion of a product, and that
of a prime number, which is not a product of other numbers ; and thence also that of a
number as the product of a determinate system of prime factors. We have here the
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 445
elements of a theory in many respects analogous to algebra: an equation is to be
solved — that is, we have to find the integer values (if any) which satisfy the equation ;
and so in other cases : the congruence notation, although of the very highest importance,
does not affect the character of the theory.
But as already noticed we have incommensurables, and the consideration of these
gives rise to a new universe of theory. We may take into consideration any surd
number such as \/2, and so consider numbers of the form a + b \/2, («, and b any positive
or negative integer numbers not excluding zero) ; calling these integer numbers, every
problem which before presented itself in regard to integer numbers in the original and
ordinary sense of the word presents itself equally in regard to integer numbers in this
new sense of the word ; of course all definitions must be altered accordingly : an ordinary
integer, which is in the ordinary sense of the word a prime number, may very well be
the product of two integers of the form a+W2, and consequently not a prime number
in the new sense of the word. Among the incommensurables which can be thus
introduced into the Theory of Numbers (and which was in fact first so introduced) we
have the imaginary i of ordinary analysis : viz. we may consider numbers a 4- bi (a and b
ordinary positive or negative integers, not excluding zero), and, calling these integer
numbers, establish in regard to them a theory analogous to that which exists for
ordinary real integers. The point which I wish to bring out is that the imaginary i
does not in the Theory of Numbers occupy a unique position, such as it does in analysis
and geometry ; it is in the Theory of Numbers one out of an indefinite multitude of
incommensurables.
I said that I would speak to you, not of the utility of mathematics in any of
the questions of common life or of physical science, but rather of the obligations of
mathematics to these different subjects. The consideration which thus presents itself is
in a great measure that of the history of the development of the different branches
of mathematical science in connexion with the older physical sciences, Astronomy and
Mechanics: the mathematical theory is in the first instance suggested by some question
of common life or of physical science, is pursued and studied quite independently thereof,
and perhaps after a long interval comes in contact with it, or with quite a different
question. Geometry and algebra must, I think, be considered as each of them originating
in connexion with objects or questions of common life — geometry, notwithstanding its
name, hardly in the measurement of land, but rather from the contemplation of such
forms as the straight line, the circle, the ball, the top (or sugar-loaf) : the Greek
geometers appropriated for the geometrical forms corresponding to the last two of these,
the words a-fyalpa and KWVOS, our sphere and cone, and they extended the word cone
to mean the complete figure obtained by producing the straight lines of the surface
both ways indefinitely. And so algebra would seem to have arisen from the sort of easy
puzzles in regard to numbers which may be made, either in the picturesque forms of
the Bija-Ganita with its maiden with the beautiful locks, and its swarms of bees amid
the fragrant blossoms, and the one queen-bee left humming around the lotus flower;
or in the more prosaic form in which a student has presented to him in a modern
text-book a problem leading to a simple equation.
446 PRESIDENTIAL ADDRESS TO THE [784
The Greek geometry may be regarded as beginning with Plato (B.C. 430 — 347) :
the notions of geometrical analysis, loci, and the conic sections are attributed to him,
and there are in his Dialogues many very interesting allusions to mathematical
questions : in particular the passage in the Thecetetus, where he affirms the incommen
surability of the sides of certain squares. But the earliest extant writings are those
of Euclid (B.C. 285) : there is hardly anything in mathematics more beautiful than
his wondrous fifth book ; and he has also in the seventh, eighth, ninth and tenth
books fully and ably developed the first principles of the Theory of Numbers, including
the theory of incommensurables. We have next Apollonius (about B.C. 247), and
Archimedes (B.C. 287 — 212), both geometers of the highest merit, and the latter of
them the founder of the science of statics (including therein hydrostatics) : his dictum
about the lever, his " Evprjfca," and the story of the defence of Syracuse, are well
known. Following these we have a worthy series of names, including the astronomers
Hipparchus (B.C. 150) and Ptolemy (A.D. 125), and ending, say, with Pappus (A.D. 400),
but continued by their Arabian commentators, and the Italian and other European
geometers of the sixteenth century and later, who pursued the Greek geometry.
The Greek arithmetic was, from the want of a proper notation, singularly
cumbrous and difficult ; and it was for astronomical purposes superseded by the
sexagesimal arithmetic, attributed to Ptolemy, but probably known before his time.
The use of the present so-called Arabic figures became general among Arabian
writers on arithmetic and astronomy about the middle of the tenth century, but was
not introduced into Europe until about two centuries later. Algebra among the Greeks
is represented almost exclusively by the treatise of Diophantus (A.D. 150), in fact a
work on the Theory of Numbers containing questions relating to square and cube
numbers, and other properties of numbers, with their solutions ; this has no historical
connexion with the later algebra, introduced into Italy from the East by Leonardi
Bonacci of Pisa (A.D. 1202 — 1208) and successfully cultivated in the fifteenth and
sixteenth centuries by Lucas Paciolus, or de Burgo, Tartaglia, Cardan, and Ferrari.
Later on, we have Vieta (1540 — 1603), Harriot, already referred to, Wallis, and others.
Astronomy is of course intimately connected with geometry ; the most simple facts
of observation of the heavenly bodies can only be stated in geometrical language : for
instance, that the stars describe circles about the pole-star, or that the different
positions of the sun among the fixed stars in the course of the year form a circle.
For astronomical calculations it was found necessary to determine the arc of a circle
by means of its chord : the notion is as old as Hipparchus, a work of whom is referred
to as consisting of twelve books on the chords of circular arcs ; we have (A.D. 125)
Ptolemy's Almagest, the first book of which contains a table of arcs and chords with
the method of construction ; and among other theorems on the subject he gives there
the theorem afterwards inserted in Euclid (Book VI. Prop. D) relating to the rectangle
contained by the diagonals of a quadrilateral inscribed in a circle. The Arabians made
the improvement of using in place of the chord of an arc the sine, or half chord, of
double the arc ; and so brought the theory into the form in which it is used in modern
trigonometry : the before-mentioned theorem of Ptolemy, or rather a particular case of
it, translated into the notation of sines, gives the expression for the sine of the sum
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 447
of two arcs in terms of the sines and cosines of the component arcs ; and it is thus
the fundamental theorem on the subject. We have in the fifteenth and sixteenth
centuries a series of mathematicians who with wonderful enthusiasm and perseverance
calculated tables of the trigonometrical or circular functions, Purbach, Miiller or
Regiomontanus, Copernicus, Reinhold, Maurolycus, Vieta, and many others ; the
tabulations of the functions tangent and secant are due to Reinhold and Maurolycus
respectively.
Logarithms were invented, not exclusively with reference to the calculation of
trigonometrical tables, but in order to facilitate numerical calculations generally ; the
invention is due to John Napier of Merchiston, who died in 1618 at 67 years of age ;
the notion was based upon refined mathematical reasoning on the comparison of the
spaces described by two points, the one moving with a uniform velocity, the other with
a velocity varying according to a given law. It is to be observed that Napier's
logarithms were nearly but not exactly those which are now called (sometimes Napierian,
but more usually) hyperbolic logarithms — those to the base e; and that the change to
the base 10 (the great step by which the invention was perfected for the object in view)
was indicated by Napier but actually made by Henry Briggs, afterwards Savilian
Professor at Oxford (d. 1630). But it is the hyperbolic logarithm which is mathematically
important. The direct function ex or exp. as, which has for its inverse the hyperbolic
logarithm, presented itself, but not in a prominent way. Tables were calculated of the
logarithms of numbers, and of those of the trigonometrical functions.
The circular functions and the logarithm were thus invented each for a practical
purpose, separately and without any proper connexion with each other. The functions
are connected through the theory of imaginaries and form together a group of the utmost
importance throughout mathematics : but this is mathematical theory ; the obligation
of mathematics is for the discovery of the functions.
Forms of spirals presented themselves in Greek architecture, and the curves were
considered mathematically by Archimedes; the Greek geometers invented some other
curves, more or less interesting, but recondite enough in their origin. A curve which
might have presented itself to anybody, that described by a point in the circumference
of a rolling carriage-wheel, was first noticed by Mersenne in 1615, and is the curve
afterwards considered by Roberval, Pascal, and others under the name of the Roulette,
otherwise the Cycloid. Pascal (1623 — 1662) wrote at the age of seventeen his Essais
pour les Coniques in seven short pages, full of new views on these curves, and in
which he gives, in a paragraph of eight lines, his theorem of the inscribed hexagon.
Kepler (1571 — 1630) by his empirical determination of the laws of planetary
motion, brought into connexion with astronomy one of the forms of conic, the ellipse,
and established a foundation for the theory of gravitation. Contemporary with him for
most of his life, we have Galileo (1564 — 1642), the founder of the science of dynamics;
and closely following upon Galileo we have Isaac Newton (1643 — 1727) : the Philosophic
naturalis Principia Mathematica known as the Principia was first published in 1687.
The physical, statical, or dynamical questions which presented themselves before
the publication of the Principia were of no particular mathematical difficulty, but it
448 PRESIDENTIAL ADDRESS TO THE [784
is quite otherwise with the crowd of interesting questions arising out of the theory
of gravitation, and which, in becoming the subject of mathematical investigation, have
contributed very much to the advance of mathematics. We have the problem of two
bodies, or what is the same thing, that of the motion of a particle about a fixed
centre of force, for any law of force ; we have also the (mathematically very interesting)
problem of the motion of a body attracted to two or more fixed centres of force ;
then, next preceding that of the actual solar system — the problem of three bodies ;
this has ever been and is far beyond the power of mathematics, and it is in the
lunar and planetary theories replaced by what is mathematically a different problem,
that of the motion of a body under the action of a principal central force and a
disturbing force : or (in one mode of treatment) by the problem of disturbed elliptic
motion. I would remark that we have here an instance in which an astronomical
fact, the observed slow variation of the orbit of a planet, has directly suggested a
mathematical method, applied to other dynamical problems, and which is the basis of
very extensive modern investigations in regard to systems of differential equations.
Again, immediately arising out of the theory of gravitation, we have the problem of
finding the attraction of a solid body of any given form upon a particle, solved by
Newton in the case of a homogeneous sphere, but which is far more difficult in the
next succeeding cases of the spheroid of revolution (very ably treated by Maclaurin)
and of the ellipsoid of three unequal axes : there is perhaps no problem of mathe
matics which has been treated by as great a variety of methods, or has given rise to
so much interesting investigation as this last problem of the attraction of an ellipsoid
upon an interior or exterior point. It was a dynamical problem, that of vibrating
strings, by which Lagrange was led to the theory of the representation of a function
as the sum of a series of multiple sines and cosines ; and connected with this we
have the expansions in terms of Legendre's functions Pn, suggested to him by the
question just referred to of the attraction of an ellipsoid ; the subsequent investigations
of Laplace on the attractions of bodies differing slightly from the sphere led to the
functions of two variables called Laplace's functions. I have been speaking of ellipsoids,
but the general theory is that of attractions, which has become a very wide branch
of modern mathematics ; associated with it we have in particular the names of Gauss,
Lejeune-Dirichlet, and Green ; and I must not omit to mention that the theory is now
one relating to w-dimensional space. Another great problem of celestial mechanics, that
of the motion of the earth about its centre of gravity, in the most simple case, that
of a body not acted upon by any forces, is a very interesting one in the mathematical
point of view.
I may mention a few other instances where a practical or physical question has
connected itself with the development of mathematical theory. I have spoken of two
map projections — the stereographic, dating from Ptolemy ; and Mercator's projection,
invented by Edward Wright about the year 1600: each of these, as a particular case
of the orthomorphic projection, belongs to the theory of the geometrical representation
of an imaginary variable. I have spoken also of perspective, and of the representation
of solid figures employed in Monge's descriptive geometry. Monge, it is well known, is
the author of the geometrical theory of the curvature of surfaces and of curves of
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 449
curvature : he was led to this theory by a problem of earthwork ; from a given area,
covered with earth of uniform thickness, to carry the earth and distribute it over an
equal given area, with the least amount of cartage. For the solution of the
corresponding problem in solid geometry he had to consider the intersecting normals
of a surface, and so arrived at the curves of curvature. (See his " Memoire sur les
Deblais et les Remblais," Mem. de I'Acad., 1781.) The normals of a surface are, again,
a particular case of a doubly infinite system of lines, and are so connected with the
modern theories of congruences and complexes.
The undulatory theory of light led to Fresnel's wave-surface, a surface of the
fourth order, by far the most interesting one which had then presented itself. A
geometrical property of this surface, that of having tangent planes each touching it
along a plane curve (in fact, a circle), gave to Sir W. R. Hamilton the theory of
conical refraction. The wave-surface is now regarded in geometry as a particular case
of Kummer's quartic surface, with sixteen conical points and sixteen singular tangent
planes.
My imperfect acquaintance as well with the mathematics as the physics prevents
me from speaking of the benefits which the theory of Partial Differential Equations
has received from the hydrodynamical theory of vortex motion, and from the great
physical theories of heat, electricity, magnetism, and energy.
It is difficult to give an idea of the vast extent of modern mathematics. This
word " extent " is not the right one : I mean extent crowded with beautiful detail —
not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful
country seen at first in the distance, but which will bear to be rambled through and
studied in every detail of hillside and valley, stream, rock, wood, and flower. But, as
for anything else, so for a mathematical theory — beauty can be perceived, but not
explained. As for mere extent, I can perhaps best illustrate this by speaking of the
dates at which some of the great extensions have been made in several branches of
mathematical science.
As regards geometry, I have already spoken of the invention of the Cartesian
coordinates (1637). This gave to geometers the whole series of geometric curves of
higher order than the conic sections : curves of the third order, or cubic curves ; curves
of the fourth order, or quartic curves; and so on indefinitely. The first fruits of it
were Newton's Enumeratio linearum tertii ordinis, and the extremely interesting
investigations of Maclaurin as to corresponding points on a cubic curve. This was at
once enough to show that the new theory of cubic curves was a theory quite as
beautiful and far more extensive than that of conies. And I must here refer to
Euler's remark in the paper " Sur une contradiction apparente dans la the'orie des
courbes planes" (Berlin Memoirs, 1748), in regard to the nine points of intersection
of two cubic curves (viz. that when eight of the points are given the ninth point is
thereby completely determined): this is not only a fundamental theorem in cubic curves
(including in itself Pascal's theorem of the hexagon inscribed in a conic), but it
introduces into plane geometry a new notion — that of the point-system, or system of
the points of intersection of two curves.
c. xi. 57
450 PRESIDENTIAL ADDRESS TO THE [784
A theory derived from the conic, that of polar reciprocals, led to the general
notion of geometrical duality — viz. that in plane geometry the point and the line are
correlative figures ; and founded on this we have Pllicker's great work, the Theorie der
algebraischen Curven (Bonn, 1839), in which he establishes the relation which exists
between the order and class of a curve and the number of its different point- and
line-singularities (Pllicker's six equations). It thus appears that the true division of
curves is not a division according to order only, but according to order and class, and
that the curves of a given order and class are again to be divided into families
according to their singularities : this is not a mere subdivision, but is really a widening
of the field of investigation; each such family of curves is in itself a subject as wide
as the totality of the curves of a given order might previously have appeared.
We unite families by considering together the curves of a given Geschlecht, or
deficiency ; and in reference to what I shall have to say on the Abelian functions,
I must speak of this notion introduced into geometry by Riemann in the memoir
" Theorie der Abel'schen Functionen," Crelle, t. Liv. (1857). For a curve of a given order,
reckoning cusps as double points, the deficiency is equal to the greatest number
^(n— l)(n — 2) of the double points which a curve of that order can have, less the
number of double points which the curve actually has. Thus a conic, a cubic with
one double point, a quartic with three double points, &c., are all curves of the
deficiency 0 ; the general cubic is a curve, and the most simple curve, of the
deficiency 1 ; the general quartic is a curve of deficiency 3 ; and so on. The deficiency
is usually represented by the letter p. Riemann considers the general question of the
rational transformation of a plane curve : viz. here the coordinates, assumed to be
homogeneous or trilinear, are replaced by any rational and integral functions, homo
geneous of the same degree in the new coordinates ; the transformed curve is in
general a curve of a different order, with its own system of double points ; but the
deficiency p remains unaltered ; and it is on this ground that he unites together and
regards as a single class the whole system of curves of a given deficiency p. It must
not be supposed that all such curves admit of rational transformation the one into
the other : there is the further theorem that any curve of the class depends, in the
case of a cubic, upon one parameter, but for p > 1 upon 3p — 3 parameters, each such
parameter being unaltered by the rational transformation ; it is thus only the curves
having the same one parameter, or 3p — 3 parameters, which can be rationally
transformed the one into the other.
Solid geometry is a far wider subject : there are more theories, and each of them
is of greater extent. The ratio is not that of the numbers of the dimensions of the
spaces considered, or, what is the same thing, of the elementary figures — point and
line in the one case ; point, line and plane in the other case — belonging to these spaces
respectively, but it is a very much higher one. For it is very inadequate to say that
in plane geometry we have the curve, and in solid geometry the curve and surface:
a more complete statement is required for the comparison. In plane geometry we
have the curve, which may be regarded as a singly infinite system of points, and also
as a singly infinite system of lines. In solid geometry we have, first, that which under
one aspect is the curve, and under another aspect the developable, and which may be
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 451
regarded as a singly infinite system of points, of lines, or of planes; secondly, the
surface, which may be regarded as a doubly infinite system of points or of planes, and
also as a special triply infinite system of lines (viz. the tangent-lines of the surface
are a special complex) : as distinct particular cases of the former figure, we have the
plane carve and the cone ; and as a particular case of the latter figure, the ruled
surface or singly infinite system of lines; we have besides the congruence, or doubly
infinite system of lines, and the complex, or triply infinite system of lines. But, even
if in solid geometry we attend only to the curve and the surface, there are crowds
of theories which have scarcely any analogues in plane geometry. The relation of a
curve to the various surfaces which can be drawn through it, or of a surface to the
various curves that can be drawn upon it, is different in kind from that which in
plane geometry most nearly corresponds to it, the relation of a system of points to
the curves through them, or of a curve to the points upon it. In particular, there is
nothing in plane geometry corresponding to the theory of the curves of curvature of a
surface. To the single theorem of plane geometry, a right line is the shortest distance
between two points, there correspond in solid geometry two extensive and difficult
theories — that of the geodesic lines upon a given surface, and that of the surface of
minimum area for any given boundary. Again, in solid geometry we have the interesting
and difficult question of the representation of a curve by means of equations ; it is not
every curve, but only a curve which is the complete intersection of two surfaces, which
can be properly represented by two equations (x, y, z, w)m = 0, (x, y, z, w)n = 0, in
quadriplanar coordinates ; and in regard to this question, which may also be regarded as
that of the classification of curves in space, we have quite recently three elaborate
memoirs by Nb'ther, Halphen, and Valentiner respectively.
In w-dimensional geometry, only isolated questions have been considered. The field
is simply too wide ; the comparison with each other of the two cases of plane geometry
and solid geometry is enough to show how the complexity and difficulty of the theory
would increase with each successive dimension.
In Transcendental Analysis, or the Theory of Functions, we have all that has been
done in the present century with regard to the general theory of the function of an
imaginary variable by Gauss, Cauchy, Puiseux, Briot, Bouquet, Liouville, Riemann, Fuchs,
Weierstrass, and others. The fundamental idea of the geometrical representation of
an imaginary variable x + iy, by means of the point having x, y for its coordinates,
belongs, as I mentioned, to Gauss; of this I have already spoken at some length.
The notion has been applied to differential equations; in the modern point of view,
the problem in regard to a given differential equation is, not so much to reduce the
differential equation to quadratures, as to determine from it directly the course of the
integrals for all positions of the point representing the independent variable: in
particular, the differential equation of the second order leading to the hypergeometric
series F (a, /3, 7, x) has been treated in this manner, with the most interesting results;
the function so determined for all values of the parameters (a, @, 7) is thus becoming
a known function. I would here also refer to the new notion in this part of analysis
introduced by Weierstrass— that of the one-valued integer function, as defined by an
57—2
452 PRESIDENTIAL ADDRESS TO THE [784
infinite series of ascending powers, convergent for all finite values, real or imaginary, of
the variable x or l/x — c, and so having the one essential singular point to = x or x = c,
as the case may be : the memoir is published in the Berlin Abhandhmgen, 1876.
But it is not only general theory: I have to speak of the various special functions
to which the theory has been applied, or say the various known functions.
For a long time the only known transcendental functions were the circular functions
sine, cosine, &c. ; the logarithm — i.e. for analytical purposes the hyperbolic logarithm
to the base e; and, as implied therein, the exponential function ex. More completely
stated, the group comprises the direct circular functions sin, cos, &c. ; the inverse
circular functions sin"1 or arc sin, &c. ; the exponential function, exp. ; and the inverse
exponential, or logarithmic, function, log.
Passing over the very important Eulerian integral of the second kind or gamma-
function, the theory of which has quite recently given rise to some very interesting
developments — and omitting to mention at all various functions of minor importance, —
we come (1811 — 1829) to the very wide groups, the elliptic functions and the single
theta-functions. I give the interval of date so as to include Legendre's two systematic
works, the Exercices de Calcul Integral (1811 — 1816) and the Theorie des Fonctions
Elliptiques (1825 — 1828); also Jacobi's Fundamenta nova theorice Functionum Ellipticarum
(1829), calling to mind that many of Jacobi's results were obtained simultaneously by
Abel. I remark that Legendre started from the consideration of the integrals depending
on a radical \/X, the square root of a rational and integral quartic function of a
variable x ; for this he substituted a radical A<£, = Vl — k2 sin2 <£, and he arrived at
his three kinds of elliptic integrals F<f>, E<b, 11$, depending on the argument or
amplitude </>, the modulus k, and also the last of them on a parameter n; the
function F is properly an inverse function, and in place of it Abel and Jacobi each
of them introduced the direct functions corresponding to the circular functions sine
and cosine, Abel's functions called by him (f>, f, F, and Jacobi's functions sinam, cosam,
Aam, or as they are also written sn, en, dn. Jacobi, moreover, in the development of
his theory of transformation obtained a multitude of formula? containing q, a tran
scendental function of the modulus defined by the equation q = e~nK'IK, and he was
also led by it to consider the two new functions H, @, which (taken each separately
with two different arguments) are in fact the four functions called elsewhere by him
©!, @2) ©s, ©45 these are the so-called theta-functions, or, when the distinction is necessary,
the single theta-functions. Finally, Jacobi using the transformation sin </> = sinam u,
expressed Legendre's integrals of the second and third kinds as integrals depending on
the new variable u, denoting them by means of the letters Z, II, and connecting
them with his own functions H and © : and the elliptic functions sn, en, dn are
expressed with these, or say with @1} ©2, ©3, ©4, as fractions having a common
denominator.
It may be convenient to mention that Hermite in 1858, introducing into the
theory in place of q the new variable a> connected with it by the equation q = e>™
(so that a> is in fact =iK'/K), was led to consider the three functions <£eo, -^rco, ^&>,
which denote respectively the values of v4/&, y/V and \/kk' regarded as functions of a).
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 453
A theta-function, putting the argument = 0, and then regarding it as a function of a>,
is what Professor Smith in a valuable memoir, left incomplete by his death, calls an
omega-function, and the three functions </>&>, -\/r&>, ^&> are his modular functions.
The proper elliptic functions sn, en, dn form a system very analogous to the
circular functions sine and cosine (say they are a sine and two separate cosines),
having a like addition-theorem, viz. the form of this theorem is that the sn, en and
dn of x + y are each of them expressible rationally in terms of the sn, en and dn
of x and of the sn, en and dn of y; and, in fact, reducing itself to the system of
the circular functions in the particular case k — 0. But there is the important
difference of form that the expressions for the sn, en and dn of x + y are fractional
functions having a common denominator: this is a reason for regarding these functions
as the ratios of four functions A, B, C, D, the absolute magnitudes of which are and
remain indeterminate (the functions sn, en, dn are in fact quotients [@1} ©2, @3]-7-@4
of the four theta-functions, but this is a further result in nowise deducible from the
addition-equations, and which is intended to be for the moment disregarded; the
remark has reference to what is said hereafter as to the Abelian functions). But
there is in regard to the functions sn, en, dn (what has no analogue for the circular
functions), the whole theory of transformation of any order n prime or composite, and,
as parts thereof, the whole theory of the modular and multiplier equations ; and this
theory of transformation spreads itself out in various directions, in geometry, in the
Theory of Equations, and in the Theory of Numbers. Leaving the theta-functions out
of consideration, the theory of the proper elliptic functions sn, en, dn is at once seen
to be a very wide one.
I assign to the Abeliari functions the date 1826 — 1832. Abel gave what is called
his theorem in various forms, but in its most general form in the Memoire sur une
propriete generate d'une classe tres-etendue de Fonctions Transcendantes (1826), presented
to the French Academy of Sciences, and crowned by them after the author's death,
in the following year. This is in form a theorem of the integral calculus, relating to
integrals depending on an irrational function y determined as a function of ac by any
algebraical equation F(x, y} = 0 whatever : the theorem being that a sum of any
number of such integrals is expressible by means of the sum of a determinate
number p of like integrals, this number p depending on the form of the equation
p (x} y) = Q which determines the irrational y (to fix the ideas, remark that considering
this equation as representing a curve, then p is really the deficiency of the curve;
but as already mentioned, the notion of deficiency dates only from 1857) : thus in
applying the theorem to the case where y is the square root of a function of the
fourth order, we have in effect Legendre's theorem for elliptic integrals F<f> + Fty
expressed by means of a single integral F/J,, and not a theorem applying in form to
the elliptic functions sn, en, dn. To be intelligible I must recall that the integrals
belonging to the case where y is the square root of a rational and integral function
of an order exceeding four are (in distinction from the general case) termed hyper-
elliptic integrals : viz. if the order be 5 or 6, then these are of the class p = 2 ; if
the order be 7 or 8, then they are of the class p = 3, and so on ; the general Abelian
integral of the class p = 2 is a hyperelliptic integral : but if p = 3, or any greater
454 PRESIDENTIAL ADDRESS TO THE [784
value, then the hyperelliptic integrals are only a particular case of the Abelian integrals
of the same class. The further step was made by Jacobi in the short but very
important memoir " Considerationes generales de transcendentibus Abelianis," Crelle,
t. ix. (1832): viz. he there shows for the hyperelliptic integrals of any class (but the
conclusion may be stated generally) that the direct functions to which Abel's theorem
has reference are not functions of a single variable, such as the elliptic sn, en, or dn,
but functions of p variables. Thus, in the case p = 2, which Jacobi specially considers,
it is shown that Abel's theorem has reference to two functions \(u, v\ \i(u, v) each
of two variables, and gives in effect an addition-theorem for the expression of the
functions X (u + u', v + v'), X: (u + u', v + v') algebraically in terms of the functions X (u, v),
\(u, v), \(u, v'), XjCw', v').
It is important to remark that Abel's theorem does not directly give, nor does
Jacobi assert that it gives, the addition-theorem in a perfect form. Take the case
p = 1 : the result from the theorem is that we have a function X (u), which is such
that X (u + v) can be expressed algebraically in terms of X (u) and X (v). This is of
course perfectly correct, sn (u + v) is expressible algebraically in terms of sn u, sn v, but
the expression involves the radicals Vl — sn2w, Vl — k2 sn2 u, v 1 — sn2 v, Vl — kz sn2 v ; but
it does not give the three functions sn, en, dn, or in anywise amount to the statement
that the sn, en and dn u of u + v are expressible rationally in terms of the sn, en
and dn of u and of v. In the case p = l, the right number of functions, each of
one variable, is 3, but the three functions sn, en and dn are properly considered as
the ratios of 4 functions ; and so, in general, the right number of functions, each of p
variables, is 4^ — 1, and these may be considered as the ratios of 4? functions. But
notwithstanding this last remark, it may be considered that the notion of the Abelian
functions of p variables is established, and the addition-theorem for these functions in
effect given by the memoirs (Abel 1826, Jacobi 1832) last referred to.
We have next for the case p = 2, which is hyperelliptic, the two extremely
valuable memoirs, Gb'pel, "Theoria transcendentium Abelianarum primi ordinis adum-
bratio Ia3va," Crelle, t. xxxv. (1847), and Koserihain, "Memoire sur les fonctions de
deux variables et a quatre periodes qui sont les inverses des integrates ultra-elliptiques
de la premiere classe " (1846), Paris, Mem. Savans fit/rang, t. XI. (1851), each of them
establishing on the analogy of the single theta-functions the corresponding functions
of two variables, or double theta-functions, and in connexion with them the theory
of the Abelian functions of two variables. It may be remarked that in order of
simplicity the theta-functions certainly precede the Abelian functions.
Passing over some memoirs by Weierstrass which refer to the general hyper
elliptic integrals, p any value whatever, we come to Riemann, who died 1866, at the
age of forty : collected edition of his works, Leipzig, 1876. His great memoir on the
Abelian and theta-functions is the memoir already incidentally referred to, "Theorie
der Abel'schen Functionen," Crelle, t. LIV. (1857) ; but intimately connected therewith
we have his Inaugural Dissertation (Gottingen, 1851), Grundlagen fur eine allgemeine
Theorie der Functionen einer verdnderlichen complexen Qrosse : his treatment of the
problem of the Abelian functions, and establishment for the purpose of this theory
of the multiple theta-functions, are alike founded on his general principles of the
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 455
theory of the functions of a variable complex magnitude x -V iy, and it is this which
would have to be gone into for any explanation of his method of dealing with the
problem.
Riemann, starting with the integrals of the most general form, and considering
the inverse functions corresponding to these integrals — that is, the Abelian functions
of p variables — defines a theta-function of p variables, or £>-tuple theta-function, as the
sum of a ^>-tuply infinite series of exponentials, the general term of course depending
on the p variables ; and he shows that the Abelian functions are algebraically con
nected with theta-functions of the proper arguments. The theory is presented in the
broadest form ; in particular as regards the theta-functions, the 4>p functions are not
even referred to, and there is no development as to the form of the algebraic relations
between the two sets of functions.
In the Theory of Equations, the beginning of the century may be regarded as an
epoch. Immediately preceding it, we have Lagrange's Traite des Equations Numeriques
(1st ed. 1798), the notes to which exhibit the then position of the theory. Immediately
following it, the great work by Gauss, the Disquisitiones Arithmetical (1801), in which
he establishes the theory for the case of a prime exponent n, of the binomial equation
xn — 1 = 0 : throwing out the factor as — I, the equation becomes an equation of the
order n — 1, and this is decomposed into equations the orders of which are the prime
factors of n — 1. In particular, Gauss was thereby led to the remarkable geometrical
result that it was possible to construct geometrically — that is, with only the ruler and
compass — the regular polygons of 17 sides and 257 sides respectively. We have then
(1826 — 1829) Abel, who, besides his demonstration of the impossibility of the solution
of a quintic equation by radicals, and his very important researches on the general
question of the algebraic solution of equations, established the theory of the class of
equations since called Abelian equations. He applied his methods to the problem of
the division of the elliptic functions, to (what is a distinct question) the division of
the complete functions, and to the very interesting special case of the lemniscate.
But the theory of algebraic solutions in its most complete form was established by
Galois (born 1811, killed in a duel 1832), who for this purpose introduced the notion
of a group of substitutions; and to him also are due some most valuable results in
relation to another set of equations presenting themselves in the theory of elliptic
functions — viz. the modular equations. In 1835 we have Jerrard's transformation of the
general quintic equation. In 1870 an elaborate work, Jordan's Traitt des Substitutions
et des equations algebriques : a mere inspection of the table of contents of this would
serve to illustrate my proposition as to the great extension of this branch of mathematics.
The Theory of Numbers was, at the beginning of the century, represented by
Legendre's Theorie des Nombres (1st ed. 1798), shortly followed by Gauss' Disquisitiones
Arithmetical (1801). This work by Gauss is, throughout, a theory of ordinary real
numbers. It establishes the notion of a congruence ; gives a proof of the theorem of
reciprocity in regard to quadratic residues ; and contains a very complete theory of
binary quadratic forms (a, b, c)(x, y}-, of negative and positive determinant, including
456 PRESIDENTIAL ADDRESS TO THE [784
the theory, there first given, of the composition of such forms. It gives also the
commencement of a like theory of ternary quadratic forms. It contains also the theory
already referred to, but which has since influenced in so remarkable a manner the
whole theory of numbers — the theory of the solution of the binomial equation xn — 1 = 0 :
it is, in fact, the roots or periods of roots derived from these equations which form
the incommensurables, or unities, of the complex theories which have been chiefly
worked at; thus, the i of ordinary analysis presents itself as a root of the equation
#* — 1 = 0. It was Gauss himself who, for the development of a real theory — that of
biquadratic residues — found it necessary to use complex numbers of the before-mentioned
form, a + bi (a and b positive or negative real integers, including zero), and the theory
of these numbers was studied and cultivated by Lejeune-Dirichlet. We have thus a
new theory of these complex numbers, side by side with the former theory of real
numbers: everything in the real theory reproducing itself, prime numbers, congruences,
theories of residues, reciprocity, quadratic forms, &c., but with greater variety and
complexity, and increased difficulty of demonstration. But instead of the equation
tf4 — 1 = 0, we may take the equation ^—1 = 0: we have here the complex numbers
a + bp composed with an imaginary cube root of unity, the theory specially considered
by Eisenstein : again a new theory, corresponding to but different from that of the
numbers a + bi. The general case of any prime value of the exponent n, and with
periods of roots, which here present themselves instead of single roots, was first con
sidered by Kummer: viz. if n — l=ef, and 77^ i]i,...,rie are the e periods, each of them
a sum of f roots, of the equation xn — 1 = 0, then the complex numbers considered
are the numbers of the form a^ + a2rj2 + ... + aetje (a1} a2,...,ae positive or negative
ordinary integers, including zero): / may be =1, and the theory for the periods thus
includes that for the single roots.
We have thus a new and very general theory, including within itself that of the
complex numbers a + bi and a + bp. But a new phenomenon presents itself; for these
special forms the properties in regard to prime numbers corresponded precisely with
those for real numbers; a non-prime number was in one way only a product of prime
factors ; the power of a prime number has only factors which are lower powers of the
same prime number: for instance, if p be a prime number, then, excluding the obvious
decomposition p.p2, we cannot have p3 = & product of two factors A, B. In the general
case this is not so, but the exception first presents itself for the number 23 ; in the
theory of the numbers composed with the 23rd roots of unity, we have prime
numbers p, such that ps = AB. To restore the theorem, it is necessary to establish
the notion of ideal numbers ; a prime number p is by definition not the product of
two actual numbers, but in the example just referred to the number p is the product
of two ideal numbers having for their cubes the two actual numbers A, B, respectively,
and we thus have p3 = AB. It is, I think, in this way that we most easily get some
notion of the meaning of an ideal number, but the mode of treatment (in Kummer's
great memoir, "Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten com-
plexen Zahlen in ihre Primfactoren," Crelle, t. xxxv. 1847) is a much more refined
one ; an ideal number, without ever being isolated, is made to manifest itself in the
properties of the prime number of which it is a factor, and without reference to the
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 457
theorem afterwards arrived at, that there is always some power of the ideal number
which is an actual number. In the still later developments of the Theory of Numbers
by Dedekind, the units, or incommensurables, are the roots of any irreducible equation
having for its coefficients ordinary integer numbers, and with the coefficient unity for
the highest power of x. The question arises, What is the analogue of a whole
number ? thus, for the very simple case of the equation #2 + 3 = 0, we have as a whole
number the apparently fractional form £(l + i\/3) which is the imaginary cube root
of unity, the p of Eisenstein's theory. We have, moreover, the (as far as appears)
wholly distinct complex theory of the numbers composed with the congruence-imaginaries
of Galois : viz. these are imaginary numbers assumed to satisfy a congruence which is
not satisfied by any real number ; for instance, the congruence a? — 2 = 0 (mod 5) has
no real root, but we assume an imaginary root i, the other root is then = — i, and
we then consider the system of complex numbers a + bi (mod 5), viz. we have thus
the 52 numbers obtained by giving to each of the numbers a, b, the values 0, 1, 2, 8, 4,
successively. And so in general, the consideration of an irreducible congruence jF(#) = 0
(mod p) of the order n, to any prime modulus p, gives rise to an imaginary con
gruence root i, and to complex numbers of the form a + bi + ci? + . . . + kin~l, where
a, b, k, ... &c., are ordinary integers each =0, 1, 2, ... , p — I.
As regards the theory of forms, we have in the ordinary theory, in addition to
the binary and ternary quadratic forms, which have been very thoroughly studied, the
quaternary and higher quadratic forms (to these last belong, as very particular cases,
the theories of the representation of a number as a sum of four, five or more squares),
and also binary cubic and quartic forms, and ternary cubic forms, in regard to all
of which something has been done ; the binary quadratic forms have been studied in the
theory of the complex numbers a + bi.
A seemingly isolated question in the Theory of Numbers, the demonstration of
Fermat's theorem of the impossibility for any exponent X greater than 3, of the equation
^.A _l_ y\ = z\^ nas given rise to investigations of very great interest and difficulty.
Outside of ordinary mathematics, we have some theories which must be referred
to : algebraical, geometrical, logical. It is, as in many other cases, difficult to draw
the line ; we do in ordinary mathematics use symbols not denoting quantities, which
we nevertheless combine in the way of addition and multiplication, a + b, and ab, and
which may be such as not to obey the commutative law ab = ba: in particular, this is
or may be so in regard to symbols of operation; and it could hardly be said that
any development whatever of the theory of such symbols of operation did not belong
to ordinary algebra. But I do separate from ordinary mathematics the system of
multiple algebra or linear associative algebra, developed in the valuable memoir by the
late Benjamin Peirce, Linear Associative Algebra (1870, reprinted 1881 in the American
Journal of Mathematics, vol. iv., with notes and addenda by his son, C. S. Peirce) ; we
here consider symbols A, B, &c. which are linear functions of a determinate number
of letters or units i, j, k, I, &c., with coefficients which are ordinary analytical magni
tudes, real or imaginary, viz. the coefficients are in general of the form x + iy, where
C. XI. 58
458 PRESIDENTIAL ADDRESS TO THE [784
i is the before-mentioned imaginary or V — 1 of ordinary analysis. The letters i, j, &c.,
are such that every binary combination i2, ij, ji, &c., (the ij being in general not =ji),
is equal to a linear function of the letters, but under the restriction of satisfying
the associative law : viz. for each combination of three letters ij . k is = i .jk, so that
there is a determinate and unique product of three or more letters ; or, what is the
same thing, the laws of combination of the units i, j, k, are defined by a multiplication
table giving the values of i'2, ij, ji, &c. ; the original units may be replaced by linear
functions of these units, so as to give rise, for the units finally adopted, to a multi
plication table of the most simple form ; and it is very remarkable, how frequently in
these simplified forms we have nilpotent or idempotent symbols (t2 = 0, or i2 = i, as the
case may be), and symbols i, j, such that ij=ji=Q; and consequently how simple are
the forms of the multiplication tables which define the several systems respectively.
I have spoken of this multiple algebra before referring to various geometrical
theories of earlier date, because I consider it as the general analytical basis, and the
true basis, of these theories. I do not realise to myself directly the notions of the
addition or multiplication of two lines, areas, rotations, forces, or other geometrical,
kinematical, or mechanical entities; and I would formulate a general theory as follows:
consider any such entity as determined by the proper number of parameters a, b, c (for
instance, in the case of a finite line given in magnitude and position, these might be
the length, the coordinates of one end, and the direction-cosines of the line considered
as drawn from this end) ; and represent it by or connect it with the linear function
ai + bj + ck + &c., formed with these parameters as coefficients, and with a given set of
units, i, j, k, &c. Conversely, any such linear function represents an entity of the kind
in question. Two given entities are represented by two linear functions; the sum of
these is a like linear function representing an entity of the same kind, which may
be regarded as the sum of the two entities; and the product of them (taken in a
determined order, when the order is material) is an entity of the same kind, which
may be regarded as the product (in the same order) of the two entities. We thus
establish by definition the notion of the sum of the two entities, and that of the
product (in a determinate order, when the order is material) of the two entities. The
value of the theory in regard to any kind of entity would of course depend on the
choice of a system of units, i, j, k,..., with such laws of combination as would give a
geometrical or kinematical or mechanical significance to the notions of the sum and
product as thus defined.
Among the geometrical theories referred to, we have a theory (that of Argand,
Wan-en, and Peacock) of imaginaries in plane geometry; Sir W. R. Hamilton's very
valuable and important theory of Quaternions; the theories developed in Grassmann's
Ausdehnungslehre, 1841 and 1802 ; Clifford's theory of Biquaternions ; and recent extensions
of Grassmann's theory to non-Euclidian space, by Mr Homersham Cox. These different
theories have of course been developed, not in anywise from the point of view in
which I have been considering them, but from the points of view of their several
authors respectively.
The literal symbols ac, y, &c., used in Boole's Laws of Thought (1854) to represent
things as subjects of our conceptions, are symbols obeying the laws of algebraic com-
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 459
bination (the distributive, commutative, and associative laws) but which are such that
for any one of them, say x, we have x — #2 = 0, this equation not implying (as in ordinary
algebra it would do) either x=0 or else x=\. In the latter part of the work relating
to the Theory of Probabilities, there is a difficulty in making out the precise meaning
of the symbols ; and the remarkable theory there developed has. it seems to me, passed
out of notice, without having been properly discussed. A paper by the same author,
" Of Propositions numerically definite " (Camb. Phil. Trans. 1869), is also on the border
land of logic and mathematics. It would be out of place to consider other systems
of mathematical logic, but I will just mention that Mr C. S. Peirce in his "Algebra of
Logic," American Math. Journal, vol. in., establishes a notation for relative terms, and
that these present themselves in connexion with the systems of units of the linear
associative algebra.
Connected with logic, but primarily mathematical and of the highest importance,
we have Schubert's Alzdhlende Geometrie (1878). The general question is, How many
curves or other figures are there which satisfy given conditions ? for example, How
many conies are there which touch each of five given conies ? The class of questions
in regard to the conic was first considered by Chasles, and we have his beautiful
theory of the characteristics /*, v, of the conies which satisfy four given conditions;
questions relating to cubics and quartics were afterwards considered by Maillard and
Zeuthen ; and in the work just referred to the theory has become a very wide one.
The noticeable point is that the symbols used by Schubert are in the first instance,
not numbers, but mere logical symbols : for example, a letter g denotes the condition
that a line shall cut a given line ; g* that it shall cut each of two given lines ; and so
in other cases ; and these logical symbols are combined together by algebraical laws :
they first acquire a numerical signification when the number of conditions becomes equal
to the number of parameters upon which the figure in question depends.
In all that I have last said in regard to theories outside of ordinary mathematics, I
have been still speaking on the text of the vast extent of modern mathematics. In
conclusion I would say that mathematics have steadily advanced from the time of the
Greek geometers. Nothing is lost or wasted ; the achievements of Euclid, Archimedes,
and Apollonius are as admirable now as they were in their own days. Descartes' method
of coordinates is a possession for ever. But mathematics have never been cultivated
more zealously and diligently, or with greater success, than in this century — in the last
half of it, or at the present time : the advances made have been enormous, the actual
field is boundless, the future full of hope. In regard to pure mathematics we may
most confidently say : —
Yet I doubt not through the ages one increasing purpose runs,
And the thoughts of men are widened with the process of the suns.
58—2
460 [785
785.
CURVE.
[From the Encyclopaedia Britannica, Ninth Edition, vol. vi. (1877), pp. 716 — 728.]
THIS subject is treated here from an historical point of view, for the purpose of
showing how the different leading ideas in the theory were successively arrived at and
developed.
A curve is a line, or continuous singly infinite system of points. We consider in
the first instance, and chiefly, a plane curve described according to a law. Such a curve
may be regarded geometrically as actually described, or kinematically as in course of
description by the motion of a point ; in the former point of view, it is the locus
of all the points which satisfy a given condition ; in the latter, it is the locus of a
point moving subject to a given condition. Thus the most simple and earliest known
curve, the circle, is the locus of all the points at a given distance from a fixed
centre, or else the locus of a point moving so as to be always at a given distance
from a fixed centre. (The straight line and the point are not for the moment regarded
as curves.)
Next to the circle we have the conic sections, the invention of them attributed
to Plato (who lived 430 to 347 B.C.); the original definition of them as the sections
of a cone was by the Greek geometers who studied them soon replaced by a proper
definition in piano like that for the circle, viz. a conic section (or as we now say a
"conic") is the locus of a point such that its distance from a given point, the focus,
is in a given ratio to its (perpendicular) distance from a given line, the directrix ;
or it is the locus of a point which moves so as always to satisfy the foregoing con
dition. Similarly any other property might be used as a definition ; an ellipse is the
locus of a point such that the sum of its distances from two fixed points (the foci)
is constant, &c., &c.
The Greek geometers invented other curves ; in particular, the " conchoid," which
is the locus of a point such that its distance from a given line, measured along the
785]
CURVE.
461
line drawn through it to a fixed point, is constant ; and the " cissoid " which is the
locus of a point such that its distance from a fixed point is always equal to the
intercept (on the line through the fixed point) between a circle passing through the
fixed point and the tangent to the circle at the point opposite to the fixed point.
Obviously the number of such geometrical or kinematical definitions is infinite. In a
machine of any kind, each point describes a curve ; a simple but important instance
is the " three-bar curve," or locus of a point in or rigidly connected with a bar
pivotted on to two other bars which rotate about fixed centres respectively. Every curve
thus arbitrarily defined has its own properties : and there was not any principle of
classification.
The principle of classification first presented itself in the Geometrie of Descartes
(1637). The idea was to represent any curve whatever by means of a relation between
the coordinates (x, y) of a point of the curve, or say to represent the curve by means
of its equation.
Descartes takes two lines xx, yy', called axes of coordinates, intersecting at a point
0 called the origin (the axes are usually at right angles to each other, and for the
y
N
M
present they are considered as being so) ; and he determines the position of a point
P by means of its distances OM (or NP) = x, and MP (or ON) = y, from these two
axes respectively ; where x is regarded as positive or negative according as it is in
the sense Ox or Ox from 0 ; and similarly y as positive or negative according as it
is in the sense Oy or Oy' from 0 ; or, what is the same thing,
x y
In the quadrant xy, or N.E., we have + +
J> y J) * * )) ™~~ 1~
S.E. „ + -
s.w.
xy
x'y'
Any relation whatever between (x, y) determines a curve, and conversely every
curve whatever is determined by a relation between (x, y).
462 CUKVE. [785
Observe that the distinctive feature is in the exclusive use of such determination
of a curve by means of its equation. The Greek geometers were perfectly familiar with
/y»2 n »2
the property of an ellipse which in the Cartesian notation is —+^=1, the equation
of the curve ; but it was as one of a number of properties, and in no wise selected
out of the others for the characteristic property of the curve *.
We obtain from the equation the notion of an algebraical or geometrical as opposed
to a transcendental curve, viz. an algebraical or geometrical curve is a curve having an
equation F (x, y) = 0, where F (x, y} is a rational and integral algebraical function of the
coordinates (x, y}] and in what follows we attend throughout (unless the contrary is
stated) only to such curves. The equation is sometimes given, and may conveniently
be used, in an irrational form, but we always imagine it reduced to the foregoing
rational and integral form, and regard this as the equation of the curve. And we
have hence the notion of a curve of a given order, viz. the order of the curve is
equal to that of the term or terms of highest order in the coordinates (x, y} con
jointly in the equation of the curve ; for instance, xy — 1 = 0 is a curve of the second
order.
It is to be noticed here that the axes of coordinates may be any two lines at
right angles to each other whatever ; and that the equation of a curve will be different
according to the selection of the axes of coordinates ; but the order is independent
of the axes, and has a determinate value for any given curve.
We hence divide curves according to their order, viz. a curve is of the first order,
second order, third order, &c., according as it is represented by an equation of the
first order, aoc+by + c = 0, or say (*$X y, 1) = 0; or by an equation of the second order,
aaf + Ihxy + by" + 2fy + 2gx + c = 0, say (*]£#, y, I)2 = 0 ; or by an equation of the third
order, &c. ; or, what is the same thing, according as the equation is linear, quadric,
cubic, &c.
A curve of the first order is a right line ; and conversely every right line is a
curve of the first order.
* There is no exercise more profitable for a student than that of tracing a curve from its equation, or
say rather that of so tracing a considerable number of curves. And he should make the equations for him
self. The equation should be in the first instance a purely numerical one, where y is given or can be
found as an explicit function of x ; here, by giving different numerical values to x, the corresponding values
of y may be found ; and a sufficient number of points being thus determined, the curve is traced by drawing
a continuous line through these points. The next step should be to consider an equation involving literal
coefficients; thus, after such curves as y=x3, y=x (x- 1) (x- 2), y = (x- 1) Jx- 2, &c., he should proceed to
trace such curves as y = (x-a)(x-b)(x-c), y = (x-a) >Jj; - b, <fec., and endeavour to ascertain for what different
relations of equality or inequality between the coefficients the curve will assume essentially or notably distinct
forms. The purely numerical equations will present instances of nodes, cusps, inflexions, double tangents,
asymptotes, &c., — specialities which he should be familiar with before he has to consider their general theory.
And he may then consider an equation such that neither coordinate can be expressed as an explicit function
of the other of them (practically, an equation such as x3 + y3 - 3xy = 0, which requires the solution of a cubic
equation, belongs to this class) ; the problem of tracing the curve here frequently requires special methods,
and it may easily be such as to require and serve as an exercise for the powers of an advanced algebraist.
785] CURVE. 463
A curve of the second order is a conic, or as it is also called a quadric ; and
conversely every conic, or quadric, is a curve of the second order.
A curve of the third order is called a cubic ; one of the fourth order a quartic ;
and so on.
A curve of the order m has for its equation (*]£#, y, l)m = 0; and when the
coefficients of the function are arbitrary, the curve is said to be the general curve of
the order m. The number of coefficients is ^(m+ l)(m + 2) ; but there is no loss of
generality if the equation be divided by one coefficient so as to reduce the coefficient
of the corresponding term to unity, hence the number of coefficients may be reckoned
as ^ (m +l)(m + 2) — 1, that is, |w(w + 3); and a curve of the order m may be
made to satisfy this number of conditions ; for example, to pass through J m (m + 3)
points.
It is to be remarked that an equation may break up; thus a quadric equation
may be (ax + by + c) (a'x + b'y + c') = 0, breaking up into the two equations ax + by + c = 0,
a'x + b'y + c' = 0, viz. the original equation is satisfied if either of these is satisfied.
Each of these last equations represents a curve of the first order, or right line ; and
the original equation represents this pair of lines, viz. the pair of lines is considered
as a quadric curve. But it is an improper quadric curve ; and in speaking of curves
of the second or any other given order, we frequently imply that the curve is a
proper curve represented by an equation which does not break up.
The intersections of two curves are obtained by combining their equations; viz.
the elimination from the two equations of y (or x) gives for x (or y) an equation
of a certain order, say the resultant equation; and then to each value of x (or y)
satisfying this equation there corresponds in general a single value of y (or x\ and
consequently a single point of intersection ; the number of intersections is thus equal
to the order of the resultant equation in x (or y}.
Supposing that the two curves are of the orders m, n, respectively, then the order
of the resultant equation is in general and at most = mn ; in particular, if the curve
of the order n is an arbitrary line (w = l), then the order of the resultant equation
is =m; and the curve of the order m meets therefore the line in m points. But
the resultant equation may have all or any of its roots imaginary, and it is thus not
always that there are m real intersections.
The notion of imaginary intersections, thus presenting itself, through algebra, in
geometry, must be accepted in geometry— and it in fact plays an all-important part in
modern geometry. As in algebra we say that an equation of the with order has m
roots, viz. we state this generally without in the first instance, or it may be without
ever, distinguishing whether these are real or imaginary; so in geometry we say that
a curve of the mth order is met by an arbitrary line in m points, or rather we
thus, through algebra, obtain the proper geometrical definition of a curve of the mth
order, as a curve which is met by an arbitrary line in m points (that is, of course,
in m, and not more than m, points).
The theorem of the m intersections has been stated in regard to an arbitrary
line ; in fact, for particular lines the resultant equation may be or appear to be of
464 CURVE. [785
an order less than m ; for instance, taking ra = 2, if the hyperbola ocy — 1 = 0 be cut
by the line y = /3, the resultant equation in x is (3x — 1 = 0, and there is apparently
only the intersection (oc = -, y=@\; but the theorem is, in fact, true for every line
whatever : a curve of the order m meets every line whatever in precisely m points.
We have, in the case just referred to, to take account of a point at infinity on the
line 2/ = /3; the two intersections are the point # = > y = $> and the point at infinity
on the line y = J3.
It is moreover to be noticed that the points at infinity may be all or any of
them imaginary, and that the points of intersection, whether finite or at infinity, real
or imaginary, may coincide two or more of them together, and have to be counted
accordingly ; to support the theorem in its universality, it is necessary to take account
of these various circumstances.
The foregoing notion of a point at infinity is a very important one in modern
geometry; and we have also to consider the paradoxical statement that in plane
geometry, or say as regards the plane, infinity is a right line. This admits of an easy
illustration in solid geometry. If with a given centre of projection, by drawing from
it lines to every point of a given line, we project the given line on a given plane,
the projection is a line, i.e., this projection is the intersection of the given plane with
the plane through the centre and the given line. Say the projection is always a
line, then if the figure is such that the two planes are parallel, the projection is
the intersection of the given plane by a parallel plane, or it is the system of points
at infinity on the given plane, that is, these points at infinity are regarded as situate
on a given line, the line infinity of the given plane*.
Reverting to the purely plane theory, infinity is a line, related like any other
right line to the curve, and thus intersecting it in m points, real or imaginary, distinct
or coincident.
Descartes in the Geometric defined and considered the remarkable curves called
after him ovals of Descartes, or simply Cartesians, which will be again referred to,
The next important work, founded on the Geometrie, was Sir Isaac Newton's Enumeratio
linearum tertii ordinis (1706), establishing a classification of cubic curves founded chiefly
on the nature of their infinite branches, which was in some details completed by
Stirling, Murdoch, and Cramer; the work contains also the remarkable theorem (to be
again referred to), that there are five kinds of cubic curves giving by their projections
every cubic curve whatever.
Various properties of curves in general, and of cubic curves, are established in
Maclaurin's memoir, "De linearum geometricarum proprietatibus generalibus Tractatus"
(posthumous, say 1746, published in the 6th edition of his Algebra). We have in it
a particular kind of correspondence of two points on a cubic curve, viz. two points
correspond to each other when the tangents at the two points again meet the cubic
in the same point.
* More generally, in solid geometry infinity is a plane, — its intersection with any given plane being the
right line which is the infinity of this given plane.
785] CURVE. 465
The Geometric Descriptive by Monge was written in the year 1794 or 1795
(7th edition, Paris, 1847), and in it we find stated, in piano with regard to the circle,
and in three dimensions with regard to a surface of the second order, the fundamental
theorem of reciprocal polars, viz. '' Given a surface of the second order and a circum
scribed conic surface which touches it .... then if the conic surface moves so that its
summit is always in the same plane, the plane of the curve of contact passes always
through the same point." The theorem is here referred to partly on account of its
bearing on the theory of imaginaries in geometry. It is, in Brianchon's memoir " Sur
les surfaces du second degre"' (Jour. Polyt., t. vi., 1806), shown how for any given
position of the summit the plane of contact is determined, or reciprocally ; say the
plane XY is determined when the point P is given, or reciprocally ; and it is noticed
that when P is situate in the interior of the surface the plane XY does not cut
the surface ; that is, we have a real plane XY intersecting the surface in the imaginary
curve of contact of the imaginary circumscribed cone having for its summit a given
real point P inside the surface.
Stating the theorem in regard to a conic, we have a real point P (called the
pole) and a real line XY (called the polar), the line joining the two (real or imaginary)
points of contact of the (real or imaginary) tangents drawn from the point to the conic ;
and the theorem is that when the point describes a line the line passes through a
point, this line and point being polar and pole to each other. The term " pole " was
first used by Servois, and " polar " by Gergonne (Gerg., t. I. and ill., 1810 — 13) ; and
from the theorem we have the method of reciprocal polars for the transformation of
geometrical theorems, used already by Brianchon (in the memoir above referred to) for
the demonstration of the theorem called by his name, and in a similar manner by
various writers in the earlier volumes of Gergonne. We are here concerned with the
method less in itself than as leading to the general notion of duality. And, bearing
in a somewhat similar manner also on the theory of imaginaries in geometry (but the
notion presents itself in a more explicit form), there is the memoir by Gaultier, on
the graphical construction of circles and spheres (Jour. Polyt., t. ix., 1813). The well-
known theorem as to radical axes may be stated as follows. Consider two circles
partially drawn so that it does not appear whether the circles, if completed, would or
would not intersect in real points, say two arcs of circles ; then we can. by means of
a third circle drawn so as to intersect in two real points each of the two arcs,
determine a right line, which, if the complete circles intersect in two real points, passes
through the points, and which is on this account regarded as a line passing through
two (real or imaginary) points of intersection of the two circles. The construction in
fact is, join the two points in which the third circle meets the first arc, and join
also the two points in which the third circle meets the second arc, and from the
point of intersection of the two joining lines, let fall a perpendicular on the line
joining the centre of the two circles; this perpendicular (considered as an indefinite
line) is what Gaultier terms the " radical axis of the two circles " ; it is a line
determined by a real construction and itself always real ; and by what precedes it is
the line joining two (real or imaginary, as the case may be) intersections of the given
circles.
c. xi. 59
466 CURVE. [785
The intersections which lie on the radical axis are two out of the four inter
sections of the two circles. The question as to the remaining two intersections did
not present itself to Gaultier, but it is answered in Poncelet's Traite des proprietes
projectives (1822), where we find (p. 49) the statement, "deux circles places arbitraire-
ment sur un plan...ont ide'alement deux points imaginaires communs a 1'mfini"; that
is, a circle qua curve of the second order is met by the line infinity in two points ;
but, more than this, they are the same two points for any circle whatever. The
points in question have since been called (it is believed first by Dr Salmon) the
circular points at infinity, or they may be called the circular points; these are also
frequently spoken of as the points /, J\ and we have thus the circle characterized
as a conic which passes through the two circular points at infinity ; the number of
conditions thus imposed upon the conic is = 2, and there remain three arbitrary con
stants, which is the right number for the circle. Poncelet throughout his work makes
continual use of the foregoing theories of imaginaries and infinity, and also of the
before-mentioned theory of reciprocal polars.
Poncelet's two memoirs " Sur les centres des moyennes harmoniques," and " Sur la
theorie generale des polaires reciproques," although presented to the Paris Academy in
1824 were only published (Crelle, t. ill. and IV., 1828, 1829), subsequent to the memoir
by Gergonne, " Considerations philosophiques sur les elemens de la science de I'e'tendue "
(Gerg., t. xvi., 1825 — 26). In this memoir by Gergonne, the theory of duality is very
clearly and explicitly stated ; for instance, we find " dans la ge/ometrie plane, a chaque
theoreme il en repond ne'cessairement un autre qui s'en de'duit en e'changeant simple-
raent entre eux les deux mots points et droites; tandis que dans la geometric de
1'espace ce sont les mots points et plans qu'il faut echanger entre eux pour passer d'un
theoreme a son correlatif " ; and the plan is introduced of printing correlative theorems,
opposite to each other, in two columns. There was a reclamation as to priority by
Poncelet in the Bulletin Universel reprinted with remarks by Gergonne (Gerg., t. xix.,
1827), and followed by a short paper by Gergonne, " Rectifications de quelques thdoremes,
&c.," which is important as first introducing the word class. We find in it explicitly
the two correlative definitions : — " a plane curve is said to be of the mth degree (order)
when it has with a line m real or ideal intersections," and " a plane curve is said to
be of the wth class when from any point of its plane there can be drawn to it m real
or ideal tangents."
It may be remarked that in Poncelet's memoir on reciprocal polars, above referred
to, we have the theorem that the number of tangents from a point to a curve of
the1 order m, or say the class of the curve, is in general and at most = ra(w— 1),
and that he mentions that this number is subject to reduction when the curve has
double points or cusps.
The theorem of duality as regards plane figures may be thus stated : — two figures
may correspond to each other in such manner that to each point and line in either
figure there corresponds in the other figure a line and point respectively. It is to
be understood that the theorem extends to all points or lines, drawn or not drawn ;
thus if in the first figure there are any number of points on a line drawn or not
drawn, the corresponding lines in the second figure, produced if necessary, must meet
785] CURVE. 467
in a point. And we thus see how the theorem extends to curves, their points and
tangents : if there is in the first figure a curve of the order m, any line meets it
in m points ; and hence from the corresponding point in the second figure there must
be to the corresponding curve m tangents; that is, the corresponding curve must be
of the class m.
Trilinear coordinates (to be again referred to) were first used by Bobillier in the
memoir, " Essai sur un nouveau mode de recherche des proprie'tes de 1'e'tendue "
(Gerg., t. xvin., 1827 — 28). It is convenient to use these rather than Cartesian coordi
nates. We represent a curve of the order m by an equation (*$#, y, z)rn=0, the
function on the left-hand being a homogeneous rational and integral function of the
order m of the three coordinates (x, y, z); clearly the number of constants is the
same as for the equation (*§x, y, l)m = 0 in Cartesian coordinates.
The theory of duality is considered and developed, but chiefly in regard to its
metrical applications, by Chasles in the "Me'moire de geometric sur deux principes
gendraux de la science, la dualite et 1'homographie," which forms a sequel to the
" Apei^u historique sur 1'origine et le developpement des methodes en geometric "
(Mem. de Brux., t. XL, 1837).
We now come to Plucker ; his " six equations " were given in a short memoir in
Crelle (1842) preceding his great work, the Theorie der algebraischen Curven (1844).
Plucker first gave a scientific dual definition of a curve, viz. " A curve is a locus
generated by a point, and enveloped by a line, — the point .moving continuously along
the line, while the line rotates continuously about the point " ; the point is a point
(ineunt) of the curve, the line is a tangent of the curve.
And, assuming the above theory of geometrical imaginaries, a curve such that m
of its points are situate in an arbitrary line is said to be of the order m; a curve
such that n of its tangents pass through an arbitrary point is said to be of the
class n ; as already appearing, this notion of the order and the class of a curve is, how
ever, due to Gergonne. Thus the line is a curve of the order 1 and the class 0 ;
and corresponding dually thereto, we have the point as a curve of the order 0 and the
class 1.
Plucker moreover imagined a system of line-coordinates (tangential coordinates).
The Cartesian coordinates (x, y) and trilinear coordinates (x, y, z) are point-coordinates
for determining the position of a point ; the new coordinates, say (£, 77, % ), are line-
coordinates for determining the position of a line. It is possible, and (not so much
for any application thereof as in order to more fully establish the analogy between
the two kinds of coordinates) important, to give independent quantitative definitions
of the two kinds of coordinates ; but we may also derive the notion of line-coordinates
from that of point-coordinates ; viz. taking f~x + yy + %z — 0 to be the equation of a
line, we say that (£, 77, f) are the line-coordinates of this line. A linear relation
«£ + by + c% = 0 between these coordinates determines a point, viz. the point whose
point-coordinates are (a, b, c); in fact, the equation in question af + bij +c£=0 expresses
that the equation %x + ?)y + £z= 0, where (x, y, z} are current point-coordinates, is
satisfied on writing therein x, y, z = a, b, c ; or that the line in question passes through
KQ o
<J «.' —
468 CURVE. [785
the point (a, b, c). Thus (£, r\, £) are the line-coordinates of any line whatever; but
when these, instead of being absolutely arbitrary, are subject to the restriction
a% + by + c£ = 0, this obliges the line to pass through a point (a, b, c) ; and the last-
mentioned equation a% + brj + c£ = 0 is considered as the line-equation of this point.
A line has only a point-equation, and a point has only a line-equation ; but any other
curve has a point-equation and also a line-equation ; the point-equation (*]£#, y, z}m = 0
is the relation which is satisfied by the point-coordinates (x, y, z) of each point of
the curve; and similarly the line-equation (*$£, 77, £)n = 0 is the relation which is
satisfied by the line-coordinates (£, y, £) of each line (tangent) of the curve.
There is in analytical geometry little occasion for any explicit use of line-coordinates ;
but the theory is very important ; it serves to show that, in demonstrating by point-
coordinates any purely descriptive theorem whatever, we demonstrate the correlative
theorem ; that is, we do not demonstrate the one theorem, and then (as by the method
of reciprocal polars) deduce from it the other, but we do at one and the same time
demonstrate the two theorems; our (x, y, z) instead of meaning point-coordinates may
mean line-coordinates, and the demonstration is then in every step of it a demonstration
of the correlative theorem.
The above dual generation explains the nature of the singularities of a plane
curve. The ordinary singularities, arranged according to a cross division, are
Proper. Improper.
r, • ^ • , . . (1. The stationary point, 2. The double point, or node;
romt-smgulanties — \ . .
( cusp, or spmode ;
T. . , ... (3. The stationary tangent, 4. The double tangent: —
Line-singularities — 4 . . . J °
( or inflexion ;
arising as follows : —
1. The cusp : the point as it travels along the line may come to rest, and then
reverse the direction of its motion.
3. The stationary tangent: the line may in the course of its rotation come to
rest, and then reverse the direction of its rotation.
2. The node : the point may in the course of its motion come to coincide with
a former position of the point, the two positions of the line not in general coinciding.
4. The double tangent: the line may in the course of its motion come to coin
cide with a former position of the line, the two positions of the point not in general
coinciding.
It may be remarked that we cannot with a real point and line obtain the node
with two imaginary tangents (conjugate or isolated point, or acnode), nor again the real
double tangent with two imaginary points of contact ; but this is of little consequence,
since in the general theory the distinction between real and imaginary is not
attended to.
The singularities (1) and (3) have been termed proper singularities, and (2) and
(4) improper; in each of the first-mentioned cases there is a real singularity, or
785] CURVE. 469
peculiarity in the motion ; in the other two cases there is not ; in (2) there is not
when the point is first at the node, or when it is secondly at the node, any peculiarity
in the motion ; the singularity consists in the point coming twice into the same
position; and so in (4) the singularity is in the line coming twice into the same
position. Moreover (1) and (2) are, the former a proper singularity, and the latter an
improper singularity, as regards the motion of the point ; and similarly (3) and (4) are,
the former a proper singularity, and the latter an improper singularity, as regards the
motion of the line.
But as regards the representation of a curve by an equation, the case is very
different.
First, if the equation be in point-coordinates, (3) and (4) are in a sense not
singularities at all. The curve (*]£#, y, z)m = 0, or general curve of the order m, has
double tangents and inflexions; (2) presents itself as a singularity, for the equations
dx(*~$x, V, *)w = 0, dy(*$x, y, z)m = 0, 4(*$>> y> z)m = 0, implying (*$>, y, z)m = 0, are
not in general satisfied by any values (a, b, c) whatever of (x, y, z), but if such
values exist, then the point (a, b, c) is a node or double point ; and (1) presents
itself as a further singularity or sub-case of (2), a cusp being a double point for which
the two tangents become coincident.
In line-coordinates all is reversed : — (1) and (2) are not singularities ; (3) pre
sents itself as a sub-case of (4).
The theory of compound singularities will be referred to further on.
In regard to the ordinary singularities, we have
m, the order,
n „ class,
B „ number of double points,
« „ „ cusps,
T „ „ double tangents,
K „ „ inflexions;
and this being so, Pliicker's " six equations " are
(1) n - w(m-l)-2S-3/c,
(2) i = 3m(wi-2)-68-8«,
(3) T =£m (m-2)(m2-9)-(m2-m-6)(23+3*) + 28(8-l) + 6SA: + |/c(Ac-l),
(4) m= w(w-l)-2r-3t,
(5) K = 3»(n-2)-6T-8t,
(6) B = $n (n - 2) (n2 - 9) - (n* - n - 6) (2r + 3t) + 2r (T - 1) + Grt + f t (i - 1).
470 CURVE. [785
It is easy to derive the further forms —
(7) i - K = 3 (n - m),
(8) 2(r-S) = (n-m)(n+m-9),
(9) £m(m + 3)-S-2« =%n (n + 3) - T- 2i,
(10) £ (m - 1) (m - 2) - 8 - * = \ (n - 1) (n - 2) - r - t,
(11, 12) m2 - 2S - 3/t = n2 - 2r - 3t, = 7ft + n,
the whole system being equivalent to three equations only: and it may be added that,
using a to denote the equal quantities 3m, + 1 and 3w -I- K, everything may be expressed
in terms of m, n, a. We have
K = a — 3n,
i = a — 3m,
28 = ??i2-m + 8?? -3a,
2T = n2 - n + 8m - 3a.
It is implied in Pliicker's theorem that, m, n, 8, K, r, i signifying as above in
regard to any curve, then in regard to the reciprocal curve n, m, r, i, 8, K will have
the same significations, viz. for the reciprocal curve these letters denote respectively
the order, class, number of nodes, cusps, double tangents, and inflexions.
The expression ^m(m + 3) — 8 — 2/c is that of the number of the disposable con
stants in a curve of the order m with 8 nodes and K cusps (in fact that there shall
be a node is 1 condition, a cusp 2 conditions): and the equation (9) thus expresses
that the curve and its reciprocal contain each of them the same number of disposable
constants.
For a curve of the order m, the expression ^m(m — 1) — 8 — K is termed the
" deficiency " (as to this more hereafter) ; the equation (10) expresses therefore that
the curve and its reciprocal have each of them the same deficiency.
The relations m2 — 28 — 3/c = ?i2 — 2r — 3t, = m + n, present themselves in the theory
of envelopes, as will appear further on.
With regard to the demonstration of Pliicker's equations it is to be remarked
that we are not able to write down the equation in point-coordinates of a curve of
the order ra, having the given numbers 8 and K of nodes and cusps. We can only
use the general equation (*]£#, y, z)m — 0, say for shortness u=0, of a curve of the
mth order, which equation, so long as the coefficients remain arbitrary, represents a
curve without nodes or cusps. Seeking then, for this curve, the values n, t, T of the
class, number of inflexions, and number of double tangents, — first, as regards the class,
this is equal to the number of tangents which can be drawn to the curve from an
arbitrary point, or what is the same thing, it is equal to the number of the points
of contact of these tangents. The points of contact are found as the intersections of
the curve u = 0 by a curve depending on the position of the arbitrary point, and
called the "first polar" of this point; the order of the first polar is = m — 1, and
785] CURVE. 471
the number of intersections is thus =m(m— 1). But it can be shown, analytically or
geometrically, that if the given curve has a node, the first polar passes through this
node, which therefore counts as two intersections : and that if the curve has a cusp,
the first polar passes through the cusp, touching the curve there, and hence the cusp
counts as three intersections. But, as is evident, the node or cusp is not a point of
contact of a proper tangent from the arbitrary point; we have, therefore, for a node
a diminution 2, and for a cusp a diminution 3, in the number of the intersections ;
and thus, for a curve with 8 nodes and K cusps, there is a diminution 28 + 3/e, and
the value of n is n= m(m — 1) — 28 — 3/e.
Secondly, as to the inflexions, the process is a similar one ; it can be shown that
the inflexions are the intersections of the curve by a derivative curve called (after
Hesse, who first considered it) the Hessian, defined geometrically as the locus of a
point such that its conic polar in regard to the curve breaks up into a pair of lines,
and which has an equation H = 0, where H is the determinant formed with the second
differential coefficients of u in regard to the variables (x, y, z); H=0 is thus a curve
of the order 3(m— 2), and the number of inflexions is =3m(m— 2). But if the given
curve has a node, then not only the Hessian passes through the node, but it has
there a node the two branches at which touch respectively the two branches of the
curve, and the node thus counts as six intersections ; so if the curve has a cusp,
then the Hessian not only passes through the cusp, but it has there a cusp through
which it again passes, that is, there is a cuspidal branch touching the cuspidal branch
of the curve, and besides a simple branch passing through the cusp, and hence the
cusp counts as eight intersections. The node or cusp is not an inflexion, and we have
thus for a node a diminution 6, and for a cusp a diminution 8, in the number of
the intersections ; hence for a curve with 8 nodes and K cusps, the diminution is
= 68 + 8tc, and the number of inflexions is i = 3m (ra — 2) — 68 — 8/c.
Thirdly, for the double tangents; the points of contact of these are obtained as the
intersections of the curve by a curve H = 0, which has not as yet been geometrically
defined, but which is found analytically to be of the order (m — 2) (m- — 9) ; the
number of intersections is thus = m(m — 2)(m2- 9) ; but if the given curve has a node
then there is a diminution = 4 (ra2 — m — 6), and if it has a cusp then there is a
diminution = 6 (m2 — m- 6), where, however, it is to be noticed that the factor
(ra2 — m — 6) is in the case of a curve having only a node or only a cusp the number
of the tangents which can be drawn from the node or cusp to the curve, and is used
as denoting the number of these tangents, and ceases to be the correct expression
if the number of nodes and cusps is greater than unity. Hence, in the case of a
curve which has 8 nodes and K cusps, the apparent diminution 2 (ra2 — m — 6) (28 + 3/c) is
too great, and it has in fact to be diminished by 2 {28 (8- 1) + 68* + f K (K- 1)}, or the
half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and
9 for each pair of cusps. We have thus finally an expression for 2r, =m(m— 2)(m2— 9)— &c.;
or dividing the whole by 2, we have the expression for r given by the third of
Plitcker's equations.
It is obvious that we cannot by consideration of the equation u — 0 in point-
coordinates obtain the remaining three of Pliicker's equations ; they might be obtained
472
CUKVE.
[785
in a precisely analogous manner by means of the equation v = 0 in line-coordinates,
but they follow at once from the principle of duality, viz. they are obtained by the
mere interchange of m, 8, K with n, r, i respectively.
To complete Plticker's theory it is necessary to take account of compound singu
larities ; it might be possible, but it is at any rate difficult, to effect this by considering
the curve as in course of description by the point moving along the rotating line ;
and it seems easier to consider the compound singularity as arising from the variation
of an actually described curve with ordinary singularities. The most simple case is
when three double points come into coincidence, thereby giving rise to a triple point ;
and a somewhat more complicated one is when we have a cusp of the second kind,
or node-cusp arising from the coincidence of a node, a cusp, an inflexion, and a double
tangent, as shown in the annexed figure, which represents the singularities as on the
point of coalescing. The general conclusion (see Cay ley, Quart. Math. Jour. t. vn., 1866,
[374], " On the higher singularities of a plane curve ") is that every singularity whatever
may be considered as compounded of ordinary singularities, say we have a singularity = 8'
nodes, K cusps, T' double tangents, and i inflexions. So that, in fact, Pliicker's equations
properly understood apply to a curve with any singularities whatever.
By means of Plticker's equations we may form a table —
m
n
8
K
T
t
0
1
0
0
1
0
0
0
—
—
2
2
0
0
0
0
3
6
0
0
0
9
4
1
0
0
3
3
0
1
0
1
4
12
0
0
28
24
10
1
0
16
18
9
0
1
10
16
8
2
0
8
12
7
1
1
4
10
6
0
2
1
8
6
3
0
4
6
5
2
1
2
4
4
1
2
1
2
3
0
3
1
0
785] CURVE. 473
The table is arranged according to the value of m ; and we have in = 0, n = 1, the
point; m = l, n = Q, the line; m = 2, n = 2, the conic; of w=3, the cubic, there are
three cases, the class being 6, 4, or 3, according as the curve is without singularities,
or as it has 1 node, or 1 cusp ; and so of m = 4, the quartic, there are nine cases,
where observe that in two of them the class is = 6, — the reduction of class arising from
two cusps or else from three nodes. The nine cases may be also grouped together
into four, according as the number of nodes and cusps (S + «) is =0, 1, 2, or 3.
The cases may be divided into sub-cases, by the consideration of compound singu
larities ; thus when m = 4, n = Q, = 3, the three nodes may be all distinct, which is the
general case, or two of them may unite together into the singularity called a tacnode,
or all three may unite together into a triple point, or else into an oscnode.
We may further consider the inflexions and double tangents, as well in general as
in regard to cubic and quartic curves.
The expression for the number of inflexions 3m (m — 2) for a curve of the order
m was obtained analytically by Plticker, but the theory was first given in a complete
form by Hesse in the two papers "Ueber die Elimination, u.s.w.," and "Ueber die
Wendepuncte der Curveri dritter Ordnung" (Grelle, t. xxvm., 1844); in the latter of
these the points of inflexion are obtained as the intersections of the curve u = 0
with the Hessian, or curve A = 0, where A is the determinant formed with the second
derived functions of u. We have in the Hessian the first instance of a covariant of
a ternary form. The whole theory of the inflexions of a cubic curve is discussed
in a very interesting manner by means of the canonical form of the equation
#3 + 2/3 + zz + Qlocyz= 0 ; and in particular a proof is given of Pliicker's theorem that the
nine points of inflexion of a cubic curve lie by threes in twelve lines.
It may be noticed that the nine inflexions of a cubic curve are three real, six
imaginary; the three real inflexions lie in a line, as was known to Newton and
Maclaurin. For an acnodal cubic the six imaginary inflexions disappear, and there
remain three real inflexions lying in a line. For a crunodal cubic, the six inflexions
which disappear are two of them real, the other four imaginary, and there remain two
imaginary inflexions and one real inflexion. For a cuspidal cubic the six imaginary
inflexions and two of the real inflexions disappear, and there remains one real inflexion.
A quartic curve has 24 inflexions; it was conjectured by Salmon, and has been
verified recently by Zeuthen, that at most 8 of these are real.
The expression |m(m — 2) (m2 — 9) for the number of double tangents of a curve
of the order m was obtained by Plticker only as a consequence of his first, second,
fourth, and fifth equations. An investigation by means of the curve H = 0, which by
its intersections with the given curve determines the points of contact of the double
tangents, is indicated by Cayley, "Recherches sur 1'elimination et la theorie des courbes",
(Crelle, t. xxxiv., 1847), [53] : and in part carried out by Hesse in the memoir " Ueber
Curven dritter Ordnung" (Crelle, t. XXXVT., 1848). A better process was indicated by
Salmon in the " Note on the double tangents to plane curves," Phil. Mag. 1858 ;
considering the m — 2 points in which any tangent to the curve again meets the
c. XL 60
474 CURVE. [785
curve, he showed how to form the equation of a curve of the order (m — 2), giving
by its intersection with the tangent the points in question; making the tangent touch
this curve of the order (m — 2), it will be a double tangent of the original curve.
See Cayley, "On the Double Tangents of a Plane Curve", (Phil. Trans, t. CXLVIIL,
1859), [260], and Dersch (Math. Ann. t. vn., 1874). The solution is still in so far
incomplete that we have no properties of the curve II = 0, to distinguish one such
curve from the several other curves which pass through the points of contact of the
double tangents.
A quartic curve has 28 double tangents, their points of contact determined as the
intersections of the curve by a curve II = 0 of the order 14, the equation of which
in a very elegant form was first obtained by Hesse (1849). Investigations in regard
to them are given by Pliicker in the Theorie der algebraischen Gurven, and in two
memoirs by Hesse and Steiner (Crelle, t. XLV., 1855), in respect to the triads of double
tangents which have their points of contact on a conic, and other like relations. It
was assumed by Pliicker that the number of real double tangents might be 28, 16,
8, 4, or 0, but Zeuthen has recently found that the last case does not exist.
The Hessian A has just been spoken of as a co variant of the form u ; the
notion of invariants and covariants belongs rather to the form u than to the curve
u = 0 represented by means of this form ; and the theory may be very briefly referred
to. A curve u = 0 may have some invariantive property, viz. a property independent
of the particular axes of coordinates used in the representation of the curve by its
equation ; for instance, the curve may have a node, and in order to this, a relation,
say A = 0, must exist between the coefficients of the equation ; supposing the axes of
coordinates altered, so that the equation becomes u' = 0, and writing A' = 0 for the
relation between the new coefficients, then the relations -4=0, A' = 0, as two different
expressions of the same geometrical property, must each of them imply the other ;
this can only be the case when A, A' are functions differing only by a constant factor,
or say, when A is an invariant of u. If, however, the geometrical property requires
two or more relations between the coefficients, say A = 0, B = Q, &c., then we must
have between the new coefficients the like relations, A' = 0, B' = Q, &c., and the two
systems of equations must each of them imply the other ; when this is so, the system
of equations, A = 0, B = 0, &c., is said to be invariantive, but it does not follow that
A, B, &c., are of necessity invariants of u. Similarly, if we have a curve U—0 derived
from the curve u = 0 in a manner independent of the particular axes of coordinates,
then from the transformed equation u = 0 deriving in like manner the curve U' — 0,
the two equations U = 0, U' = 0 must each of them imply the other ; and when this
is so, U will be a covariant of u. The case is less frequent, but it may arise, that
there are covariant systems £7 = 0, F=0, &c., and U' = 0, F' = 0, &c., each implying the
other, but where the functions U, V, &c., are not of necessity covariants of u.
The theory of the invariants and covariants of a ternary cubic function u has been
studied in detail, and brought into connexion with the cubic curve u = 0 ; but the
theory of the invariants and covariants for the next succeeding case, the ternary quartic
function, is still very incomplete.
785] CURVE. 475
In further illustration of the Pluckerian dual generation of a curve, we may con
sider the question of the envelope of a variable curve. The notion is very probably
older, but it is at any rate to be found in Lagrange's Thforie des f auctions analytiques
(1798) ; it is there remarked that the equation obtained by the elimination of the
parameter a from an equation f(x, y, a) = 0 and the derived equation in respect to a
is a curve, the envelope of the series of curves represented by the equation / (x, y, a) = 0
in question. To develope the theory, consider the curve corresponding to any particular
value of the parameter ; this has with the consecutive curve (or curve belonging to
the consecutive value of the parameter) a certain number of intersections, and of
common tangents, which may be considered as the tangents at the intersections ; and
the so-called envelope is the curve which is at the same time generated by the points
of intersection and enveloped by the common tangents ; we have thus a dual gener
ation. But the question needs to be further examined. Suppose that in general the
variable curve is of the order m with 8 nodes and K cusps, and therefore of the class
n with T double tangents and i inflexions, m, n, 8, K, T, i being connected by the
Pluckerian equations, — the number of nodes or cusps may be greater for particular values
of the parameter, but this is a speciality which may be here disregarded. Considering
the variable curve corresponding to a given value of the parameter, or say simply the
variable curve, the consecutive curve has then also 8 and K nodes and cusps, con
secutive to those of the variable curve ; and it is easy to see that among the
intersections of the two curves we have the nodes each counting twice, and the cusps
each counting three times; the number of the remaining intersections is = m2— 2S— 3/c.
Similarly among the common tangents of the two curves we have the double tangents
each counting twice, and the stationary tangents each counting three times, and the
number of the remaining common tangents is =?i2— 2r — 3t (=m2 — 28 — 3/e, inasmuch
as each of these numbers is as was seen = m + n). At any one of the m2 — 28 — 3/c
points the variable curve and the consecutive curve have tangents distinct from yet
innnitesimally near to each other, and each of these two tangents is also infinitesimally
near to one of the n2 — 2r — '3i common tangents of the two curves ; whence, attending
only to the variable curve, and considering the consecutive curve as coming into actual
coincidence with it, the n- — 2r — '3i common tangents are the tangents to the variable
curve at the m- — 28 — 3/c points respectively, and the envelope is at the same time
generated by the m- — 28 — 3/c points, and enveloped by the n2 — 2r — 3t tangents ; we
have thus a dual generation of the envelope, which only differs from Pliicker's dual
generation, in that in place of a single point and tangent we have the group of
m2 — 28 — 3/c points and n- — 2r — 3t tangents.
The parameter which determines the variable curve may be given as a point upon
a given curve, or say as a parametric point; that is, to the different positions of the
parametric point on the given curve correspond the different variable curves, and the
nature of the envelope will thus depend on that of the given curve ; we have thus
the envelope as a derivative curve of the given curve. Many well-known derivative
curves present themselves in this manner ; thus the variable curve may be the normal
(or line at right angles to the tangent) at any point of the given curve ; the inter
section of the consecutive normals is the centre of curvature ; and we have the evolute
60—2
476 CURVE. [785
as at once the locus of the centre of curvature and the envelope of the normal. It
may be added that the given curve is one of a series of curves, each cutting the
several normals at right angles. Any one of these is a " parallel " of the given curve ;
and it can be obtained as the envelope of a circle of constant radius having its centre
on the given curve. We have in like manner, as derivatives of a given curve, the
caustic, catacaustic, or diacaustic, as the case may be, and the secondary caustic, or
curve cutting at right angles the reflected or refracted rays.
We have in much that precedes disregarded, or at least been indifferent to, reality;
it is only thus that the conception of a curve of the mth order, as one which is
met by every right line in in points, is arrived at ; and the curve itself, and the line
which cuts it, although both are tacitly assumed to be real, may perfectly well be
imaginary. For real figures we have the general theorem that imaginary intersections, &c.,
present themselves in conjugate pairs: hence, in particular, that a curve of an even
order is met by a line in an even number (which may be = 0) of points ; a curve
of an odd order in an odd number of points, hence in one point at least ; it will be seen
further on that the theorem may be generalized in a remarkable manner. Again, when
there is in question only one pair of points or lines, these, if coincident, must be real ;
thus, a line meets a cubic curve in three points, one of them real, the other two real
or imaginary ; but if two of the intersections coincide they must be real, and we have
a line cutting a cubic in one real point and touching it in another real point. It
may be remarked that this is a limit separating the two cases where the intersec
tions are all real, and where they are one real, two imaginary.
Considering always real curves, we obtain the notion of a branch ; any portion
capable of description by the continuous motion of a point is a branch ; and a curve
consists of one or more branches. Thus the curve of the first order or right line
consists of one branch ; but in curves of the second order, or conies, the ellipse and
the parabola consist each of one branch, the hyperbola of two branches. A branch
is either re-entrant, or it extends both ways to infinity, and in this case, we may
regard it as consisting of two legs (crura, Newton), each extending one way to infinity,
but without any definite separation. The branch, whether re-entrant or infinite, may
have a cusp or cusps, or it may cut itself or another branch, thus having or giving
rise to crunodes; an acnode is a branch by itself, — it may be considered as an
indefinitely small re-entrant branch. A branch may have inflexions and double tangents,
or there may be double tangents which touch two distinct branches ; there are also
double tangents with imaginary points of contact, which are thus lines having no visible
connexion with the curve. A re-entrant branch not cutting itself may be everywhere
convex, and it is then properly said to be an oval ; but the term oval may be used
more generally for any re-entrant branch not cutting itself; and we may thus speak
of a once indented, twice indented oval, &c., or even of a cuspidate oval. Other
descriptive names for ovals and re-entrant branches cutting themselves may be used
when required ; thus, in the last-mentioned case a simple form is that of a figure of
eight; such a form may break up into two ovals, or into a doubly indented oval or
hour-glass. A form which presents itself is when two ovals, one inside the other,
unite, so as to give rise to a crunode — in default of a better name this may be called,
785] CURVE. 47
after the curve of that name, a Iima9on. Names may also be used for the different
forms of infinite branches, but we have first to consider the distinction of hyperbolic
and parabolic. The leg of an infinite branch may have at the extremity a tangent ;
this is an asymptote of the curve, and the leg is then hyperbolic ; or the leg may
tend to a fixed direction, but so that the tangent goes further and further off to
infinity, and the leg is then parabolic ; a branch may thus be hyperbolic or parabolic
as to its two legs ; or it may be hyperbolic as to one leg, and parabolic as to the
other. The epithets hyperbolic and parabolic are of course derived from the conies hyper
bola and parabola respectively. The nature of the two kinds of branches is best under
stood by considering them as projections, in the same way as we in effect consider the
hyperbola and the parabola as projections of the ellipse. If a line ft cut an arc aa', so
that the two segments ab, ba' lie on opposite sides of the line, then projecting the
figure so that the line ft goes off to infinity, the tangent at b is projected into the
asymptote, and the arc ab is projected into a hyperbolic leg touching the asymptote
at one extremity; the arc ba' will at the same time be projected into a hyperbolic
leg touching the same asymptote at the other extremity (and on the opposite side),
but so that the two hyperbolic legs may or may not belong to one and the same
branch. And wre thus see that the two hyperbolic legs belong to a simple inter
section of the curve by the line infinity. Next, if the line ft touch at b the arc aa'
so that the two portions ab', ba lie on the same side of the line ft, then projecting
the figure as before, the tangent at b, that is, the line ft itself, is projected to infinity ;
the arc ab is projected into a parabolic leg, and at the same time the arc ba' is
projected into a parabolic leg, having at infinity the same direction as the other leg,
but so that the two legs may or may not belong to the same branch. And we thus
see that the two parabolic legs represent a contact of the line infinity with the
curve, — the point of contact being of course the point at infinity determined by the
common direction of the two legs. It will readily be understood how the like con
siderations apply to other cases, — for instance, if the line ft is a tangent at an inflexion,
passes through a cruriode, or touches one of the branches of a crunode, &c. ; thus, if
the line ft passes through a crunode we have pairs of hyperbolic legs belonging to
two parallel asymptotes. The foregoing considerations also show (what is very important)
how different branches are connected together at infinity, and lead to the notion of
a complete branch, or circuit.
The two legs of a hyperbolic branch may belong to different asymptotes, and in
this case we have the forms which Newton calls inscribed, circumscribed, ambigene, &c. ;
or they may belong to the same asymptote, and in this case we have the serpentine
form, where the branch cuts the asymptote, so as to touch it at its two extremities
on opposite sides, or the conchoidal form, where it touches the asymptote on the same
side. The two legs of a parabolic branch may converge to ultimate parallelism, as in
the conic parabola, or diverge to ultimate parallelism, as in the semi-cubical parabola
y2 = x3, and the branch is said to be convergent, or divergent, accordingly ; or they
may tend to parallelism in opposite senses, as in the cubical parabola y=a?. As
mentioned with regard to a branch generally, an infinite branch of any kind may have
cusps, or, by cutting itself or another branch, may have or give rise to a crunode, &c.
478 CURVE. [785
We may now consider the various forms of cubic curves, as appearing by Newton's
Enumeratio, and by the figures belonging thereto. The species are reckoned as 72,
which are numbered accordingly 1 to 72; but to these should be added 10a, 13a, 22a,
and 226. It is not intended here to consider the division into species, nor even com
pletely that into genera, but only to explain the principle of classification. It may
be remarked generally that there are at most three infinite branches, and that there
may besides be a re-entrant branch or oval.
The genera may be arranged as follows : —
1, 2, 3, 4 redundant hyperbolas,
5, 6 defective hyperbolas,
7, 8 parabolic hyperbolas,
9 hyperbolisms of hyperbola,
10 „ „ ellipse,
11 „ „ parabola,
12 trident curve,
13 divergent parabolas,
>
14 cubic parabola ;
and, thus arranged, they correspond to the different relations of the line infinity to the
curve. First, if the three intersections by the line infinity are all distinct, we have
the hyperbolas ; if the points are real, the redundant hyperbolas, with three hyperbolic
branches ; but if only one of them is real, the defective hyperbolas, with one hyperbolic
branch. Secondly, if two of the intersections coincide, say if the line infinity meets
the curve in a onefold point and a twofold point, both of them real, then there is
always one asymptote : the line infinity may at the twofold point touch the curve, and
we have the parabolic hyperbolas ; or the twofold point may be a singular point, —
viz. a crunode giving the hyperbolisms of the hyperbola ; an acnode, giving the hyper
bolisms of the ellipse ; or a cusp, giving the hyperbolisms of the parabola. As regards
the so-called hyperbolisms, observe that (besides the single asymptote) we have in the
case of those of the hyperbola two parallel asymptotes ; in the case of those of the
ellipse the two parallel asymptotes become imaginary, that is, they disappear, and in
the case of those of the parabola they become coincident, that is, there is here an
ordinary asymptote, and a special asymptote answering to a cusp at infinity. Thirdly,
the three intersections by the line infinity may be coincident and real; or say we
have a threefold point : this may be an inflexion, a crunode, or a cusp, that is, the
line infinity may be a tangent at an inflexion, and we have the divergent parabolas:
a tangent at a crunode to one branch, and we have the trident curve ; or lastly, a
tangent at a cusp, and we have the cubical parabola.
It is to be remarked that the classification mixes together non-singular and singular
curves, in fact, the five kinds presently referred to : thus the hyperbolas and the
divergent parabolas include curves of every kind, the separation being made in the
785] CURVE. 479
species ; the hyperbolisms of the hyperbola and ellipse, and the trident curve, are nodal ;
the hyperbolisms of the parabola, and the cubical parabola, are cuspidal. The divergent
parabolas are of five species which respectively belong to and determine the five kinds
of cubic curves; Newton gives (in two short paragraphs without any development) the
remarkable theorem that the five divergent parabolas by their shadows generate and
exhibit all the cubic curves.
The five divergent parabolas are curves each of them symmetrical with regard to
an axis. There are two non-singular kinds, the one with, the other without, an oval,
but each of them has an infinite (as Newton describes it) campaniform branch ; this
cuts the axis at right angles, being at first convex, but ultimately concave, towards
the axis, the two legs continually tending to become at right angles to the axis. The
oval may unite itself with the infinite branch, or it may dwindle into a point, and
we have the crunodal and the acnodal forms respectively ; or if simultaneously the oval
dwindles into a point and unites itself to the infinite branch, we have the cuspidal
form. Drawing a line to cut any one of these curves and projecting the line to infinity,
it would not be difficult to show how the line should be drawn in order to obtain a
curve of any given species. We have herein a better principle of classification; con
sidering cubic curves, in the first instance, according to singularities, the curves are
non-singular, nodal (viz. crunodal or acnodal), or cuspidal; and we see further that
there are two kinds of non-singular curves, the complex and the simplex. There is
thus a complete division into the five kinds, the complex, simplex, crunodal, acnodal,
and cuspidal. Each singular kind presents itself as a limit separating two kinds of
inferior singularity; the cuspidal separates the crunodal and the acnodal, and these last
separate from each other the complex and the simplex.
The whole question is discussed very fully and ably by Mobius in the memoir
"Ueber die Grundformen der Linien dritter Ordnung" (Abh. der K. Sachs. Ges. zu
Leipzig, t. I., 1852; Ges. Werke, t. L). The author considers not only plane curves, but also
cones, or, what is almost the same thing, the spherical curves which are their sections
by a concentric sphere. Stated in regard to the cone, we have there the fundamental
theorem that there are two different kinds of sheets : viz. the single sheet, not sepa
rated into two parts by the vertex (an instance is afforded by the plane considered
as a cone of the first order generated by the motion of a line about a point), and
the double or twin-pair sheet, separated into two parts by the vertex (as in the cone
of the second order). And it then appears that there are two kinds of non-singular
cubic cones, viz. the simplex, consisting of a single sheet, and the complex, consisting
of a single sheet and a twin-pair sheet ; and we thence obtain (as for cubic curves) the
crunodal, the acnodal, and the cuspidal kinds of cubic cones. It may be mentioned
that the single sheet is a sort of wavy form, having upon it three lines of inflexion,
and which is met by any plane through the vertex in one or in three lines; the
twin-pair sheet has no lines of inflexion, and resembles in its form a cone on an
oval base.
In general a cone consists of one or more single or twin-pair sheets, and if we
consider the section of the cone by a plane, the curve consists of one or more com
plete branches, or say circuits, each of them the section of one sheet of the cone ;
480 CURVE. [785
thus, a cone of the second order is one twin-pair sheet, and any section of it is one
circuit composed, it may be, of two branches. But although we thus arrive by pro
jection at the notion of a circuit, it is not necessary to go out of the plane, and
we may (with Zeuthen, using the shorter term circuit for his complete branch} define a
circuit as any portion (of a curve) capable of description by the continuous motion
of a point, it being understood that a passage through infinity is permitted. And we
then say that a curve consists of one or more circuits ; thus the right line, or curve
of the first order, consists of one circuit; a curve of the second order consists of one
circuit ; a cubic curve consists of one circuit or else of two circuits.
A circuit is met by any right line always in an even number, or always in an
odd number, of points, and it is said to be an even circuit or an odd circuit
accordingly ; the right line is an odd circuit, the conic an even circuit. And we have
then the theorem, two odd circuits intersect in an odd number of points ; an odd and
an even circuit, or two even circuits, in an even number of points. An even circuit
not cutting itself divides the plane into two parts, the one called the internal part,
incapable of containing any odd circuit, the other called the external part, capable of
containing an odd circuit.
We may now state in a more convenient form the fundamental distinction of the
kinds of cubic curve. A non-singular cubic is simplex, consisting of one odd circuit,
or it is complex, consisting of one odd circuit and one even circuit. It may be added
that there are on the odd circuit three inflexions, but on the even circuit no inflexion ;
it hence also appears that from any point of the odd circuit there can be drawn to
the odd circuit two tangents, and to the even circuit (if any) two tangents, but that
from a point of the even circuit there cannot be drawn (either to the odd or the
even circuit) any real tangent ; consequently, in a simplex curve the number of tangents
from any point is two ; but in a complex curve the number is four, or none, — four if
the point is on the odd circuit, none if it is on the even circuit. It at once appears
from inspection of the figure of a non- singular cubic curve, which is the odd and
which the even circuit. The singular kinds arise as before ; in the crunodal and the
cuspidal kinds the whole curve is an odd circuit, but in the acnodal kind the acnode
must be regarded as an even circuit.
The analogous question of the classification of quartics (in particular non-singular
quartics and nodal quartics) is considered in Zeuthen's memoir " Sur les differentes
formes des courbes planes du quatrieme ordre " (Math. Ann. t. VIL, 1874). A non-
singular quartic has only even circuits ; it has at most four circuits external to each
other, or two circuits one internal to the other, and in this last case the internal
circuit has no double tangents or inflexions. A very remarkable theorem is established
as to the double tangents of such a quartic : — distinguishing as a double tangent of
the first kind a real double tangent which either twice touches the same circuit, or
else touches the curve in two imaginary points, the number of the double tangents
of the first kind of a non-singular quartic is =4; it follows that the quartic has at
most 8 real inflexions. The forms of the non-singular quartics are very numerous, but
it is not necessary to go further into the question.
785] CUKVE. 481
We may consider in relation to a curve, not only the line infinity, but also the
circular points at infinity ; assuming the curve to be real, these present themselves
always conjointly ; thus a circle is a conic passing through the two circular points,
and is thereby distinguished from other conies. Similarly a cubic through the two
circular points is termed a circular cubic; a quartic through the two points is termed
a circular quartic, and if it passes twice through each of them, that is, has each of
them for a node, it is termed a bicircular quartic. Such a quartic is of course binodal
(m = 4, 8 = 2, K = 0) ; it has not in general, but it may have, a third node, or a cusp.
Or again, we may have a quartic curve having a cusp at each of the circular points :
such a curve is a " Cartesian," it being a complete definition of the Cartesian to say
that it is a bicuspidal quartic curve (m = 4, 8 = 0, K = 2), having a cusp at each of the
circular points. The circular cubic and the bicircular quartic, together with the Cartesian
(being in one point of view a particular case thereof), are interesting curves which
have been much studied, generally, and in reference to their focal properties.
The points called foci presented themselves in the theory of the conic, and were
well known to the Greek geometers, but the general notion of a focus was first
established by Plucker, in the memoir "Ueber solche Puncte die bei Curven einer
hoheren Ordnung den Brennpuncten der Kegelschnitte entsprechen," (Crelle, t. x., 1833).
We may from each of the circular points draw tangents to a given curve ; the inter
section of two such tangents (belonging of course to the two circular points respectively)
is a focus. There will be from each circular point X tangents (A,, a number depending
on the class of the curve and its relation to the line infinity and the circular points,
= 2 for the general conic, 1 for the parabola, 2 for a circular cubic or a bicircular
quartic, &c.); the \ tangents from the one circular point and those from the other
circular point intersect in X real foci (viz. each of these is the only real point on
each of the tangents through it), and in X2 — X imaginary foci; each pair of real foci
determines a pair of imaginary foci (the so-called antipoints of the two real foci), and
the £X(X — 1) pairs of real foci thus determine the X2 — X imaginary foci. There are
in some cases points termed centres, or singular or multiple foci (the nomenclature is
unsettled), which are the intersections of improper tangents from the two circular points
respectively; thus, in the circular cubic, the tangents to the curve at the two circular
points respectively (or two imaginary asymptotes of the curve) meet in a centre.
The notions of distance and of lines at right angles are connected with the circular
points ; and almost every construction of a curve by means of lines of a determinate
length, or at right angles to each other, and (as such) mechanical constructions by
means of linkwork, give rise to curves passing the same definite number of times
through the two circular points respectively, or say to circular curves, and in which
the fixed centres of the construction present themselves as ordinary, or as singular,
foci. Thus the general curve of three-bar motion (or locus of the vertex of a triangle,
the other two vertices whereof move on fixed circles) is a tricircular sextic, having
besides three nodes (m = 6, 8 = 3 + 3 + 3, =9), and having the centres of the fixed circles
each for a singular focus ; there is a third singular focus, and we have thus the remark
able theorem (due to Mr S. Roberts) of the triple generation of the curve by means
of the three several pairs of singular foci.
C. XL 61
482 CURVE. [785
Again, the normal, qua line at right angles to the tangent, is connected with the
circular points, and these accordingly present themselves in the before-mentioned theories
of evolutes and parallel curves.
We have several recent theories which depend on the notion of correspondence :
two points whether in the same plane or in different planes, or on the same curve
or in different curves, may determine each other in such wise that to any given
position of the first point there correspond a! positions of the second point, and to
any given position of the second point a positions of the first point ; the two points
have then an (a, a') correspondence ; and if a, a' are each = 1, then the two points
have a (1, 1) or rational correspondence. Connecting with each theory the author's
name, the theories in question are — Riemann, the rational transformation of a plane
curve ; Cremona, the rational transformation of a plane ; and Chasles, correspondence of
points on the same curve, and united points. The theory first referred to, with the
resulting notion of Geschlecht, or deficiency, is more than the other two an essential
part of the theory of curves, but they will all be considered.
Riemann's results are contained in the memoirs on " Theorie der Abel'schen
Functionen," (Crelle, t. Liv., 1857); and we have next Clebsch, "Ueber die Singularitaten
algebraischer Curven," (Crelle, t. LXV., 1865), and Cayley, "On the Transformation of
Plane Curves," (Proc. Lond. Math. Soc. t. I., 1865, [384]). The fundamental notion of
the rational transformation is as follows : —
Taking u, X, Y, Z to be rational and integral functions (X, Y, Z all of the same
order) of the coordinates (x, y, z), and u, X' , Y', Z' rational and integral functions
(X', F, Z' all of the same order) of the coordinates (x', y', z'}, we transform a given
curve u = 0, by the equations x' : y' : z' = X : Y : Z, thereby obtaining a transformed
curve u' = 0, and a converse set of equations x : y : z = X' : Y' : Z' ; viz. assuming
that this is so, the point (x, y, z} on the curve u = 0 and the point (x, y', /) on
the curve u' = 0 will be points having a (1, 1) correspondence. To show how this is,
observe that to a given point (x, y, z) on the curve u = 0 there corresponds a single
point (x, y', z'} determined by the equations x : y' : z = X : Y : Z ; from these equations
and the equation u = 0 eliminating x, y, z we obtain the equation u' — Q of the trans
formed curve. To a given point (x ', y', z') not on the curve w' = 0 there corresponds,
not a single point, but the system of points (x, y, z) given by the equations
x : y' : z' = X : Y : Z, viz. regarding x, y', z as constants (and to fix the ideas,
assuming that the curves X=0, F = 0, Z=0 have no common intersections), these are
the points of intersection of the curves X : Y : Z= x' : y' : z, but no one of these
points is situate on the curve u = Q. If, however, the point (x, y', z'} is situate on
the curve u' = 0, then one point of the system of points in question is situate on the
curve u = 0, that is, to a given point of the curve u' = 0 there corresponds a single
point of the curve w = 0; and hence also this point must be given by a system of
equations such as x : y : z = X' : Y : Z'.
It is an old and easily proved theorem that, for a curve of the order m, the
number B + K of nodes and cusps is at most =\ (ra-l)(m-2); for a given curve the
deficiency of the actual number of nodes and cusps below this maximum number, viz.
785] CURVE. 483
|- (m— 1) (ra — 2) — 8 — K, is the " Geschlecht," or "deficiency," of the curve, say this is
= D. When D = 0, the curve is said to be unicursal, when = 1, bicursal, and so on.
The general theorem is that two curves corresponding rationally to each other have
the same deficiency. In particular, a curve and its reciprocal have this rational or
(1, 1) correspondence, and it has been already seen that a curve and its reciprocal
have the same deficiency.
A curve of a given order can in general be rationally transformed into a curve
of a lower order ; thus a curve of any order for which D = 0, that is, a unicursal
curve, can be transformed into a line ; a curve of any order having the deficiency 1
or 2 can be rationally transformed into a curve of the order D + 2, deficiency D ; and
a curve of any order deficiency = or > 3 can be rationally transformed into a curve of
the order D + 3, deficiency D.
Taking x' ', y', z as coordinates of a point of the transformed curve, and in its
equation writing x' : y' : z' = 1 : 9 : <f> we have <f> a certain irrational function of 6, and
the theorem is that the coordinates x, y, z of any point of the given curve can be
expressed as proportional to rational and integral functions of 0, (f>, that is, of 6 and
a certain irrational function of 6.
In particular, if D = 0, that is, if the given curve be unicursal, the transformed
curve is a line, (f> is a mere linear function of 6, and the theorem is that the
coordinates x, y, z of a point of the unicursal curve can be expressed as proportional
to rational and integral functions of 6 ; it is easy to see that for a given curve of
the order m, these functions of 0 must be of the same order m.
If -0 = 1, then the transformed curve is a cubic ; it can be shown that in a cubic,
the axes of coordinates being properly chosen, <f> can be expressed as the square root
of a quartic function of 6; and the theorem is that the coordinates x, y, z of a
point of the bicursal curve can be expressed as proportional to rational and integral
functions of 6, and of the square root of a quartic function of 6.
And so if D— 2, then the transformed curve is a nodal quartic ; <j> can be ex
pressed as the square root of a sextic function of 0, and the theorem is, that the
coordinates x, y, z of a point of the tricursal curve can be expressed as proportional
to rational and integral functions of 6, and of the square root of a sextic function
of 6. But when D = 3, we have no longer the like law, viz. <£ is not expressible as
the square root of an octic function of 6.
Observe that the radical, square root of a quartic function, is connected with the
theory of elliptic functions, and the radical, square root of a sextic function, with that
of the first kind of Abelian functions, but that the next kind of Abelian functions
does not depend on the radical, square root of an octic function.
It is a form of the theorem for the case D — 1, that the coordinates x, y, z of
a point of the bicursal curve, or in particular the coordinates of a point of the cubic,
can be expressed as proportional to rational and integral functions of the elliptic
functions sn u, en u, dn u ; in fact, taking the radical to be Vl — & . I — &&*, and writing
61—2
484 CURVE. [785
6 = sn u, the radical becomes = en u . dn u ; and we have expressions of the form in
question.
It will be observed that the equations x' : y' : z' = X : Y : Z before-mentioned do
not of themselves lead to the other system of equations x : y : z = X' : Y' : Z', and
thus that the theory does not in anywise establish a (1, 1) correspondence between the
points (x, y, z) and (x', y', z') of two planes or of the same plane ; this is the corre
spondence of Cremona's theory.
In this theory, given in the memoirs " Sulle trasformazioni geometriche delle
figure piane," Mem. di Bologna, t. II. (1863), and t. v. (1865), we have a system of
equations x : y' : z = X : Y : Z which does lead to a system x : y : z = X' : Y' : Z',
where, as before, X, Y, Z denote rational and integral functions, all of the same order,
of the coordinates x, y, z, and X', Y', Z' rational and integral functions, all of the
same order, of the coordinates x, y', z', and there is thus a (1, 1) correspondence given
by these equations between the two points (x, y, z) and (x', y', z'}. To explain this,
observe that starting from the equations x : y' : z' = X : Y : Z, to a given point
(x, y, z} there corresponds one point (x', y', z'), but that if n be the order of the
functions X, Y, Z, then to a given point x, y', z there would, if the curves X = 0,
F=0, Z=0 had no common intersections, correspond n2 points (x, y, z). If, however,
the functions are such that the curves X = 0, F= 0, Z=Q have k common inter
sections, then among the n2 points are included these k points, which are fixed points
independent of the point (x ', y', z'} ; so that, disregarding these fixed points, the number
of points (x, y, z) corresponding to the given point (x, y, z') is = ?i2-&; and in
particular if k=n?— 1, then we have one corresponding point; and hence the original
system of equations x' : y' : z' = X : Y : Z must lead to the equivalent system
x : y : z = X' : Y' : Z' ; and in this system by the like reasoning the functions must
be such that the curves X' = 0, Y' = 0, Z' = 0 have n'2 - 1 common intersections. The
most simple example is in the two systems of equations x' : y : z' = yz : zx : xy and
x : y : z = y'z : z'x : x'y' ; where yz = 0, z<c = 0, xy = Q are conies (pairs of lines) having
three common intersections, and where obviously either system of equations leads to
the other system. In the case where X, Y, Z are of an order exceeding 2, the
required number n2 — 1 of common intersections can only occur by reason of common
multiple points on the three curves; and assuming that the curves X = 0, F=0, Z=0
have «! + «2 + «3 + . . . + an_j common intersections, where the «j points are ordinary points,
the «2 points are double points, the a3 points are triple points, &c., on each curve, we
have the condition
«! + 4a2 + 9«3 + ... + (n- I)2 o^ = n2 - 1 ;
but to this must be joined the condition
«! + 3a2 + 6«3 + ... +£ (n - 1) (n - 2) o^ = \n (n + 3) - 2,
(without which the transformation would be illusory); and the conclusion is that
«i> «2> ••-, <*n-i may be any numbers satisfying these two equations. It may be added
that the two equations together give
785] CURVE. 485
which expresses that the curves X = 0, F=0, Z=0 are unicursal. The transformation
may be applied to any curve u = 0, which is thus rationally transformed into a curve
u=0, by a rational transformation such as is considered in Riemann's theory; hence
the two curves have the same deficiency.
Coming next to Chasles, the principle of correspondence is established and used
by him in a series of memoirs relating to the conies which satisfy given conditions,
and to other geometrical questions, contained in the Comptes Rendus, t. LVIII. et seq.
(1864 to the present time). The theorem of united points in regard to points in a
right line was given in a paper, June — July 1864, and it was extended to unicursal
curves in a paper of the same series (March 1866), " Sur les courbes planes ou a
double courbure dont les points peuvent se determiner individuellement — application du
principe de correspondance dans la thdorie de ces courbes."
The theorem is as follows: if in a unicursal curve two points have an (a, /:?)
correspondence, then the number of united points (or points each corresponding to
itself) is = a + /3. In fact, in a unicursal curve the coordinates of a point are given
as proportional to rational and integral functions of a parameter, so that any point
of the curve is determined uniquely by means of this parameter; that is, to each
point of the curve corresponds one value of the parameter, and to each value of the
parameter one point on the curve ; and the (a, /3) correspondence between the two
points is given by an equation of the form (*$#, l)a (<£, 1)0=0 between their para
meters 6 and </> ; at a united point <£ = 6, and the value of Q is given by an equation
of the order a + /3. The extension to curves of any given deficiency D was made in
the memoir of Cayley, " On the correspondence of two points on a curve," — Proc.
Lond. Math. Soc. t. I. (1866), [385], — viz. taking P, P' as the corresponding points in an
(a, a') correspondence on a curve of deficiency D, and supposing that when P is given
the corresponding points P' are found as the intersections of the curve by a curve ©
containing the coordinates of P as parameters, and having with the given curve k
intersections at the point P, then the number of united points is a = a+ a'+ 2kD; and
more generally, if the curve © intersect the given curve in a set of points P' each
p times, a set of points Q' each q times, &c., in such manner that the points (P, P'),
the points (P, Q'), &c., are pairs of points corresponding to each other according to
distinct laws ; then if (P, P') are points having an (a, a') correspondence with a number
= a of united points, (P, Q') points having a (/8, /3') correspondence with a number =6
of united points, and so on, the theorem is that we have
The principle of correspondence, or say rather the theorem of united points, is a
most powerful instrument of investigation, which may be used in place of analysis for the
determination of the number of solutions of almost every geometrical problem. We can
by means of it investigate the class of a curve, number of inflexions, &c., — in fact,
Pliicker's equations; but it. is necessary to take account of special solutions; thus, in one
of the most simple instances, in finding the class of a curve, the cusps present them
selves as special solutions.
486 CURVE. [785
Imagine a curve of order m, deficiency D, and let the corresponding points P, P'
be such that the line joining them passes through a given point 0 ; this is an
(m—I, m— 1) correspondence, and the value of k is =1, hence the number of united
points is = 2m — 2 + 2D ; the united points are the points of contact of the tangents from
0 and (as special solutions) the cusps, and we have thus the relation w + /e=2ra— 2 + 2D ;
or, writing D = -|(ra— l)(m — 2) — 8 — K, this is n = m(m — 1) — 2S — 3*, which is right.
The principle in its original form as applying to a right line was used throughout
by Chasles in the investigations on the number of the conies which satisfy given
conditions, and on the number of solutions of very many other geometrical problems.
There is one application of the theory of the (a, a') correspondence between two
planes which it is proper to notice.
Imagine a curve, real or imaginary, represented by an equation (involving, it may
be, imaginary coefficients) between the Cartesian coordinates u, u ; then, writing
u = x + iy, u' = x' + iy', the equation determines real values of (x, y), and of (x', y'),
corresponding to any given real values of (x', y) and (x, y) respectively; that is, it
establishes a real correspondence (not of course a rational one) between the points
(x, y) and (x, y'} ; for example in the imaginary circle u2 + u'2 = (a + bi)2, the corre
spondence is given by the two equations x2 - y'2 + x'2 — y'2 = a2 — b2, xy + x'y' = ab. We
have thus a means of geometrical representation for the portions, as well imaginary
as real, of any real or imaginary curve. Considerations such as these have been used
for determining the series of values of the independent variable, and the irrational
functions thereof in the theory of Abelian integrals, but the theory seems to be worthy
of further investigation.
The researches of Chasles (Comptes Rendus, t. LVIIL, 1864, et seq.) refer to the
conies which satisfy given conditions. There is an earlier paper by De Jonquieres,
" Theoremes gdneVaux concernant les courbes ge'ometriques planes d'un ordre quelconque,"
Liouv. t. vi. (1861), which establishes the notion of a system of curves (of any order)
of the index N, viz. considering the curves of the order n which satisfy \n (n + 3) — 1
conditions, then the index N is the number of these curves which pass through a
given arbitrary point. But Chasles in the first of his papers (February 1864), con
sidering the conies which satisfy four conditions, establishes the notion of the two
characteristics (/*, v) of such a system of conies, viz. /z is the number of the conies
which pass through a given arbitrary point, and v is the number of the conies which
touch a given arbitrary line. And he gives the theorem, a system of conies satisfying
four conditions, and having the characteristics (/j,, v) contains 2i> — /j, line-pairs (that is,
conies, each of them a pair of lines), and 2/i — v point-pairs (that is, conies, each of
them a pair of points, — coniques infiniment aplaties), which is a fundamental one in
the theory. The characteristics of the system can be determined when it is known
how many there are of these two kinds of degenerate conies in the system, and how
often each is to be counted. It was thus that Zeuthen (in the paper Nyt Bydrag,
" Contribution to the Theory of Systems of Conies which satisfy four Conditions,"
Copenhagen, 1865, translated with an addition in the Nouvelles Annales) solved the
question of finding the characteristics of the systems of conies which satisfy four
785] CURVE. 487
conditions of contact with a given curve or curves ; and this led to the solution of
the further problem of finding the number of the conies which satisfy five conditions
of contact with a given curve or curves (Cayley, Comptes Rendus, t. LXIII., 1866, [377]),
and "On the Curves which satisfy given Conditions" (Phil. Trans, t. CLVIIL, 1868, [406]).
It may be remarked that although, as a process of investigation, it is very con
venient to seek for the characteristics of a system of conies satisfying 4 conditions,
yet what is really determined is in every case the number of the conies which satisfy
5 conditions ; the characteristics of the system (4>p) of the conies which pass through
4p points are (op), (4p, II), the number of the conies which pass through 5 points,
and which pass through 4 points and touch 1 line : and so in other cases. Similarly
as regards cubics, or curves of any other order : a cubic depends on 9 constants, and
the elementary problems are to find the number of the cubics (9p), (8p, II), &c., which
pass through 9 points, pass through 8 points and touch 1 line, &c. ; but it is in the
investigation convenient to seek for the characteristics of the systems of cubics (8p), &c.,
which satisfy 8 instead of 9 conditions.
The elementary problems in regard to cubics are solved very completely by Maillard
in his These, Recherche des caracteristiques des systemes elementaires des courbes planes du
troisieme ordre (Paris, 1871). Thus, considering the several cases of a cubic
No. of consts.
1. With a given cusp 5,
2. „ cusp on given line 6,
3. „ cusp 7,
4. „ a given node 6,
5. „ node on given line 7,
6. „ node 8,
7. non-singular 9,
he determines in every case the characteristics (p, v) of the corresponding systems of
cubics (4>p), (Sp, II), &c. The same problems, or most of them, and also the elementary
problems in regard to quartics are solved by Zeuthen, who in the elaborate memoir
"Almindelige Egenskaber, &c.," Danish Academy, t. x. (1873), considers the problem in
reference to curves of any order, and applies his results to cubic and quartic curves.
The methods of Maillard and Zeuthen are substantially identical ; in each case the
question considered is that of finding the characteristics (/*, v) of a system of curves
by consideration of the special or degenerate forms of the curves included in the
system. The quantities which have to be considered are very numerous. Zeuthen in
the case of curves of any given order establishes between the characteristics ytt, v, and
18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 23
independent equations), involving (besides the 20 quantities) other quantities relating
to the various forms of the degenerate curves, which supplementary terms he determines,
partially for curves of any order, but completely only for quartic curves. It is in the
discussion and complete enumeration of the special or degenerate forms of the curves,
488 CURVE. [785
and of the supplementary terms to which they give rise, that the great difficulty of
the question seems to consist ; it would appear that the 24 equations are a complete
system, and that (subject to a proper determination of the supplementary terms) they
contain the solution of the general problem.
The remarks which follow have reference to the analytical theory of the degenerate
curves which present themselves in the foregoing problem of the curves which satisfy
given conditions.
A curve represented by an equation in point-coordinates may break up : thus if
PI, P2)... be rational and integral functions of the coordinates (so, y, z) of the orders
ra^ m2)... respectively, we have the curve P^'P/*... = 0, of the order m, =alml+OL2mz+...,
composed of the curve P1 = 0 taken ^ times, the curve P2 = 0 taken «2 times, &c.
Instead of the equation P1a'P2a2... =0, we may start with an equation u = 0, where
u is a function of the order m containing a parameter 6, and for a particular value
say 0 = 0, of the parameter reducing itself to P^Pf* ____ Supposing 6 indefinitely
small, we have what may be called the penultimate curve, and when 6 = 0 the ultimate
curve. Regarding the ultimate curve as derived from a given penultimate curve, we
connect with the ultimate curve, and consider as belonging to it, certain points called
" summits " on the component curves Px = 0, P2 = 0, respectively ; a summit 2 is a point
such that, drawing from an arbitrary point 0 the tangents to the penultimate curve,
we have 02 as the limit of one of these tangents. The ultimate curve together with
its summits may be regarded as a degenerate form of the curve u = 0. Observe that
the positions of the summits depend on the penultimate curve u = 0, viz. on the values
of the coefficients in the terms multiplied by 6, #2, . . . ; they are thus in some measure
arbitrary points as regards the ultimate curve P^Pf* ... = 0.
It may be added that we have summits only on the component curves P3 = 0, of
a multiplicity ax > 1 ; the number of summits on such a curve is in general = (c^2— 0.1) mf.
Thus assuming that the penultimate curve is without nodes or cusps, the number of
the tangents to it is =m? — m, = (<xlm1 + a2m2+ ...)2 — (a1m1 + a2m2 +...), taking Pj = 0 to
have &j nodes and /^ cusps, and therefore its class Tij to be = m^ — m1— 2^ — 3/Cj, &c.,
the expression for the number of tangents to the penultimate curve is
= (tti2 - «!> Wi2 + (cc22 - Oj) w22 + . . . + 2«1a2m1™2 + . . . + aa fa + 28l + 3^) + «2 (na + 282 + 3/ca) + ...
where a term 2a1a2ra1rw2 indicates tangents which are in the limit the lines drawn to the
intersections of the curves Pj = 0, P2 = 0 each line 2«]o2 times; a term ^(^ + 2Sj + 3/Cj)
tangents which are in the limit the proper tangents to Px = 0 each «! times, the lines
to its nodes each 2ax times, and the lines to its cusps each 3^ times; the remaining
terms (o^2 — o^) m? + («22 — e^) m.22 + . . . indicate tangents which are in the limit the lines
drawn to the several summits, that is, we have (a^ — oil) ra^ summits on the curve
Pa = 0, &C.
There is of course a precisely similar theory as regards line-coordinates; taking
IIj , ILj , &c., to be rational and integral functions of the coordinates (f , 77, £), we con
nect with the ultimate curve I!^'!!/2... = 0, and consider as belonging to it certain
lines, which for the moment may be called "axes," tangents to the component curves
785] CURVE. 480
H! = 0, II2 = 0 respectively. Considering an equation in point-coordinates, we may have
among the component curves right lines ; and, if in order to put these in evidence, we
take the equation to be Lj* . . . Pja> . . . = 0, where Zj = 0 is a right line, P! = 0 a curve
of the second or any higher order, then the curve will contain as part of itself
summits not exhibited in this equation, but the corresponding line-equation will be
A/1 ... IT/1 ... =0, where Aj = 0, ... are the equations of the summits in question, 11! = 0,
&c., are the line-equations corresponding to the several point-equations Pl = 0, &c. ; and
this curve will contain as part of itself axes not exhibited by this equation, but which
are the lines L^ = 0, . . . of the equation in point-coordinates.
In conclusion a little may be said as to curves of double curvature, otherwise
twisted curves, or curves in space. The analytical theory by Cartesian coordinates was
first considered by Clairaut, Recherches sur les courbes d double courbure (Paris, 1731).
Such a curve may be considered as described by a point, moving in a line which at
the same time rotates about the point in a plane which at the same time rotates
about the line ; the point is a point, the line a tangent, and the plane an osculating
plane, of the curve ; moreover the line is a generating line, and the plane a tangent
plane, of a developable surface or torse, having the curve for its edge of regression.
Analogous to the order and class of a plane curve we have the order, rank, and class,
of the system (assumed to be a geometrical one), viz. if an arbitrary plane contains
TO points, an arbitrary line meets r lines, and an arbitrary point lies in n planes, of
the system, then m, r, n are the order, rank, and class respectively. The system has
singularities, and there exist between m, r, n and the numbers of the several singularities
equations analogous to Plticker's equations for a plane curve.
It is a leading point in the theory that a curve in space cannot in general be
represented by means of two equations U=0, F=0; the two equations represent
surfaces, intersecting in a curve ; but there are curves which are not the complete inter
section of any two surfaces; thus we have the cubic in space, or skew cubic, which is
the residual intersection of two quadric surfaces which have a line in common ; the
equations U=0, V=0 of the two quadric surfaces represent the cubic curve, not by
itself, but together with the line.
c. xi. 62
490 [786
786.
EQUATION.
[From the Encyclopaedia Britannica, Ninth Edition, vol. vill. (1878), pp. 497 — 509.]
THE present article includes Determinant and Theory of Equations; and it may
be proper to explain the relation to each other of the two subjects. Theory of
Equations is used in its ordinary conventional sense to denote the theory of a single
equation of any order in one unknown quantity ; that is, it does not include the
theory of a system or systems of equations of any order between any number of
unknown quantities. Such systems occur very frequently in analytical geometry and
other parts of mathematics, but they are hardly as yet the subject-matter of a
distinct theory ; and even Elimination, the transition-process for passing from a system
of any number of equations involving the same number of unknown quantities to a
single equation in one unknown quantity, hardly belongs to the Theory of Equations
in the above restricted sense. But there is one case of a system of equations which
precedes the Theory of Equations, and indeed presents itself at the outset of algebra,
that of a system of simple (or linear) equations. Such a system gives rise to the
function called a Determinant, and it is by means of these functions that the solution
of the equations is effected. We have thus the subject Determinant as nearly
equivalent to (but somewhat more extensive than) that of a system of linear equations ;
and we have the other subject, Theory of Equations, used in the restricted sense
above referred to, and as not including Elimination.
Determinant.
1. A sketch of the history of determinants is given under [the Article] Algebra; it
thereby appears that the algebraical function called a determinant presents itself in
the solution of a system of simple equations, and we have herein a natural source of
the theory. Thus, considering the equations
a x + b y + c 2 = d ,
a' x + b' y + c' z = d' ,
a"x + b"y + c"z = d",
786]
EQUATION.
491
and proceeding to solve them by the so-called method of cross multiplication, we
multiply' the equations by factors selected in such a manner that, upon adding the
results, the whole coefficient of y becomes = 0 and the whole coefficient of z becomes
= 0 ; the factors in question are b'c" — b"c, b"c — be", be' — b'c (values which, as at once
seen, have the desired property) ; we thus obtain an equation which contains on the
left-hand side only a multiple of SB, and on the right-hand side a constant term ;
the coefficient of x has the value
a (b'c" - b"c') + a' (b"c - be") + a" (be' - b'c),
and this function, represented in the form
a , b , c
a, b', c'
a", b", c"
is said to be a determinant ; or, the number of elements being 32, it is called a
determinant of the third order. It is to be noticed that the resulting equation is
a , b , c
a', b', c
a", b", c"
x = d , b , c
d', b', c'
d", b", c"
where the expression on the right-hand side is the like function with d, d', d" in
place of a, a', a" respectively, and is of course also a determinant. Moreover, the
functions b'c" — b"c', b"c — be", be' — b'c used in the process are themselves the determ
inants of the second order
b' , c' , b", c" , b, c
b", c" b , c b', c'
We have herein the suggestion of the rule for the derivation of the determinants of
the orders 1, 2, 3, 4, &c., each from the preceding one, viz. we have
a
= a,
a,
b
= a
b'
-a'
b
,
a',
b'
a
, b,
c
= a
b' ,
c'
+ a'
b", c"
+ a"
b, c ,
a'
, b',
c'
b",
c"
b, c
&', c'
a"
, b",
c"
a
, b
, c , d
= a
V , c
, d'
— a
b"
, c"
, d"
+ a"
b'", c'", d'" •
-a'" b , c , d
a'
, b'
, c', d'
b"
c"
, d"
b'", c'", d'"
b , c , d
b', cf, d'
a"
, b"
, c", d"
V"
, c"', d"
b
, c
, d
b' , c' , d'
b", c", d"
a'", b'", c"', d'"
and so on, the terms being all + for a determinant of an odd order, but alternately
+ and — for a determinant of an even order.
62—2
492 EQUATION. [786
2. It is easy, by induction, to arrive at the general results : —
A determinant of the order n is the sum of the 1 . 2 . 3 . . . n products which can
be formed with n elements out of n- elements arranged in the form of a square, no
two of the n elements being in the same line or in the same column, and each
such product having the coefficient + unity.
The products in question may be obtained by permuting in every possible manner
the columns (or the lines) of the determinant, and then taking for the factors the
n elements in the dexter diagonal. And we thence derive the rule for the signs,
viz. considering the primitive arrangement of the columns as positive, then an arrange
ment obtained therefrom by a single interchange (inversion, or derangement) of two
columns is regarded as negative ; and so in general an arrangement is positive or
negative according as it is derived from the primitive arrangement by an even or an
odd number of interchanges. This implies the theorem that a given arrangement
can be derived from the primitive arrangement only by an odd number, or else only
by an even number of interchanges, — a theorem the verification of which may be easily
obtained from the theorem (in fact, a particular case of the general one), an arrange
ment can be derived from itself only by an even number of interchanges. And this
being so, each product has the sign belonging to the corresponding arrangement of
the columns ; in particular, a determinant contains with the sign + the product of the
elements in its dexter diagonal. It is to be observed that the rule gives as many
positive as negative arrangements, the number of each being =^.l.2...n.
The rule of signs may be expressed in a different form. Giving to the columns
in the primitive arrangement the numbers 1, 2, 3, ... , n, to obtain the sign belonging
to any other arrangement we take, as often as a lower number succeeds a higher one,
the sign — , and, compounding together all these minus signs, obtain the proper sign,
-f or — as the case may be.
Thus, for three columns, it appears by either rule that 123, 231, 312 are positive;
132, 321, 213 are negative; and the developed expression of the foregoing determinant
of the third order is
= ab'c' — ab"c + a'b"c — a be" + a" be — a'b'c.
3. It further appears that a determinant is a linear function* of the elements
of each column thereof, and also a linear function of the elements of each line
thereof; moreover, that the determinant retains the same value, only its sign being
altered, when any two columns are interchanged, or when any two lines are inter
changed ; more generally, when the columns are permuted in any manner, or when
the lines are permuted in any manner, the determinant retains its original value, with
the sign + or — according as the new arrangement (considered as derived from the
primitive arrangement) is positive or negative according to the foregoing rule of signs.
* The expression, a linear function, is here used in its narrowest sense, a linear function without con
stant term; what is meant is, that the determinant is in regard to the elements a, a', a", ... of any
column or line thereof, a function of the form Aa + A'a' + A"a" + ... , without any term independent of
«, a', a", ....
786]
EQUATION.
493
It at once follows that, if two columns are identical, or if two lines are identical,
the value of the determinant is = 0. It may be added that, if the lines are con
verted into columns, and the columns into lines, in such a way as to leave the dexter
diagonal unaltered, the value of the determinant is unaltered ; the determinant is in
this case said to be transposed.
4. By what precedes it appears that there exists a function of the ?i2 elements,
linear as regards the terms of each column (or say, for shortness, linear as to each
column), and such that only the sign is altered when any two columns are inter
changed; these properties completely determine the function, except as to a common
factor which may multiply all the terms. If, to get rid of this arbitrary common
factor, we assume that the product of the elements in the dexter diagonal has the
coefficient +1, we have a complete definition of the determinant; and it is interesting
to show how from these properties, assumed for the definition of the determinant, it
at once appears that the determinant is a function serving for the solution of a
system of linear equations. Observe that the properties show at once that if any
column is =0 (that is, if the elements in the column are each = 0), then the
determinant is = 0 ; and further that, if any two columns are identical, then the
determinant is = 0.
5. Reverting to the system of linear equations written down at the beginning
of this article, consider the determinant
a x + b y + c z — d , b , c
a x + b' y + c' z — d' , b' , c'
a"x + V'y + c"z-d", b", c"
it appears that this is
= x a , b , c
a, b', c'
a", b", c"
viz. the second and the third terms each vanishing, it is
+y
b , b , c
b', b', c
b", b", c"
+ 2
c , b , c
c' , b' , c'
c", b", c"
—Id, b , c
• d' , b' , c
d", b", c"
= x i a , 6 , c
a' , b' , c'
a", b", c"
d , b , c
d', b', c'
d", b", c"
But if the linear equations hold good, then the first column of the original determ
inant is = 0, and therefore the determinant itself is = 0 ; that is, the linear equations
give
= 0;
a ,
b ,
c
-
d,
b,
c
a',
b',
c'
d',
b',
c
a",
b",
c"
d",
b",
c"
which is the result obtained above.
494
EQUATION.
[786
We might in a similar way find the values of y and z, but there is a more
symmetrical process. Join to the original equations the new equation
ax + fiy + yz = 8 ;
a like process shows that, the equations being satisfied, we have
a , 0, y, 8 =0;
a , b , c , d
a', b' , c' , d'
a", b", c", d"
or, as this may be written,
a , b , c , d
a', b' , c' , d'
a", b", c", d"
— 8 a , b , c
a', b', c'
a", b", c"
= 0;
which, considering 8 as standing herein for its value ax + @y + <yz, is a consequence of
the original equations only. We have thus an expression for ax + Py+<yz, an arbitrary
linear function of the unknown quantities x, y, z; and by comparing the coefficients
of a, /3, 7 on the two sides respectively, we have the values of x, y, z\ in fact, these
quantities, each multiplied by
a , b , c ,
a, b', c
a", b", c"
are in the first instance obtained in the forms
a , b , c , d
a', b', c', d'
a", b", c", d"
a , b , c , d
a', b', c', d'
a", b", c", d"
a , b , c , d
a', b' , c' , d'
a", b", c", d"
but these are
=
b , c , d
>
c , d , a
t
d , a , b
b', c', d'
c' , d' , of
d', a', b'
b", c", d"
c", d", a"
d", a", b"
or, what is the same thing,
=
b , c , d
i
c , a , d
>
a , b , d
b', c', d'
c', a', d'
a', b', d'
b", c", d"
c", a", d"
a", b", d"
respectively.
786]
EQUATION.
495
» » »
;> j) »
I
» » »
a , b , c
a', b', c'
a", b", c"
a , /3 , 7
*', P, 7
a", £", 7"
6. Multiplication of two determinants of the same order. — The theorem is obtained
very easily from the last preceding definition of a determinant. It is most simply
expressed thus —
(a, «', a"), 08, P, ft"}, (y, y', y"}
(a , b , c )
(a', V, c')
(a", b", c")
where the expression on the left side stands for a determinant, the terms of the first line
being (a, b, c) (a, a', a"), that is, aa + bet' + ca", (a, b, c) (B, ft', ft"\ that is, aft + bft' + eft",
(a, b, c) (y, y', y"), that is, ay + by' + cy" ; and similarly the terms in the second and
third lines are the like functions with (a', b', c') and (a", b", c") respectively.
There is an apparently arbitrary transposition of lines and columns ; the result would
hold good if on the left-hand side we had written (a, ft, y), (a, ft', y), (a", ft", y"),
or what is the same thing, if on the right-hand side we had transposed the second
determinant; and either of these changes would, it might be thought, increase the
elegance of the form, but, for a reason which need not be explained*, the form actually
adopted is the preferable one.
To indicate the method of proof, observe that the determinant on the left-hand
side, qua linear function of its columns, may be broken up into a sum of (33 =) 27
determinants, each of which is either of some such form as
a , a , b
a', a', b'
a", a", b"
where the term afty is not a term of the a/fy-determinant, and its coefficient (as a
determinant with two identical columns) vanishes ; or else it is of a form such as
a , b , c
a , b' , c'
a", b", c"
that is, every term which does not vanish contains as a factor the a&c-determinant
last written down ; the sum of all other factors + aft'y" is the a/fy-determinant of
the formula; and the final result then is, that the determinant on the left-hand side
is equal to the product on the right-hand side of the formula.
7. Decomposition of a determinant into complementary determinants. Consider, for
simplicity, a determinant of the fifth order, 5 = 2 + 3, and let the top two lines be
a , b , c , d , e ,
a' , b' , c' , d' , e';
* The reason is the connexion with the corresponding theorem for the multiplication of two matrices.
496
EQUATION.
[786
then, if we consider how these elements enter into the determinant, it is at once seen
a,
, &c.,
that they enter only through the determinants of the second order i
I «', V
which can be formed by selecting any two columns at pleasure. Moreover, representing
the remaining three lines by
a" , b" , c" , d" , e" ,
a'", b'", c'", d'", e'" ,
„"" u"' „"" j'"> mi
a , o , c , d , e ,
it is further seen that the factor which multiplies the determinant formed with any
two columns of the first set is the determinant of the third order formed with the
complementary three columns of the second set; and it thus appears that the determ
inant of the fifth order is a sum of all the products of the form
+ a, b
a', b'
c" , d" , e"
c'", d'", e'"
c"", d"", e""
the sign + being in each case such that the sign of the term + ah' . c"d'"e"" obtained
from the diagonal elements of the component determinants may be the actual sign of
this term in the determinant of the fifth order; for the product written down the
sign is obviously +.
Observe that for a determinant of the nth order, taking the decomposition to
be l+(n — 1), we fall back upon the equations given at the commencement, in order
to show the genesis of a determinant.
a, b
formed out of the elements of the original determ-
a, b
8. Any determinant
inant, by selecting the lines and columns at pleasure, is termed a minor of the
original determinant; and when the number of lines and columns, or order of the
determinant, is n — l, then such determinant is called a first minor; the number of
the first minors is = ?i2, the first minors, in fact, corresponding to the several elements
of the determinant — that is, the coefficient therein of any term whatever is the corre
sponding first minor. The first minors, each divided by the determinant itself, form a
system of elements inverse to the elements of the determinant.
A determinant is symmetrical when every two elements symmetrically situated in
regard to the dexter diagonal are equal to each other; if they are equal and opposite
(that is, if the sum of the two elements be = 0), this relation not extending to the
diagonal elements themselves, which remain arbitrary, then the determinant is skew;
but if the relation does extend to the diagonal terms (that is, if these are each = 0),
then the determinant is skew symmetrical; thus the determinants
v, - fj,
0, X
JL, - X, 0
are respectively symmetrical, skew, and skew symmetrical.
a, h, g
>
a, v, — /j,
)
h, b, f
-v, b, X
/•
/*, — X, c
786] EQUATION. 497
The theory admits of very extensive algebraic developments, and applications in
algebraical geometry and other parts of mathematics; but the fundamental properties
of the functions may fairly be considered as included in what precedes.
Theory of Equations.
9. In the subject "Theory of Equations," the term equation is used to denote an
equation of the form xn — p1xn~1+ ... ± pn = 0, where pl, p*,...,pn are regarded as known,
and x as a quantity to be determined ; for shortness, the equation is written f(x) = 0.
The equation may be numerical; that is, the coefficients plt pi,...,pn are then
numbers, — understanding by number a quantity of the form a+fti (a. and @ having
any positive or negative real values whatever, or say each of these is regarded as
susceptible of continuous variation from an indefinitely large negative to an indefinitely
large positive value), and i denoting V — 1.
Or the equation may be algebraical; that is, the coefficients are not then restricted
to denote, or are not explicitly considered as denoting, numbers.
I. We consider first numerical equations. (Real theory, 10 to 14; Imaginary
theory, 15 to 18.)
10. Postponing all consideration of imaginaries, we take in the first instance the
coefficients to be real, and attend only to the real roots (if any); that is, plt pz,...,pn
are real positive or negative quantities, and a root a, if it exists, is a positive or
negative quantity such that an -p^an~l+ ... ±pn = 0, or say, /(a) = 0. The fundamental
theorems are given in the article Algebra, sections x., xin., xiv.; but there are various
points in the theory which require further development.
It is very useful to consider the curve y =/(^),_0r, what would come to the
same, the curve Ay =f(x),— but it is better to retain the first-mentioned form of
equation, drawing, if need be, the ordinate y on a reduced scale. For instance, if the
given equation be a? - 6#2 + Use- 6 "06 =0,* then the curve y = a? - Qx2 + 1 \x - 6'06 is
as shown in the figure at page 501, without any reduction of scale for the ordinate.
It is clear that, in general, y is a continuous one-valued function of x, finite for
every finite value of x, but becoming infinite when x is infinite ; i.e. assuming throughout
that the coefficient of xn is +1, then when x = oo , y = +oo- but when x = — oo , then
y = + ao or - oo , according as n is even or odd ; the curve cuts any line whatever,
and in particular it cuts the axis of x, in at most n points ; and the value of x,
at any point of intersection with the axis, is a root of the equation f(x) = 0.
If /3, a. are any two values of x (a > /3, that is, a nearer +00), then if / (/3),
/(a) have opposite signs, the curve cuts the axis an odd number of times, and
therefore at least once, between the points x = fi, x = a.\ but if f(j3), /(a) have the
same sign, then between these points the curve cuts the axis an even number of
times, or it may be not at all. That is, /(#), /(a) having opposite signs, there are
between the limits /?, a an odd number of real roots, and therefore at least one real
* The coefficients were selected so that the roots might be nearly 1, 2, 3.
c. XL 63
498 EQUATION. [786
root; but /(/3), /(a) having the same sign, there are between these limits an even
number of real roots, or it may be there is no real root. In particular, by giving
to /8, a the values — oo , + oo (or, what is the same thing, any two values sufficiently
near to these values respectively) it appears that an equation of an odd order has
always an odd number of real roots, and therefore at least one real root ; but that
an equation of an even order has an even number of real roots, or it may be no
real root.
If a be such that for x= or >cc (that is, x nearer to +00) /(#) is always +,
and /3 be such that for x— or < /3 (that is, x nearer to — oo ) f(x) is always — ,
then the real roots (if any) lie between these limits x = /3, x = a. ; and it is easy to
find by trial such two limits including between them all the real roots (if any).
11. Suppose that the positive value 8 is an inferior limit to the difference
between two real roots of the equation ; or rather (since the foregoing expression
would imply the existence of real roots) suppose that there are not two real roots
such that their difference taken positively is = or < 8 ; then, 7 being any value what
ever, there is clearly at most one real root between the limits 7 and 7 + 8 ; and by
what precedes there is such real root or there is not such real root, according as
f(y), .f(y + 8) have opposite signs or have the same sign. And by dividing in this
manner the interval /3 to a into intervals each of which is = or < 8, we should not
only ascertain the number of the real roots (if any), but we should also separate the
real roots, that is, find for each of them limits 7, 7 + 8 between which there lies this
one, and only this one, real root.
In particular cases it is frequently possible to ascertain the number of the real
roots, and to effect their separation by trial or otherwise, without much difficulty ; but
the foregoing was the general process as employed by Lagrange even in the second
edition (1808) of the Traite de la resolution des Equations Numdriques * ; the determ
ination of the limit 8 had to be effected by means of the "equation of differences"
or equation of the order \n(n— 1), the roots of which are the squares of the differences
of the roots of the given equation, and the process is a cumbrous and unsatisfactory one.
12. The great step was effected by Sturm's theorem (1835) — viz. here starting
from the function f(x), and its first derived function f (x), we have (by a process
which is a slight modification of that for obtaining the greatest common measure of
these two functions) to form a series of functions
f(x\ f (x\ f2 (x), ...,/„ (as)
of the degrees n, n—1, n — 2, . . . , 0 respectively, — the last term fn (x) being thus an
absolute constant. These lead to the immediate determination of the number of real
roots (if any) between any two given limits /3, a ; viz. supposing a > ft (that is,
a. nearer to +00), then substituting successively these two values in the series of
functions, and attending only to the signs of the resulting values, the number of the
changes of sign lost in passing from ft to a is the required number of real roots
* The third edition (1826) is a reproduction of that of 1808; the first edition has the date 1798, but
a large part of the contents is taken from memoirs of 1767 — 68 and 1770 — 71.
786] EQUATION. 499
between the two limits. In particular, taking ft, a = — oo , +QO respectively, the signs
of the several functions depend merely on the signs of the terms which contain the
highest powers of x, and are seen by inspection, and the theorem thus gives at once
the whole number of real roots.
And although theoretically, in order to complete by a finite number of operations
the separation of the real roots, we still need to know the value of the before-
mentioned limit S ; yet in any given case the separation may be effected by a limited
number of repetitions of the process. The practical difficulty is when two or more
roots are very near to each other. Suppose, for instance, that the theorem shows that
there are two roots between 0 and 10 ; by giving to x the values 1, 2, 3, ... successively,
it might appear that the two roots were between 5 and 6 ; then again that they
were between 5*3 and 5*4, then between 5'34 and 5*35, and so on until we arrive at
a separation; say it appears that between 5 '346 and 5'347 there is one root, and
between 5'348 and 5'349 the other root. But in the case in question 8 would have
a very small value, such as '002, and even supposing this value known, the direct
application of the first-mentioned process would be still more laborious.
13. Supposing the separation once effected, the determination of the single real
root which lies between the two given limits may be effected to any required degree
of approximation either by the processes of Horner and Lagrange (which are in
principle a carrying out of the method of Sturm's theorem), or by the process of
Newton, as perfected by Fourier (which requires to be separately considered).
First as to Horner and Lagrange. We know that between the limits ft, a. there
lies one, and only one, real root of the equation; /(/3) and /(a) have therefore opposite
signs. Suppose any intermediate value is 0; in order to determine by Sturm's theorem
whether the root lies between ft, 6, or between 6, a, it would be quite unnecessary
to calculate the signs of /(#), f'(0\ fz(@), • •- ', only the sign of f(0) is required: for,
if this has the same sign as f(ft}, then the root is between ft, 9; if the same sign
as f(a), then the root is between 6, a. We want to make d increase from the inferior
limit ft, at which /(#) has the sign of f(ft), so long as f(0) retains this sign, and
then to a value for which it assumes the opposite sign ; we have thus two nearer
limits of the required root, and the process may be repeated indefinitely.
Horner's method (1819) gives the root as a decimal, figure by figure; thus, if the
equation be known to have one real root between 0 and 10, it is in effect shown
say that 5 is too small (that is, the root is between 5 and 6) ; next that 5'4 is too
small (that is, the root is between 5*4 and 5'5) ; and so on to any number of
decimals. Each figure is obtained, not by the successive trial of all the figures which
precede it, but (as in the ordinary process of the extraction of a square root, which
is in fact Horner's process applied to this particular case) it is given presumptively
as the first figure of a quotient; such value may be too large, and then the next
inferior integer must be tried instead of it, or it may require to be further diminished.
And it is to be remarked that the process not only gives the approximate value a
of the root, but (as in the extraction of a square root) it includes the calculation of
the function /(a) which should be, and approximately is, =0. The arrangement of the
63—2
500 EQUATION. [786
calculations is very elegant, and forms an integral part of the actual method. It is
to be observed that after a certain number of decimal places have been obtained, a
good many more can be found by a mere division. It is in the progress tacitly
assumed that the roots have been first separated.
Lagrange's method (1767) gives the root as a continued fraction a + T -
o + c +
where a is a positive or negative integer (which may be = 0), but b, c, ... are positive
integers. Suppose the roots have been separated ; then (by trial if need be of con
secutive integer values) the limits may be made to be consecutive integer numbers:
say they are a, a + 1 ; the value of x is therefore = a + - , where y is positive and
\J
greater than 1 ; from the given equation for x, writing therein as = a + - , we form an
equation of the same order for y, and this equation will have one, and only one,
positive root greater than 1 ; hence finding for it the limits b, b 4- 1 (where b is =
or > 1), we have y = b + -, where z is positive and greater than 1; and so on —
z
that is, we thus obtain the successive denominators b, c, d,... of the continued fraction.
The method is theoretically very elegant, but the disadvantage is that it gives the
result in the form of a continued fraction, which for the most part must ultimately
be converted into a decimal. There is one advantage in the method, that a com
mensurable root (that is, a root equal to a rational fraction) is found accurately, since,
when such root exists, the continued fraction terminates.
14. Newton's method (1711), as perfected by Fourier (1831), may be roughly stated
as follows. If x = 7 be an approximate value of any root, and 7 + h the correct value,
then f(y+h) = Q, that is,
/ '
and then, if h be so small that the terms after the second may be neglected,
(ry) + hf'(y) = Q, that is, h= — j-~^ or the new approximate value is X = (Y~ f'7\>
and so on, as often as we please. It will be observed that so far nothing has been
assumed as to the separation of the roots, or even as to the existence of a real
root; 7 has been taken as the approximate value of a root, but no precise meaning
has been attached to this expression. The question arises, what are the conditions to
be satisfied by 7 in order that the process may by successive repetitions actually lead
to a certain real root of the equation ; or say that, 7 being an approximate value of
a certain real root, the new value 7 — frr may be a more approximate value.
Referring to the figure, it is easy to see that, if OC represent the assumed
value 7, then, drawing the ordinate CP to meet the curve in P, and the tangent
PC' to meet the axis in C', we shall have OC' as the new approximate value of the
root. But observe that there is here a real root OX, and that the curve beyond X
786]
EQUATION.
501
is convex to the axis; under these conditions the point C' is nearer to X than
was C; and, starting with C" instead of C, and proceeding in like manner to draw
a new ordinate and tangent, and so on as often as we please, we approximate con
tinually, and that with great rapidity, to the true value OX. But if C had been
taken on the other side of X, where the curve is concave to the axis, the new
point C' might or might not be nearer to X than was the point (7; and in this
case the method, if it succeeds at all, does so by accident only, i.e., it may happen
that C" or some subsequent point comes to be a point C, such that 00 is a proper
approximate value of the root, and then the subsequent approximations proceed in the
same manner as if this value had been assumed in the first instance, all the pre
ceding work being wasted. It thus appears that for the proper application of the
method we require more than the mere separation of the roots. In order to be able
to approximate to a certain root a, = OX, we require to know that, between OX and
some value ON, the curve is always convex to the axis: analytically, between the two
values, f(x) and /" (x) must have always the same sign. When this is so, the point
C may be taken anywhere on the proper side of X, and within the portion XN of
the axis; and the process is then the one already explained. The approximation is
in general a very rapid one. If we know for the required root OX the two limits
OM, ON such that from M to X the curve is always concave to the axis, while
from X to N it is always convex to the axis, — then, taking D anywhere in the
portion MX and (as before) C in the portion XN, drawing the ordinates DQ, CP,
and joining the points P, Q by a line which meets the axis in D', also constructing
the point C' by means of the tangent at P as before, we have for the required root
the new limits OD', OC' ; and proceeding in like manner with the points D', C', and
so on as often as we please, we obtain at each step two limits approximating more
and more nearly to the required root OX. The process as to the point D', translated
into analysis, is the ordinate process of interpolation. Suppose OD = /3, OC = a, we have
approximately
a-ft
whence, if the root is f3+h, then
'
502 EQUATION. [786
Returning for a moment to Horner's method, it may be remarked that the
correction h, to an approximate value a, is therein found as a quotient, the same or
such as the quotient f(a) -=-/' (a) which presents itself in Newton's method. The
difference is that with Horner the integer part of this quotient, is taken as the
presumptive value of h, and the figure is verified at each step. With Newton the
quotient itself, developed to the proper number of decimal places, is taken as the
value of h ; if too many decimals are taken, there would be a waste of work ; but
the error would correct itself at the next step. Of course the calculation should be
conducted without any such waste of work.
Next as to the theory of imaginaries.
15. It will be recollected that the expression number and the correlative epithet
numerical were at the outset used in a wide sense, as extending to imaginaries. This
extension arises out of the theory of equations by a process analogous to that by
which number, in its original most restricted sense of positive integer number, was
extended to have the meaning of a real positive or negative magnitude susceptible
of continuous variation.
If for a moment number is understood in its most restricted sense as meaning
positive integer number, the solution of a simple equation leads to an extension;
ax — b = 0, gives x = - , a positive fraction, and we can in this manner represent, not
G/
accurately, but as nearly as we please, any positive magnitude whatever; so an equation
ax + b = 0 gives x = , which (approximately as before) represents any negative
magnitude. We thus arrive at the extended signification of number as a continuously
varying positive or negative magnitude. Such numbers may be added or subtracted,
multiplied or divided one by another, and the result is always a number. Now from
a quadric equation we derive, in like manner, the notion of a complex or imaginary
number such as is spoken of above. The equation x2 + 1 = 0 is not (in the foregoing
sense, number = real number) satisfied by any numerical value whatever of x ; but we
assume that there is a number which we call i, satisfying the equation i? + 1 = 0 ;
and then taking a and b any real numbers, we form an expression such as a + bi,
and use the expression number in this extended sense : any two such numbers may
be added or subtracted, multiplied or divided one by the other, and the result is
always a number. And if we consider first a quadric equation a?+px + q = Q where
p and q are real numbers, and next the like equation, where p and q are any numbers
whatever, it can be shown that there exists for x a numerical value which satisfies
the equation; or, in other words, it can be shown that the equation has a numerical
root. The like theorem, in fact, holds good for an equation of any order whatever.
But suppose for a moment that this was not the case: say that there was a cubic
equation a?+px~ + qx + r = 0, with numerical coefficients, not satisfied by any numerical
value of as, we should have to establish a new imaginary j satisfying some such
equation, and should then have to consider numbers of the form a + bj, or perhaps
a + bj + cj2 (a, b, c numbers a + (Bi of the kind heretofore considered), — first we should
be thrown back on the quadric equation a?+px + q = Q, p and q being now numbers
786] EQUATION. 503
of the last-mentioned extended form — non constat that every such equation has a
numerical root — and if not, we might be led to other imaginaries k, I, &c., and so on
ad infinitum in inextricable confusion.
But in fact a numerical equation of any order whatever has always a numerical
root, and thus numbers (in the foregoing sense, number = quantity of the form a. + fti)
form (what real numbers do not) a universe complete in itself, such that starting in
it we are never led out of it. There may very well be, and perhaps are, numbers in
a more general sense of the term (quaternions are not a case in point, as the
ordinary laws of combination are not adhered to) : but in order to have to do with
such numbers (if any), we must start with them.
16. The capital theorem as regards numerical equations thus is, every numerical
equation has a numerical root ; or for shortness (the meaning being as before), every
equation has a root. Of course the theorem is the reverse of self-evident, and it
requires proof; but provisionally assuming it as true, we derive from it the general
theory of numerical equations. As the term root was introduced in the course of an
explanation, it will be convenient to give here the formal definition.
A number a such that substituted for x it makes the function xn — p1xn~l + ... ±pn
to be = 0, or say such that it satisfies the equation f(x) — 0, is said to be a root
of the equation ; that is, a being a root, we have
an -p±an~l + . . . ± pn = 0, or say /(a) = 0 ;
and it is then easily shown that x — a is a factor of the function f(x), viz. that we
have f(x) = (x-a)fl(x), where /i(#) is a function xn~^ — q1xn~2 + ... ±qn_l of the order
?i — l, with numerical coefficients qly q2,...,qn-i-
In general, a is not a root of the equation /x (x) = 0, but it may be so — i.e., /i (x)
may contain the factor as — a ; when this is so, f(x) will contain the factor (x — a)2 ;
writing then f(x) = (x — a)2/2 (x), and assuming that a is not a root of the equation
/2 (x) = 0, x = a is then said to be a double root of the equation f(x} = 0 ; and similarly
f(x) may contain the factor (x— a)3 and no higher power, and x = a is then a triple
root ; and so on.
Supposing, in general, that f(x) = (x - a)a F (x), a. being a positive integer which
may be = 1, (x — a)a the highest power of x — a which divides f(x), and F(x) being
of course of the order n — a, then the equation F (x) = 0 will have a root b which
will be different from a ; x — b will be a factor, in general a simple one, but it may
be a multiple one, of F (x), and f(x) will in this case be = (x — a)° (x — bf <£> (x\
fi a positive integer which may be =1, (x -by* the highest power of x - b in F (x)
or f(x\ and 3> (x) being of course of the order n — a.— ft. The original equation
y (#) = 0 is in this case said to have a roots each = a, ft roots each = b ; and so on
for any other factors (x — c)y, &c.
We have thus the theorem — A numerical equation of the order n has in every
case n roots, viz. there exist n numbers a, b,..., in general all distinct, but which may
arrange themselves in any sets of equal values, such that f(x} = (x - a)(x-b} (x- c)...
identically.
504 EQUATION. [786
If the equation has equal roots, these can in general be determined : and the case
is at any rate a special one which may be in the first instance excluded from con
sideration. It is therefore, in general, assumed that the equation f(x) = 0 has all its
roots unequal.
If the coefficients pi,pz,... are all or any one or more of them imaginary, then
the equation f(x) — 0, separating the real and imaginary parts thereof, may be written
F (x) + i<$> (x) = 0, where F (x}, <£ (x) are each of them a function with real coefficients ;
and it thus appears that the equation f(x) = 0, with imaginary coefficients, has not in
general any real root ; supposing it to have a real root a, this must be at once a
root of each of the equations F(x) = 0 and 4> (x) = 0.
But an equation with real coefficients may have as well imaginary as real roots,
and we have further the theorem that for any such equation the imaginary roots
enter in pairs, viz. a + fti being a root, then a. — fti will be also a root. It follows
that, if the order be odd, there is always an odd number of real roots, and therefore
at least one real root.
17. In the case of an equation with real coefficients, the question of the existence
of real roots, and of their separation, has been already considered. In the general case
of an equation with imaginary (it may be real) coefficients, the like question arises as
to the situation of the (real or imaginary) roots ; thus if, for facility of conception, we
regard the constituents a, @ of a root a + /3i as the coordinates of a point in piano,
and accordingly represent the root by such point, then drawing in the plane any closed
curve or " contour," the question is how many roots lie within such contour.
This is solved theoretically by means of a theorem of Cauchy's (1837), viz. writing
in the original equation x + iy in place of x, the function f(x + iy) becomes = P + iQ,
where P and Q are each of them a rational and integral function (with real coefficients)
of (x, y). Imagining the point (x, y) to travel along the contour, and considering the
number of changes of sign from — to + and from + to — of the fraction corresponding
to passages of the fraction through zero, that is, to values for which P becomes = 0,
disregarding those for which Q becomes = 0, the difference of these numbers gives
the number of roots within the contour.
It is important to remark that the demonstration does not presuppose the existence
of any root; the contour may be the infinity of the plane (such infinity regarded as
a contour, or closed curve), and in this case it can be shown (and that very easily)
that the difference of the numbers of changes of sign is = n ; that is, there are
within the infinite contour, or (what is the same thing) there are in all, n roots ;
thus Cauchy's theorem contains really the proof of the fundamental theorem that a
numerical equation of the nth order (not only has a numerical root, but) has precisely
n roots. It would appear that this proof of the fundamental theorem in its most
complete form is in principle identical with Gauss's last proof (1849) of the theorem,
in the form — A numerical equation of the nth order has always a root*.
* The earlier demonstrations by Euler, Lagrange, &c., relate to the case of a numerical equation with
real coefficients ; and they consist in showing that such equation has always a real quadratic divisor,
furnishing two roots, which are either real or else conjugate imaginaries a + /3?: see Lagrange's Equations
Numeriques.
786] EQUATION. 505
But in the case of a 'finite contour, the actual determination of the difference
which gives the number of real roots can be effected only in the case of a rectangular
contour, by applying to each of its sides separately a method such as that of Sturm's
theorem ; and thus the actual determination ultimately depends on a method such as
that of Sturm's theorem.
Very little has been done in regard to the calculation of the imaginary roots of
an equation by approximation ; and the question is not here considered.
18. A class of numerical equations which needs to be considered is that of the
binomial equations xn — a = 0 (a=a+j3i, a complex number). The foregoing conclusions
apply, viz. there are always n roots, which, it may be shown, are all unequal. And
these can be found numerically by the extraction of the square root, and of an ?zth
root, of real numbers, and by the aid of a table of natural sines and cosines*. For
writing
there is always a real angle \ (positive and less than 2?r), such that its cosine and
r> _
sine are = — ===== and . ___ respectively ; that is, writing for shortness V a2 + ft* = p,
we have ct + /ft = p (cos \ + i sin X), or the equation is xn — p (cos X + i sin X) ; hence
/ X \\H . • e • n/~ ( X . . X\
observing that I cos - 4- 1 sin - I = cos X + 1 sin X, a value ot x is = v p I cos - + i sin - 1 .
The formula really gives all the roots, for instead of X we may write X + 2s?r, s a
positive or negative integer, and then we have
X + 2S7T . . X + 2S7T
x = v p { cos ---- h i sin -
which has the n values obtained by giving to s the values 0, 1, 2, ..., n- 1 in succession;
the roots are, it is clear, represented by points lying at equal intervals on a circle.
But it is more convenient to proceed somewhat differently; taking one of the roots
to be 6, so that On = a, then assuming x = Qy, the equation becomes yn - 1 = 0, which
equation, like the original equation, has precisely n roots (one of them being of course
= 1). And the original equation xn - a = 0 is thus reduced to the more simple
equation arn - 1 = 0 ; and although the theory of this equation is included in the pre
ceding one, yet it is proper to state it separately.
The equation xn-l=0 has its several roots expressed in the form 1, <w, &>2, ..., wn-\
where o> may be taken = cos — + i sin — ; in fact, « having this value, any integer
Jlf it
power o)k is = cos + i sin — , and we thence have (&>*)" = cos 2-irk + i sin 2irk, = 1,
n n
that is, o)k is a root of the equation. The theory will be resumed further on.
* The square root of a + /3t can be determined by the extraction of square roots of positive real numbers,
without the trigonometrical tables.
C. XI. 64
506 EQUATION. [786
1
By what precedes, we are led to the notion (a numerical) of the radical an regarded
as an w-valued function ; any one of these being denoted by \/a, then the series of
i
values is \/a, a> \/a, ..., mn~li\/a; or we may, if we please, use \X a instead of an as
a symbol to denote the 7i-valued function.
As the coefficients of an algebraical equation may be numerical, all which follows
in regard to algebraical equations is (with, it may be, some few modifications) applicable
to numerical equations; and hence, concluding for the present this subject, it will be
convenient to pass on to algebraical equations.
II. We consider, secondly, algebraical equations (19 to 34).
19. The equation is
and we here assume the existence of roots, viz. we assume that there are n quantities
a, b, c, ... (in general all of them different, but which in particular cases may become
equal in sets in any manner), such that
or looking at the question in a different point of view, and starting with the roots
a, b, c, ... as given, we express the product of the n factors x — a, x — b, ... in the
foregoing form, and thus arrive at an equation of the order n having the n roots
a, b, c,.... In either case we have
pl = Sa, p2 = Sa6, . . . , pn = abc . . . ;
i.e., regarding the coefficients pl} p2,...,pn as given, then we assume the existence of
roots a, b, c,... such that p1 = 2a, &c. ; or, regarding the roots as given, then we write
plt p2, &c., to denote the Functions 2 a, Sa6, &c.
As already explained, the epithet algebraical is not used in opposition to numerical ;
an algebraical equation is merely an equation wherein the coefficients are not restricted
to denote, or are not explicitly considered as denoting, numbers. That the abstraction
is legitimate, appears by the simplest example; in saying that the equation xz—px + q = Q
has a root a = £ (p + ^/p2 — 4<q), we mean that writing this value for x the equation
becomes an identity, {£ (p + \fp* - 4q)}2 -p {£ (p + </p2 - 4q)} + q = 0; and the verification
of this identity in nowise depends upon p and q meaning numbers. But if it be
asked what there is beyond numerical equations included in the term algebraical
equation, or, again, what is the full extent of the meaning attributed to the term —
the latter question at any rate it would be very difficult to answer; as to the former
one, it may be said that the coefficients may, for instance, be symbols of operation.
As regards such equations, there is certainly no proof that every equation has a root,
or that an equation of the ??.th order has n roots; nor is it in any wise clear what
the precise signification of the statement is. But it is found that the assumption of
the existence of the n roots can be made without contradictory results; conclusions
786] EQUATION. 507
derived from it, if they involve the roots, rest on the same ground as the original
assumption; but the conclusion may be independent of the roots altogether, and in this
case it is undoubtedly valid ; the reasoning, although actually conducted by aid of the
assumption (and, it may be, most easily and elegantly in this manner), is really inde
pendent of the assumption. In illustration, we observe that it is allowable to express
a function of p and q as follows, — that is, by means of a rational symmetrical function
of a and 6; this can, as a fact, be expressed as a rational function of a + b and ab ;
and if we prescribe that a + b and ab shall then be changed into p and q respectively,
we have the required function of p, q. That is, we have F (a, /3) as a representation
of f(p, q), obtained as if we had p = a + b, q = ab, but without in any wise assuming
the existence of the a, b of these equations.
20. Starting from the equation
xn — p1a}n~l +...=# — a. x — b . &c.,
or the equivalent equations pl = 2a, &c., we find
an-p1an~l+ ...=0,
bn -p,bn-1 + ...=0;
(it is as satisfying these equations that a,b,... are said to be the roots of xn — p1xn~l+...=Q) ;
and conversely from the last-mentioned equations, assuming that a, b, ... are all different,
we deduce
P! = 2a, p» = 2a6, &c.,
and
xn — p^x"-1 + ... =x — a.x — 6.&c.
Observe that if, for instance, a = b, then the equations an — p1an~1 + ... =0, bn—p1bn~l + ...=0
would reduce themselves to a single relation, which would not of itself express that
a was a double root, — that is, that (x — of was a factor of xn — p^"*1 + &c. ; but by
considering b as the limit of a + h, h indefinitely small, we obtain a second equation
nan~l -(n-l) p^11'2 + . . . = 0,
which, with the first, expresses that a is a double root ; and then the whole system
of equations leads as before to the equations pl = '2la, &c. But the existence of a
double root implies a certain relation between the coefficients ; the general case is
when the roots are all unequal.
We have then the theorem that every rational symmetrical function of the roots
is a rational function of the coefficients. This is an easy consequence from the less
general theorem, every rational and integral symmetrical function of the roots is a
rational and integral function of the coefficients.
In particular, the sums of the powers 2a2, Sa3, &c., are rational and integral
functions of the coefficients.
64—2
508 EQUATION. [786
The process originally employed for the expression of other functions 2aa&0, &c.,
in terms of the coefficients is to make them depend upon the sums of powers: for
instance, 2aa63 = SaaSa3 - 2aa+P ; but this is very objectionable ; the true theory consists
in showing that we have systems of equations
Pi = 2a,
p3 =
2a26 + 32a6c,
where in each system there are precisely as many equations as there are root-functions
on the right-hand side — e.g. 3 equations and 3 functions 2abc, 2a26, Sa3. Hence in
each system the root-functions can be determined linearly in terms of the powers and
products of the coefficients :
/«-« 7
Pi,
t'Zabc = p3,
2a26 = plP*-3p3,
v2a3 =pls-3p1p2 + 3ps,
and so on. The older process, if applied consistently, would derive the originally
assumed value Sa&, =p2, from the two equations 2ta=p1, So? = p,2 — 2p2 ; i.e. we have
2a . 2a - 2a2, = p,2 - (p* - 2p2), = 2ps.
21. It is convenient to mention here the theorem that, x being determined as
above by an equation of the order n, any rational and integral function whatever of x,
or more generally any rational function which does not become infinite in virtue of
the equation itself, can be expressed as a rational and integral function of x, of the
order n - 1, the coefficients being rational functions of the coefficients of the equation.
Thus the equation gives xn a function of the form in question ; multiplying each side
by x, and on the right-hand side writing for x11 its foregoing value, we have xn+l, a
function of the form in question; and the like for any higher power of x, and therefore
also for any rational and integral function of x. The proof in the case of a rational
non-integral function is somewhat more complicated. The final result is of the form
-— — = /(ar), or say <f> (x) - -v/r (x} I (x) = 0, where <j>, ^, I are rational and integral
functions; in other words, this equation, being true if only /(#) = 0, can only be so
by reason that the left-hand side contains f(x) as a factor, or we must have identically
= M(x}f(x}. And it is, moreover, clear that the equation - = I (x)
TW
being satisfied if only f(x} = 0, must be satisfied by each root of the equation.
786] EQUATION. 509
From the theorem that a rational symmetrical function of the roots is expressible
in terms of the coefficients, it at once follows that it is possible to determine an
equation (of an assignable order) having for its roots the several values of any given
(unsymmetrical) function of the roots of the given equation. For example, in the case of
a quartic equation, having the roots (a, b, c, d), it is possible to find an equation having
the roots ab, ac, ad, be, bd, cd, being therefore a sextic equation : viz. in the product
(y - ab} (y - ac) (y - ad) (y - be) (y - bd) (y - cd),
the coefficients of the several powers of y will be symmetrical functions of a, b, c, d
and therefore rational and integral functions of the coefficients of the quartic equation ;
hence, supposing the product so expressed, and equating it to zero, we have the
required sextic equation. In the same manner can be found the sextic equation
having the roots (a — b)2, (a — c)2, (a — d)-, (b — c)2, (b - d)2, (c - d)2, which is the equation
of differences previously referred to ; and similarly we obtain the equation of differences
for a given equation of any order. Again, the equation sought for may be that having
for its n roots the given rational functions <f> (a), (f> (b), ... of the several roots of the
given equation. Any such rational function can (as was shown) be expressed as a
rational and integral function of the order n — 1 ; and, retaining x in place of any
one of the roots, the problem is to find y from the equations xn — piXn~l + ... = 0, and
y = M0xn~~l + M±xn~'i + , . . , or, what is the same thing, from these two equations to
eliminate x. This is, in fact, Tschirnhausen's transformation (1683).
22. In connexion with what precedes, the question arises as to the number of
values (obtained by permutations of the roots) of given unsymmetrical functions of the
roots, or say of a given set of letters : for instance, with roots or letters (a, b, c, d)
as before, how many values are there of the function ab -f cd, or better, how many
functions are there of this form ? The answer is 3, viz. ab + cd, ac + bd, ad+bc- or
again we may ask whether, in the case of a given number of letters, there exist
functions with a given number of values, 3-valued, 4-valued functions, &c.
It is at once seen that for any given number of letters there exist 2-valued
functions ; the product of the differences of the letters is such a function ; however
the letters are interchanged, it alters only its sign ; or say the two values are A, — A.
And if P, Q are symmetrical functions of the letters, then the general form of such
a function is P + QA ; this has only the two values P + QA, P — QA.
In the case of 4 letters there exist (as appears above) 3-valued functions : but
in the case of 5 letters there does not exist any 3-valued or 4-valued function ; and
the only 5-valued functions are those which are symmetrical in regard to four of the
letters, and can thus be expressed in terms of one letter and of symmetrical functions
of all the letters. These last theorems present themselves in the demonstration of
the non-existence of a solution of a quintic equation by radicals.
The theory is an extensive and important one, depending on the notions of sub
stitutions and of groups *.
* A substitution is the operation by which we pass from the primitive arrangement of n letters to any
other arrangement of the same letters : for instance, the substitution / — — -' means that a is to be changed
510 EQUATION. [786
23. Returning to equations, we have the very important theorem that, given the
value of any imsymmetrical function of the roots, e.g. in the case of a quartic
equation, the function ab + cd, it is in general possible to determine rationally the
value of any similar function, such as (a + 6)3 + (c + d)z.
The a priori ground of this theorem may be illustrated by means of a numerical
equation. Suppose that the roots of a quartic equation are 1, 2, 3, 4, then if it is
given that ab + cd=14, this in effect determines a, b to be 1, 2 and c, d to be 3, 4
(viz. a = l, 6=2 or a = 2, 6 = 1, and c = 3, d=4 or c = 4, d = 3) or else a, b to be
3, 4 and c, d to be 1, 2 ; and it therefore in effect determines (a + bj> + (c + d)3 to
be = 370, and not any other value ; that is, (a + 6)3 + (c + d)3, as having a single value,
must be determinable rationally. And we can in the same way account for cases of
failure as regards particular equations; thus, the roots being 1, 2, 3, 4 as before,
a26 = 2 determines a to be = 1 and 6 to be = 2 ; but if the roots had been 1, 2, 4, 16
then a"-b = 16 does not uniquely determine a, b but only makes them to be 1, 16 or
2, 4 respectively.
As to the a posteriori proof, assume, for instance,
t1 = ab+cdi y1 = (a + 6)3 + (c + d)3,
t, = ac + bd, 7/0 = (a + c )3 + (6 + d)3,
t3 = ad + be, ?/3 = (a + d)3 + (b + c)3:
then
2/i + 2/2 + 2/3, *i2/i + t,y.2 + t3y3 , t^ij, + t,zy, + t3%,
will be respectively symmetrical functions of the roots of the quartic, and therefore
rational and integral functions of the coefficients ; that is, they will be known.
Suppose for a moment that tlt tz, t3 are all known; then the equations being
linear in y1, yz, y.A these can be expressed rationally in terms of the coefficients and
of t1} tz> ts; that is, y1} y.2, y3 will be known. But observe further that y1 is obtained
as a function of tl} t.2, t3 symmetrical as regards t2, t3: it can therefore be expressed
into b, b into c, c into d, d into a. Substitutions may, of course, be represented by single letters a, 0, . . ;
(abed ' = lf is the substitution which leaves the letters unaltered. Two or more substitutions may be com
pounded together and give rise to a substitution; i.e., performing upon the primitive arrangement first the
substitution p and then upon the result the substitution a, we have the substitution a/3. Substitutions are
not commutative; thus, a£ is not in general =/3a ; but they are associative, a.p.y=a.py, so that afty has a
determinate meaning. A substitution may be compounded any number of times with itself, and we thus
have the powers a2, a3, . . , &c. Since the number of substitutions is limited, some power a" must be =1: or,
as this may be expressed, every substitution is a root of unity. A group of substitutions is a set such
that each two of them compounded together in either order gives a substitution belonging to the set; every
group includes the substitution unity, so that we may in general speak of a group 1, a, /3, ... (the number
of terms is the order of the group). The whole system of the 1.2.3...n substitutions which can be per
formed upon the n letters is obviously a group : the order of every other group which can be formed out
of these substitutions is a submultiple of this number; but it is not conversely true that a group exists
the order of which is any given submultiple of this number. In the case of a determinant the substitutions
which give rise to the positive terms form a group the order of which is =£ . 1 . 2 . 3 ..?;. For any function
of the n letters, the whole series of substitutions which leave the value of the functions unaltered form a
group; and thence also the number of values of the function is =1.2.3...n divided by the order of the group.
786] EQUATION. 511
as a rational function of ti and of t2 + t3> t2ts, and thence as a rational function of ti
and of £i + £2 + ^, ^tj + Ms + Ms, £iMs; but these last are symmetrical functions of the
roots, and as such they are expressible rationally in terms of the coefficients; that is,
y± will be expressed as a rational function of ^ and of the coefficients ; or ^ (alone,
not t2 or £3) being known, yl will be rationally determined.
24. We now consider the question of the algebraical solution of equations, or,
more accurately, that of the solution of equations by radicals.
In the case of a quadric equation a? — px + q = Q, we can by the assistance of the
sign V( ) or ( )* find an expression for # as a two-valued function of the coefficients
p, q such that, substituting this value in the equation, the equation is thereby identically
satisfied ; it has been found that this expression is
* = \ {p ± VF* - 4g|,
and the equation is on this account said to be algebraically solvable, or more accurately
solvable by radicals. Or we may by writing x = -\p-\-z, reduce the equation to
^ ~ i (p* ~ 4<?) viz. to an equation of the form z* = a ; and in virtue of its being
thus reducible we say that the original equation is solvable by radicals. And the
question for an equation of any higher order, say of the order n, is, can we by means
j^
of radicals, that is, by aid of the sign ^/( ) or ( )"*, using as many as we please
of such signs and with any values of m, find an n-valued function (or any function)
of the coefficients which substituted for sc in the equation shall satisfy it identically.
It will be observed that the coefficients p, q,... are not explicitly considered as
numbers, but even if they do denote numbers, the question whether a numerical
equation admits of solution by radicals is wholly unconnected with the before-mentioned
theorem of the existence of the n roots of such an equation. It does not even
follow that in the case of a numerical equation solvable by radicals the algebraical
solution gives the numerical solution, but this requires explanation. Consider first a
numerical quadric equation with imaginary coefficients. In the formula x=^(p ± \/p2— 4q)}
substituting for p, q their given numerical values, we obtain for x an expression of
the form x — a + $i ± \Ay + 8i, where a, /3, 7, B are real numbers. This expression sub
stituted for a; in the quadric equation would satisfy it identically, and it is thus an
algebraical solution; but there is no obvious a priori reason why ^y + Si should have
a value = c + di, where c and d are real numbers calculable by the extraction of a
root or roots of real numbers; however the case is (what there was no a priori
right to expect) that \iy + 8i has such a value calculable by means of the radical
expressions V{^72 + & ± y] '• and hence the algebraical solution of a numerical quadric
equation does in every case give the numerical solution. The case of a numerical
cubic equation will be considered presently.
25. A cubic equation can be solved by radicals. Taking for greater simplicity
the cubic in the reduced form a? + qx — r — 0, and assuming x = a + b, this will be a
solution if only 3ab = q and a3 + b3 = r, equations which give (a3 — bs)* = r- — -gTq3, a
512 EQUATION. [786
quadric equation solvable by radicals, and giving a3 — bs = Vr2 — ^qA, a two- valued
function of the coefficients : combining this with a3 4- b3 = r, we have a3 = ^ (r + Vr2 —
a two-valued function: we then have a by means of a cube root, viz.
a six- valued function of the coefficients; but then, writing q = —-) we have, as may
•jd
be shown, a + b a three-valued function of the coefficients ; and x = a + b is the
required solution by radicals. It would have been wrong to complete the solution by
writing ___^_
for then a + b would have been given as a 9-valued function having only 3 of its
values roots, and the other 6 values being irrelevant. Observe that in this last
process we make no use of the equation 3a& = q, in its original form, but use only
the derived equation 27a363 = q3, implied in, but not implying, the original form.
An interesting variation of the solution is to write x = ab (a + b), givino-
3r
a3b3 (a3 + b3) = r and 3a3b3 = q, or say a3 + b3 = — , a3b3 = ^q ; and consequently
3 __ 3
a3 = 2 (r + Vr1 - Ao3), b3 = ^ (r - Vr2 - wW1),
9 9
i.e., here a3, b3 are each of them a two-valued function, but as the only effect of
altering the sign of the quadric radical is to interchange a3, b3, they may be regarded
as each of them one-valued ; a and b are each of them 3-valued (for observe that
here only a3b3, not ab, is given) ; and ab (a + b) thus is in appearance a 9-valued
function, but it can easily be shown that it is (as it ought to be) only 3-valued.
In the case of a numerical cubic, even when the coefficients are real, substituting
their values in the expression
this may depend on an expression of the form \/y + 8i, where 7 and 8 are real
numbers (it will do so if r2 — ^q3 is a negative number), and then we cannot by
the extraction of any root or roots of real positive numbers reduce \/y + 8i to the
form c + di, c and d real numbers ; hence here the algebraical solution does not
give the numerical solution, and we have here the so-called "irreducible case" of a
cubic equation. By what precedes, there is nothing in this that might not have
been expected ; the algebraical solution makes the solution depend on the extraction
of the cube root of a negative number, and there was no reason for expecting this to
be a real number. It is well known that the case in question is that wherein the
three roots of the numerical cubic equation are all real; if the roots are two
imaginary, one real, then contrariwise the quantity under the cube root is real ; and
the algebraical solution gives the numerical one.
786] EQUATION. 513
The irreducible case is solvable by a trigonometrical formula, but this is not a
solution by radicals : it consists, in effect, in reducing the given numerical cubic (not
to a cubic of the form z* = a, solvable by the extraction of a cube root, but) to a
cubic of the form 4^ — 3x = a, corresponding to the equation 4 cos3 6 — 3 cos 6 = cos 30
which serves to determine cos 6 when cos 36 is known. The theory is applicable to
an algebraical cubic equation ; say that such an equation, if it can be reduced to
the form 4^ — 3x = a, is solvable by " trisection " — then the general cubic equation
is solvable by trisection.
26. A quartic equation is solvable by radicals : and it is to be remarked that the
existence of such a solution depends on the existence of 3-valued functions such as
ah + cd of the four roots (a, b. c, d) : by what precedes, ab + cd is the root of a cubic
equation, which equation is solvable by radicals : hence ab + cd can be found by
radicals; and since abed is a given function, ab and cd can then be found by radicals.
But by what precedes, if ab be known then any similar function, say a + b, is ob
tainable rationally ; and then from the values of a + b and ab we may by radicals
obtain the value of a or b, that is, an expression for the root of the given quartic
equation : the expression ultimately obtained is 4-valued, corresponding to the different
values of the several radicals which enter therein, and we have thus the expression
by radicals of each of the four roots of the quartic equation. But when the quartic
is numerical the same thing happens as in the cubic, and the algebraical solution
does not in every case give the numerical one.
It will be understood, from the foregoing explanation as to the quartic, how in
the next following case, that of the quintic, the question of the solvability by radicals
depends on the existence or non-existence of /> valued functions of the five roots
(a, b, c, d, e); the fundamental theorem is the one already stated, a rational function
of five letters, if it has less than 5, cannot have more than 2 values, that is, there
are no 3-valued or 4-valued functions of 5 letters : and by reasoning depending in
part upon this theorem, Abel (1824) showed that a general quintic equation is not
solvable by radicals ; and a fortiori the general equation of any order higher than 5
is not solvable by radicals.
27. The general theory of the solvability of an equation by radicals depends
fundamentally on Vandermonde's remark (1770) that, supposing an equation is solvable
by radicals, and that we have therefore an algebraical expression of x in terms of the
coefficients, then substituting for the coefficients their values in terms of the roots,
the resulting expression must reduce itself to any one at pleasure of the roots a, b, c,.. ;
thus in the case of the quadric equation, in the expression x = ^ (p + *Jp'2 — 4>q), sub
stituting for p and q their values, and observing that (a + 6)2 — 4o6 = (a — 6)2, this
becomes x = \ {a + b + ^(a — b)-}, the value being a or b according as the radical is
taken to be + (a — b) or — (a — b).
So in the cubic equation tf—px2 + qx — r=(), if the roots are a, b, c, and if w
is used to denote an imaginary cube root of unity, &>* + &> + 1 = 0, then writing for
shortness p = a + b + c, L = a + wb + o>3c, M = a + or& + we, it is at once seen that LM,
c. XL 65
•I I KQIJATION. [780
/- i M\ and therefore IIJMO (I/--M*)'1 are Mytnrnct/ricul fuwit/ioiiH of the rootH, and eori-
wqnently rational fund-ion* of the codlicicnt.H : hence
JM a rational fund/ion of the coeflieientM, which when theHe arc replaced by their
value* an fllM('l,io||H of I, In- lool.M becoWCH, according to tin; Hign jj;iven to the quudii--
radical, —A11 or M ' tnliing it - //', tin- cube root of tin; oxproHHJon h;m the I, luce
VU.ll|(iM I., in!,. <»' I, , -ui'l A/1/ divided by the name cube root hun thendon; the vulue.s
J\1 , n>'l\! , tinW ; whi'iieit fiiiully th<i cxpn'MHion
Inn. the three vidlieM
^(p \ A I M), <\ (/> I ft> A I ft>'JA/), J^ (/> -f ft>'A I dt]\J) ;
that ih, thcMo are — u, />, (! rcHpcctivoly. If the value Mn hail been taken instead of
A", then the exprcNMJon would have L.i-l the name three valiien <i, h, c. Comparing
the ..Inn. .n given lor the cubic . \ </.r r <), it, will readily be seen that the
two HollltioiiN are identical, and I hut I he IUIK-II-.H /' .,1^' under I In- radical sign
muni, (by aid of the K-L-ili-.n // 0 which HiibMintH in this case) reduce itself to
< A" /!/")"; it in only by encli radical being equal to a rational function of the
rooliN that the final expre^Hion run become eijn.il to the rootH n, l>, c, respectively.
2H. The formulie for the cubic were obtained by Lagrangc (1770 -71) from a
different point of view. Upon examining und comparing tin principal known methods
for the Holulion of algebraical equations, he found that they all ultimately depended
H pi MI (hiding M "rcMolvont" equation of which the root is n \ o>/> \ ore | (,)'<! \ ... t <y
In nit; an imaginary root ol umlv, -.1 the name order MS the equation; e.g., for the
• nine the root IM it I (.i/) I diY, n> MII imaginary cube root of unity. Kvidont ly ill-
method ^iveH for A" it quitdric equation, which in the "renolvont" <>quation in this
parlieular eiiMe.
I • i a quartie ill. formulie pi. . m thiMiiselyes in it somewhat. ditVerent form, by
rcMhon that -I- \» not a prime number. At.fempting to apply it to a quintic, we seek
for I. lie equation of which the root is (<t \ <••/> I o>Y I (•>'</ I <n'('). <•> an imaginary fifth
root of unity, or rather the fifth power (.hereof (a -f tab + wV I mY/ I «iY)", this is a
'2-k valued funellon. but if wo consider the tour values corresponding to the roots of
\ -.'. -.' . ui', <ii4. vi/. t he valuen
(<t I ... /. i (,.V | 0 ,/ | <i«YV\
((I -f «*/> + W 0 + «i'4(/ I n't'Y ,
(a + w^ -»- ft)«o + »VJ 4- ft) f)^,
M\y wynunelrieid i\> notion of (.host1, lor instance their sum, is a si\ \alued function
of the roots, and may therefore be determined by me.'ins of a sextic equation, the
786] IMITATION. 515
coefficients whereof ;nv rational functions of the coefficients of the original quintie
equation; the conclusion being that the solution of an equation of the tilth order
is made to depend upon that of an equation of the sixth order. This is, of course,
useless for the solution of the quintic equation, which, us already mentioned, does
not admit of solution hy radicals; hut the equation of the sixth order, Ijagrango's
resolvent sextie, is very important, and is intimately connected with all the later
investigations in tin- theory.
2!). It is to be remarked, in regard to the question of solvability by radicals,
that not only the coefficients are taken to be arbitrary, but it is assumed that they
are represented each by a single letter, or say rather thai they are not so expressed
in terms of other arbitrary quantities as to make a solution possible. If the
eoeilicienfs a. re not all arbitrary, for instance, if some of them are /.oro, a sextie
equation might be of the form .*•" I l>.rl \ r.r' I </ 0, and so be solvable as a cubic;
or if the coefficients of the sextie are given functions of the nix arbitrary quantities
<t, b, c, </, c, /, such that the sextie is really of the form
(of + a* 4- b) (.1 •' I r.r" | dot + ttO +/) - 0,
then it breaks up into the equations up + CM + 6 • 0, m* + Oaf + dot9 + t)iv \ /' 0, and is
consequently solvable by radicals; so also if the form is
(as — a) (a} — b) (oo — o)(oo — d) (fl — •) (0 — /) • 0,
then the equation is solvable by radicals, in this extreme case rationally. Such
cases of solvability are self-evident; but they are enough to show that the general
theorem of the non-solvability by radicals of an equation of the fifth or any higher
order does not in any wise exclude for such orders the existence of particular
equations solvable by radicals, and there are, in fact, extensive classes of equations
which are thus solvable; the binomial equations «;" — I 0 present an instance.
.'{(). It, has already been shown how the several roots of the equation ,r" — I •• 0
*-iV7T l^tVTT
'•MM In- expressed in the form COM + 1 sin , 1ml. the question is now that of
n n
the algebraical solution (or solution by radicals) of this equation. There is always a.
root =1; if <u be any other root, then obviously M, ft)",..., ro" ' are all of them roots ;
a:11 — I contains the factor //; — I, and it/ thus appears that <i>, or <<>" ' are the n — I
roots of the equation
epr-i + afH-*+ ... 4-fl. ,. i .Q;
We have, of course,
w" ' + w" " + . . . + a> + I - 0.
It is proper to distinguish tin eases n prime and n, composite; und in the
latter case there is a distinction according as the prime factors of // are simple or
multiple, liy way of illustration, suppose successively // = 15 and //=«!>; in the former
case, if a be an imaginary root of ./."'— I -• 0 (or root of ,/•- | ./• | I -.()), and ft an
imaginary root of u* — I = 0 (or root of at + a? + a? + 01 \ I 0), then <u may be
taken =a/3; the successive powers thereof, a/tf, a'p, 0\ a/J4, a'J, /tf, o/^'J, a"^", ft', a.
65—2
516 EQUATION. [786
a2/3, /32, a/33, a2/34, are the roots of xu + xiz + . . . + x + 1 = 0 ; the solution thus depends
on the solution of the equations x3 — l = 0 and x5 — 1=0. In the latter case, if a
be an imaginary root of x3 — l = Q (or root of x'2 + x + 1 = 0), then the equation a? — 1 = 0
gives a? =1, a, or a2; a?=\ gives x=l, a, or a2; and the solution thus depends on
the solution of the equations x3-l = 0, x3-a = 0, x3-o.2 = 0. The first equation has
the roots 1, a, a2; if ft be a root of either of the others, say if /33 = a, then
assuming &> = /3, the successive powers are /3, /32, a, a/3, a/82, a2, a2/3, a'/S2, which are
the roots of the equation Xs + x7 + ... + x + 1 =0.
It thus appears that the only case which need be considered is that of n a
prime number, and writing (as is more usual) r in place of o>, we have r, r2, r3,...,^1-1
as the (n— 1) roots of the reduced equation
then not only rn — 1 = 0, but also rn~l + rn~2 + ... +r + I =0.
31. The process of solution due to Gauss (1801) depends essentially on the
arrangement of the roots in a certain order, viz. riot as above, with the indices of
r in arithmetical progression, but with their indices in geometrical progression ; the
prime number n has a certain number of prime roots g, which are such that gn~l
is the lowest power of g, which is = 1 to the modulus n ; or, what is the same
thing, that the series of powers 1, g, g-,..., gn~2, each divided by n, leave (in a
different order) the remainders 1, 2, 3,..., n — I; hence giving to r in succession the
indices 1 , g, g2,..., gn~'2, we have, in a different order, the whole series of roots
In the most simple case, n = 5, the equation to be solved is
here 2 is a prime root of 5, and the order of the roots is r, r2, r*, r3. The Gaussian
process consists in forming an equation for determining the periods P1} P2, = r + r4
and r2 + 1* respectively, — these being such that the symmetrical functions Pl + P2, PXP2
are rationally determinable : in fact,
PI + Pa = - 1, PiP2 = (r + r4) (r2 + r3), = r3 + r* + r6 + r7, = r3 + r4 + r + r2, = - 1.
Pj, P2 are thus the roots of u* + u— 1=0; and taking them to be known, they
are themselves broken up into subperiods, in the present case single terms, r and r*
for Pj, r2 and r3 for P2; the symmetrical functions of these are then rationally
determined in terms of Pl and P2 ; thus r + ^ = P1, r.r*=l, or r, r4 are the roots
of u2 — P^u + 1=0. The mode of division is more clearly seen for a larger value
of n ; thus, for n = 7 a prime root is = 3, and the arrangement of the roots is
r, r3, r-, r6, r4, r5. We may form either 3 periods each of 2 terms,
PI, Ps, Ps, =r + r«, r3 + r4, r* + r5,
respectively; or else 2 periods each of 3 terms, P1} P2 = r + r2 + r4, rs + r6 + r5 respec
tively; in each case the symmetrical functions of the periods are rationally determinable;
thus in the case of the two periods P1 + P2 = — 1 , PjP2 = 3 + r + r2 + r3 + r4 + r5 + r6, = 2 ;
786] EQUATION. 517
and, the periods being known, the symmetrical functions of the several terms of each
period are rationally determined in terms of the periods, thus
r + r- + r* = P!, r . r2 + r . r*+ r2 . r* = P2, r . r- . r*=\.
The theory was further developed by Lagrange (1808), who, applying his general
process to the equation in question, xn~l + xn~~ + ...+# + 1 = 0, the roots a, b, c, ... being
the several powers of r, the indices in geometrical progression as above, showed that
the function (a + &>6 + &)2c + ...)w~1 was in this case a given function of &> with integer
coefficients. Reverting to the before-mentioned particular equation #4 + #3 + #2 + # + l = 0,
it is very interesting to compare the process of solution with that for the solution
of the general quartic the roots whereof are a, b, c, d.
Take &>, a root of the equation &>4— 1 = 0 (whence o> is =!,—!, i, or — i, at
pleasure), and consider the expression
(a + wb + &>2c + a)3dy.
The developed value of this is
= a4 + &4 + c4 + d4 + 6 (a2c2 + &2<22) + 12 (a?bd + b2ca + c-db + d2ac)
+ to {4 (asb + bsc + csd + dsa) + 12 (a~cd + b2da + c2ab + d2bc)}
+ to2 (6 (a?b2 + bn-c2 + c2d2 + d2a2) + 4 (a3c + b3d + c^a + dsb) + 24a&cd}
+ a>3 {4 (asd + b3a + c3b + d3c) + 1 2 (a2bc + b*cd + c2da + d2ab)} ;
that is, this is a 6-valued function of a, b, c, d, the root of a sextic (which is, in
fact, solvable by radicals ; but this is not here material).
If, however, a, b, c, d denote the roots r, r2, r4, r3 of the special equation, then
the expression becomes
r4 + r3 + r + r2 + 6 (1 + 1) + 12 (r2 + r4 + r3 + r)
+ o) {4(1 + 1 +1 +!)+12(r4 + r3 + r + r2)}
+ &)2 {6 (r + r2 + r4 + r3) + 4 (r2 + r4 + r3 + r )}
+ ft)3 {4 (r + r2 + r4 + r3) + 12 (r3 + r + r2 + r4)} ;
viz. this is
= -l+4o) + 14o)2-16a)3,
a completely determined value. That is, we have
(r + cor2 + o>2r4 + &>¥3)4 = - 1 + 4&> + 14&>2 - 16<a3,
which result contains the solution of the equation. If «o = l, we have (r + r2 + r4 + r3)4 = 1,
which is right ; if w = — 1, then (r + r4 — r2 — r3)4 = 25 ; if &> = i, then we have
{r-r* + i (r2-r3)}4 = - 15 + 20*'; and if a> = -i, then [r-r*-i (r2 -r3)j4 = - 15 -20i; the
solution may be completed without difficulty.
The result is perfectly general, thus : — n being a prime number, r a root of the
equation xn~l + xn~2 +...+# +1 = 0, o> a root of tu'1"1 —1=0, and g a prime root of
t?71-1 = 1 (mod. n), then
(r
OF THK
UNIVERSITY
518 EQUATION. [786
is a given function M0 + M^ + . . . + J/n_2 wn~- with integer coefficients, and by the
extraction of (n — l)th roots of this and similar expressions we ultimately obtain r
in terms of &>, which is taken to be known ; the equation xn — 1 = 0, n a prime
number, is thus solvable by radicals. In particular, if n — 1 be a power of 2, the
solution (by either process) requires the extraction of square roots only ; and it was
thus that Gauss discovered that it was possible to construct geometrically the regular
polygons of 17 sides and 257 sides respectively. Some interesting developments in
regard to the theory were obtained by Jacobi (1837); see the memoir " Ueber die
Kreistheilung, u.s.w.," Crelle, t. xxx. (1846).
The equation xn~l + , . . + x + 1 = 0 has been considered for its own sake, but it also
serves as a specimen of a class of equations solvable by radicals, considered by Abel
(1828), and since called Abelian equations, viz., for the Abelian equation of the order n,
if a? be any root, the roots are x, Ox, 6-x,..., (fr-^x (Qx being a rational function of x,
and 6nx = x); the theory is, in fact, very analogous to that of the above particular
case. A more general theorem obtained by Abel is as follows : — If the roots of an
equation of any order are connected together in such wise that all the roots can be
expressed rationally in terms of any one of them, say x\ if, moreover, Qx, Q^x being
any two of the roots, we have QQ^x = O^x, the equation will be solvable algebraically.
It is proper to refer also to Abel's definition of an irreducible equation :— an equation
$>x = 0, the coefficients of which are rational functions of a certain number of known
quantities a, b, c, ..., is called irreducible when it is impossible to express its roots
by an equation of an inferior degree, the coefficients of which are also rational functions
of a, b, c, ... (or, what is the same thing, when 4>x does not break up into factors
which are rational functions of a, b, c, ...). Abel applied his theory to the equations
which present themselves in the division of the elliptic functions, but not to the modular
equations.
32. But the theory of the algebraical solution of equations in its most complete
form was established by Galois (born October 1811, killed in a duel May 1832; see
his collected works, Liouville, t. XL, 1846). The definition of an irreducible equation
resembles Abel's, — an equation is reducible when it admits of a rational divisor,
irreducible in the contrary case; only the word rational is used in this extended
sense that, in connexion with the coefficients of the given equation, or with the
irrational quantities (if any) whereof these are composed, he considers any number of
other irrational quantities called "adjoint radicals," and he terms rational any rational
function of the coefficients (or the irrationals whereof they are composed) and of these
adjoint radicals; the epithet irreducible is thus taken either absolutely or in a relative
sense, according to the system of adjoint radicals which are taken into account. For
instance, the equation a? + z3 + a? + x+1 = 0; the left-hand side has here no rational
divisor, and the equation is irreducible ; but this function is = (a? + \x + 1)2- f^2, and
it has thus the irrational divisors a? + i (1 +Jl)x+ 1, x- + \ (1 - Jl)x + 1 ; and these,
if we adjoin the radical Jb, are rational, and the equation is no longer irreducible.
In the case of a given equation, assumed to be irreducible, the problem to solve the
equation is, in fact, that of finding radicals by the adjunction of which the equation
786] EQUATION. 519
becomes reducible; for instance, the general quadric equation a?+px + q = Q is irre
ducible, but it becomes reducible, breaking up into rational linear factors, when we
adjoin the radical J\pz — q-
The fundamental theorem is the Proposition I. of the " Me moire sur les conditions
de resolubilit^ des Equations par radicaux " ; viz. given an equation of which a, b, c, ...
are the m roots, there is always a group of permutations of the letters a, b, c, ...
possessed of the following properties : —
1. Every function of the roots invariable by the substitutions of the group is
rationally known.
2. Reciprocally, every rationally determinable function of the roots is invariable
by the substitutions of the group.
Here by an invariable function is meant not only a function of which the form is
invariable by the substitutions of the group, but further, one of which the value is
invariable by these substitutions: for instance, if the equation be <j>a; = Q, then <f)oc is
a function of the roots invariable by any substitution whatever. And in saying that
a function is rationally known, it is meant that its value is expressible rationally in
terms of the coefficients and of the adjoint quantities.
For instance, in the case of a general equation, the group is simply the system of
the 1 . 2 . 3 ... n permutations of all the roots, since, in this case, the only rationally
determinable functions are the symmetric functions of the roots.
In the case of the equation xn~l + . . . + x + 1 = 0, n a prime number,
2 H ~~ 2
where g is a prime root of n, then the group is the cyclical group abc ... k,
be ... ka, ..., kab ...j, that is, in this particular case the number of the permutations
of the group is equal to the order of the equation.
This notion of the group of the original equation, or of the group of the equation
as varied by the adjunction of a series of radicals, seems to be the fundamental one
in Galois's theory. But the problem of solution by radicals, instead of being the
sole object of the theory, appears as the first link of a long chain of questions relating
to the transformation and classification of irrationals.
Returning to the question of solution by radicals, it will be readily understood
that by the adjunction of a radical the group may be diminished ; for instance, in
the case of the general cubic, where the group is that of the six permutations, by
the adjunction of the square root which enters into the solution, the group is reduced
to abc, bca, cab ; that is, it becomes possible to express rationally, in terms of the
coefficients and of the adjoint square root, any function such as a2b + 62c + c2a which
is not altered by the cyclical substitution a into b, b into c, c into a. And hence,
to determine whether an equation of a given form is solvable by radicals, the course
of investigation is to inquire whether, by the successive adjunction of radicals, it is
520 EQUATION. [786
possible to reduce the original group of the equation so as to make it ultimately
consist of a single permutation.
The condition in order that an equation of a given prime order n may be
solvable by radicals was in this way obtained — in the first instance in the form,
scarcely intelligible without further explanation, that every function of the roots
#!, #,, ...,a}n, invariable by the substitutions #<,*+& f°r xk> must be rationally known;
and then in the equivalent form that the resolvent equation of the order 1 . 2 ... n — 2
must have a rational root. In particular, the condition in order that a quintic equation
may be solvable is that Lagrange's resolvent of the order 6 may have a rational
factor, a result obtained from a direct investigation in a valuable memoir by E. Luther,
Crelle, t. xxxiv. (1847).
Among other results demonstrated or announced by Galois may be mentioned
those relating to the modular equations in the theory of elliptic functions ; for the
transformations of the orders 5, 7, 11, the modular equations of the orders 6, 8, 12
are depressible to the orders 5, 7, 11 respectively; but for the transformation, n a
prime number greater than 11, the depression is impossible.
The general theory of Galois in regard to the solution of equations was completed,
and some of the demonstrations supplied, by Betti (1852). See also Serret's Cours
d'Algebre superieure, 2nd ed. 1854 ; 4th ed. 1877 — 78.
33. Returning to quintic equations, Jerrard (1835) established the theorem that
the general quintic equation is, by the extraction of only square and cubic roots,
reducible to the form of + ax + b = 0, or what is the same thing, to x5 + x + b = 0.
The actual reduction by means of Tschirnhausen's theorem was effected by Hermite
in connexion with his elliptic-function solution of the quintic equation (1858) in a
very elegant manner. It was shown by Cockle and Harley (1858 — 59) in connexion
with the Jerrardian form, and by Cayley (1861), that Lagrange's resolvent equation of
the sixth order can be replaced by a more simple sextic equation occupying a like
place in the theory.
The theory of the modular equations, more particularly for the case n = 5, has
been studied by Hermite, Kronecker, and Brioschi. In the case n = 5, the modular
equation of the order 6 depends, as already mentioned, on an equation of the order 5 ;
and conversely the general quintic equation may be made to depend upon this modular
equation of the order 6 ; that is, assuming the solution of this modular equation, we
can solve (not by radicals) the general quintic equation; this is Hermite's solution
of the general quintic equation by elliptic functions (1858) ; it is analogous to the
before-mentioned trigonometrical solution of the cubic equation. The theory is repro
duced and developed in Brioschi's memoir, "Ueber die Auflb'sung der Gleichungen vom
flinften Grade," Math. Annalen, t. xin. (1877 — 78).
34. The great modern work, reproducing the theories of Galois, and exhibiting
the theory of algebraic equations as a whole, is Jordan's Traite des Substitutions et
des Equations Algebriques, Paris, 1870. The work is divided into four books — book I.,
786] EQUATION. 521
preliminary, relating to the theory of congruences ; book n. is in two chapters, the first
relating to substitutions in general, the second to substitutions defined analytically, and
chiefly to linear substitutions ; book in. has four chapters, the first discussing the
principles of the general theory, the other three containing applications to algebra,
geometry, and the theory of transcendents ; lastly, book iv., divided into seven chapters,
contains a determination of the general types of equations solvable by radicals, and a
complete system of classification of these types. A glance through the index will show
the vast extent which the theory has assumed, and the form of general conclusions
arrived at ; thus, in book III., the algebraical applications comprise Abelian equations,
equations of Galois; the geometrical ones comprise Hesse's equation, Clebsch's equations,
lines on a quartic surface having a nodal line, singular points of Kummer's surface, lines
on a cubic surface, problems of contact ; the applications to the theory of transcendents
comprise circular functions, elliptic functions (including division and the modular equation),
hyperelliptic functions, solution of equations by transcendents. And on this last subject,
solution of equations by transcendents, we may quote the result, — "the solution of the
general equation of an order superior to five cannot be made to depend upon that of
the equations for the division of the circular or elliptic functions " ; and again (but with
a reference to a possible case of exception), "the general equation cannot be solved by
aid of the equations which give the division of the hyperelliptic functions into an odd
number of parts."
C. XI. 66
522 [787
787.
FUNCTION.
[From the Encyclopcedia Britannica, Ninth Edition, vol. ix. (1879), pp. 818 — 824.]
FUNCTIONALITY, in Analysis, is dependence on a variable or variables ; in the case
of a single variable u, it is the same thing to say that v depends upon u, or to say
that v is a function of u, only in the latter form of expression the mode of dependence
is embodied in the term " function." We have given or known functions such as u2
or smu, and the general notation of the form <f>u, where the letter <£ is used as a
functional symbol to denote a function of u, known or unknown as the case may be :
in each case u is the independent variable or argument of the function, but it is
to be observed that, if v be a function of u, then v like u is a variable, the values
of v regarded as known serve to determine those of u ; that is, we may conversely
regard u as a function of v. In the case of two or more independent variables, say
when w depends on or is a function of u, v, &c., or w=<f>(u, v, ...), then u, v,... are
the independent variables or arguments of the function ; frequently when one of these
variables, say u, is principally or alone attended to, it is regarded as the independent
variable or argument of the function, and the other variables v, &c., are regarded as
parameters, the values of which serve to complete the definition of the function. We
may have a set of quantities w, t, ... each of them a function of the same variables
u, v, . . . ; and this relation may be expressed by means of a single functional symbol </>,
{w, t, . . .) = (f> (u, v, ...); but, as to this, more hereafter.
The notion of a function is applicable in geometry and mechanics as well as in
analysis ; for instance, a point Q, the position of which depends upon that of a
variable point P, may be regarded as a function of the point P ; but here, sub
stituting for the points themselves the coordinates (of any kind whatever) which
determine their positions, we may say that the coordinates of Q are each of them a
function of the coordinates of P, and we thus return to the analytical notion of a
function. And in what follows a function is regarded exclusively in this point of view,
787] FUNCTION. 523
viz. the variables are regarded as numbers; and we attend to the case of a function
of one variable v =fu. But it has been remarked (see Equation) that it is not
allowable to confine the attention to real numbers; a number u must in general be
taken to be a complex number u = x + iy, a; and y being real numbers, each suscept
ible of continuous variation between the limits — oo , +00, and i denoting V — 1. In
regard to any particular function, fu, although it may for some purposes be sufficient
to know the value of the function for any real value whatever of u, yet to attend
only to the real values of u is an essentially incomplete view of the question ; to
properly know the function, it is necessary to consider u under the aforesaid imaginary
or complex form u — a; + iy.
To a given value x + iy of u there corresponds in general for v a value or values
of the like form v = as' + iy', and we obtain a geometrical notion of the meaning of
the functional relation v=fu by regarding as, y as rectangular coordinates of a point P
it) a plane II, and x, y' as rectangular coordinates of a point P' in a plane (for
greater convenience a different plane) II' ; P, P' are thus the geometrical represent
ations, or representative points, of the variables u = as + iy and u' = x + iy' respectively ;
and, according to a locution above referred to, the point P' might be regarded as a
function of the point P ; a given value of u = x + iy is thus represented by a point
P in the plane JI, and corresponding hereto we have a point or points P' in the
plane II', representing (if more than one, each of them) a value of the variable
v=ac' + iy. And, if we attend only to the values of u as corresponding to a series
of positions of the representative point P, we have the notion of the " path " of a
complex variable u = a; + iy.
Known Functions.
1. The most simple kind of function is the rational and integral function. We
have the series of powers w2, u3, ... each calculable not only for a real but also for a
complex value of u, (x + iy)2 = x- — iy- + 2ixy, (x + iy)s = a? — 3xy- + i (Sx2y — y3), &c., and
thence, if a, b, ... be real or complex numbers, the general form a + bu + cu2 + ... + kum,
of a rational and integral function of the order m. And taking two such functions,
say of the orders m and n respectively, the quotient of one of these by the other
represents the general form of a rational function of u.
The function which next presents itself is the algebraical function, and in particular
the algebraical function expressible by radicals. To take the most simple case, suppose
^
(m being a positive integer) that vin=u; v is here the irrational function =um.
Obviously, if u is real and positive, there is always a real and positive value of v,
calculable to any extent of approximation from the equation vm = u, which serves as
j_
the definition of um ; but it is known (see Equation) that, as well in this case as
in the general case where u is a complex number, there are in fact m values of the
^
function um ; and that for their determination we require the theory of the so-called
66—2
524 FUNCTION. [787
circular functions sine and cosine ; and these depend on the exponential function exp u,
-or, as it is commonly written, e11, which has for its inverse the logarithmic function
logw; these are all of them transcendental functions.
2. In a rational and integral function a + bn + cu2 + ... + kum, the number of terms
is finite, and the coefficients a, b, — k may have any values whatever, but if we
imagine a like series a + bu + cu2 + ... extending to infinity, non constat that such an
expression has any calculable value, — that is, any meaning at all ; the coefficients
a, b, c, ... must be such as, either for every value whatever of u (that is, for every
finite value) or for values included within certain limits, to make the series convergent.
It is easy to see that the values of a, b, c, ... may be such as to make the series
always convergent ; for instance, this is the case for the exponential function,
u3
taking for the moment u to be real and positive, then it is evident that however
large u may be, the successive terms will become ultimately smaller and smaller, and
the series will have a determinate calculable value. A function thus expressed by
means of a convergent infinite series is not in general algebraical, and when it is not
so, it is said to be transcendental (but observe that it is in nowise true that we
have thus the most general form of a transcendental function) ; in particular, the
exponential function above written down is not an algebraical function.
By forming the expression of expw, and multiplying together the two series, we
derive the fundamental property
exp u exp v = exp (u + v) ;
whence also
exp x exp iy = exp (x + iy),
so that exp (x + iy) is given as the product of the two series exp x and exp iy. As
regards this last, if in place of u we actually write the value iy, we find
where obviously each series is convergent and actually calculable for any real value
whatever of y. Calling the two series cosine y and sine y respectively, or in the ordinary
abbreviated notation cos y and sin y, the equation is
exp iy = cos y + i sin y ;
and if we herein for y write z, and multiply the two expressions together, observing
that the product will be = exp i (y + z), we obtain the fundamental equations
cos (y + z) = cos y cos z — sin y sin zt
sin (y + z) = sin y cos z + sin z cos y,
for the functions sine and cosine.
787] FUNCTION. 525
Taking y as an angle, and defining as usual the sine and cosine as the ratios
of the perpendicular and base respectively to the radius, the sine and cosine will be
functions of y ; and we obtain geometrically the foregoing fundamental equations for the
sine and cosine ; but in order to the truth of the foregoing equation exp iy = cos y + i sin y,
it is further necessary that the angle should be measured in circular measure, that
is, by the ratio of the arc to the radius ; so that TT denoting as usual the number
314159..., the measure of a right angle is =^TT. And this being so, the functions
sine and cosine, obtained as above by consideration of the exponential function, have
their ordinary geometrical significations.
3. The foregoing investigation was given in detail in order to the completion of
^
the theory of the irrational function um. We henceforth take the theory of the
circular functions as known, and speak of tana;, &c., as the occasion may arise.
We have
x + iy = r (cos 0 + i sin 6},
where, writing V#2+2/2 to denote the positive value of the square root, we have
r = V#2 + y2, cos 9 = . , sin 6 = ,—.— ,
V #2 + f Va? + f
and therefore also
tan 6 = ^ .
x
Treating x, y as the rectangular coordinates of a point P, r is the distance (regarded
as positive) of this point from the origin, and 9 is the inclination of r to the positive
part of the axis of x ; to fix the ideas 6 may be regarded as lying within the
limits 0, TT, or 0, — TT, according as y is positive or negative ; 9 is thus completely
determinate, except in the case, x negative, y = 0, for which 9 is = TT or — TT indifferently.
And if u = x + iy, we hence have
u* = (x + ty)» = r » ( cos + i sin
m m )
where r"1 is real and positive and s has any positive or negative integer value what
ever: but we thus obtain for um only the m values corresponding to the values
0, 1, 2, ...,m— 1 of s. More generally we may, instead of the index , take the index
m
lli
to be any rational fraction — . Supposing this to be in its least terms, and m to
be positive, the number of distinct values is always = m. If instead of — we take
m
the index to be the general real or complex quantity m, we have um, no longer an
algebraical function of u, and having in general an infinity of values.
526 FUNCTION. [787
4. The foregoing equation exp (x + y) = exp as . exp y is, in fact, the equation of
indices, ax+y = ax . ay ; exp x is thus the same thing as e?, where e denotes a properly
determined number, and putting e? equal to the series, and then writing x=l, we
have e=l + -+ - + — 3 + &c., that is, e = 2-7128... But as well theoretically as
for convenience of printing, there is considerable advantage in the use of the notation
exp u.
From the equation, exp iy = cos y + i sin y, we deduce exp (— iy) = cos y — i sin y, and
thence
cos y = % {exp (iy) + exp (- iy)},
sin y = ^-. [exp (iy) - exp (- iy)} ;
if we write herein ix instead of y we have
cos ix = ^ {exp x + exp (— x)},
*i
sin ix — ^ {exp x — exp (— x)},
z
viz. these values are
a? x4
a?
a?+i7o + -
each of them real when x is real. The functions in question 1 + -— + f — + . . .
& • 25 & • Z • 0 • 4
a?
and # + ^~ 2~~o + •••> regarded as functions of x, are termed the hyperbolic cosine and
sine, and are represented by the notations cosh# and sinh x respectively; and similarly
we have the hyperbolic tangent tanhar, &c. : although it is easy to remember that
cos ix, ^ sin ix, are, in fact, real functions of x, and to understand accordingly the formulas
wherein they occur, yet the use of these notations of the hyperbolic functions is often
convenient.
5. Writing u = exp v, then v is conversely a function of u which is called the
logarithm (hyperbolic logarithm, to distinguish it from the tabular or Briggian logarithm),
and we write v = log u, or what is the same thing, we have u = exp (log u) : and it is
clear that if u be real and positive there is always a real and positive value of logw,
in particular the real logarithm of e is =1 ; it is however to be observed that the
logarithm is not a one-valued function, but has an infinity of values corresponding to
the different integer values of a constant «; in fact,, if logw be any one of its values,
then log u + 2s7ri is also a value, for we have exp (log u + 2s7ri) = exp log u exp Zsm, or
since exp Zsiri is =1, this is =u; that is, logu + Zsm is a value of the logarithm of u.
We have
uv = exp (log uv) = exp log u . exp log v,
787] FUNCTION. 527
and hence the equation which is commonly written
log tiv = log u + log v,
but which requires the addition on one side of a term 2s-rri. And reverting to the
equation x + iy = r (cos 6 + i sin 6), or as it is convenient to write it, x + iy = r exp 16,
we hence have
log (x + iy) = log r + i (6 + 2s7r),
where log?' may be taken to denote the real logarithm of the real positive quantity ?•,
and 6 the completely determinate angle defined as already mentioned.
Reverting to the function um, we have u = exp log u, and thence um = exp (m log u),
which, on account of the infinity of values of logu, has in general (as before remarked)
an infinity of values ; if u = e, then e>n, = exp (m log e), has in general in like manner
an infinity of values, but in regarding em as identical with the one-valued function
exp m, we take loge to be = its real value, 1.
The inverse functions cos"1 x, sin"1 x, tan"1 x, are in fact logarithmic functions ; thus
in the equation exp ix = cos x + i sin x, writing first cos x = u, the equation becomes
exp i cos"1 u = u + i Vl — w2, or we have cos~1u = - log(u + i Vl — it2), and from the same
equation, writing secondly sin x = u, we have sin"1 u = - log (\/l — it? + iu). But the
1
formula for tan"1 a is a more elegant one, as not involving the radical Vl — 1<? ; we have
exp ix — exp (— ix) exp 2ix — 1
i tan x — — ^—. — • — ; — ' —
exp ix + exp (— ix) ' exp 2ix + 1 '
and thence
1 + i tan x
exp 2ix = = r— — ,
1 — i tan x
that is,
1 1 + i tan x
2i 1 — i tan x '
or, if tan x = u, then
1 . 1 + m
tan"1 tt = x-. log = r- .
The logarithm (or inverse exponential function) and the inverse circular functions
present themselves as the integrals of algebraic functions
dx ,
— = logon,
whence also
dx 1 , I +ix
1 + x* 2i I — ix
and
dx
= sm~J x.
528 FUNCTION. [787
6. Each of the functions exp u, sin u, cos u, tan u, &c., as a one-valued function
of u, is in this respect analogous to a rational function of u ; and there are further
analogies of exp u, sin u, cos u, to a rational and integral function ; and of tan u, sec u, &c.,
to a rational non-integral function.
A rational and integral function has a certain number of roots, or zeros, each of
a given multiplicity, and is completely determined (except as to a constant factor)
when the several roots and the multiplicity of each of them is given ; i.e., if a, b, c, ...
( u\p ( u\Q
be the roots, p, q, r, ... their multiplicities, then the form is -4(1 ) ( 1 — r ) ••.;
\ dj \ o J
a rational (non-integral) function has a certain number of infinities, or poles, each of
them of a given multiplicity, viz. the infinities are the roots or zeros of the rational
and integral function which is its denominator.
The function exp u has no finite roots, but an infinity of roots each = — x ; this
/ u\n
appears from the equation exp u = 1 1 + - j , where n is indefinitely large and positive.
The function sin u has the roots sir where s is any positive or negative integer, zero
included ; or, what is the same thing, its roots are 0 and + STT, s now denoting any
positive integer from 1 to oo ; each of these is a simple root, and we in fact have
/ w2 \
sin u = ull ( 1 — 2 „ ) • Similarly the roots of cos u are (s + |) TT, s denoting any positive
or negative integer, zero included, or, what is the same thing, they are + (s + ^) TT,
s now denoting any positive integer from 0 to oo ; each root is simple, and we have
/ u" \
cos u = II 1 1 + -. — -rpr- — „). Obviously tanw, as the quotient sinu-rcosw, has both roots
\ (s+£)-7rV
and infinities, its roots being the roots of sin u, its infinities the roots of cos u ; sec u
as the reciprocal of COSM has infinities only, these being the roots of cos u, &c.
/ u2 \
In the foregoing expression sin u = ull ( 1 — -., - , the product must be understood
V srtr)
u2 \
to mean the limit of Hf ( 1 — — 1 for an indefinitely large positive integer value of n,
\ o~7T /
viz. the product is first to be formed for the values s=l, 2, 3,... up to a determinate
number n, and then n is to be taken indefinitely large. If, separating the positive
(77 \ / 7/ \
1 -f — ) IV1 ( 1 I . (where
S7T/ V S7T/
in the first product s has all the positive integer values from 1 to n, and in the
second product s has all the positive integer values from 1 to m), then by making
m and n each of them indefinitely large, the function does not approximate to sinw,
unless m : n be a ratio of equality*. And similarly as regards cosw, the product
no'/l f 1 + j— y. — ) n ( -, T\— ) , m and n indefinitely large, does not approximate to
cos u, unless m : n be a ratio of equality.
* The value of the function in question Mllj" ( 1 + — J II^1 ( 1- — j , when m, n are each indefinitely
u
large, but — not =1, is =( - }n sinw.
n \mj
787] FUNCTION. 529
7. The functions sin u, cos u, are periodic, having the period 2?r, " ' (u + 2?r) = Sm (u)-
cos cos
and the half-period IT, (u + 7r) = — u ; the periodicity may be verified by means
cos cos *
of the foregoing fractional forms, but some attention is required; thus writing, as we
may do, sin u = — ^ , where s extends from — n to n, n ultimately infinite, if
for u we write U + TT, each factor of the numerator is changed into the following one
and the numerator is unaltered, save only that there is an introduced factor u + (n+l)7r
at the superior limit, and an omitted factor U — HTT at the inferior limit ; the ratio of
these, (u + n + ITT) 4- (u — HTT), for n infinite is = — 1, and we thus have, as we should
have, sin (u + TT) = — sin u.
The most general periodic function having no infinities, and each root a simple
root, and having a given period a, has the form A sin f- B cos , or, what is the
a a
, . T sin /27m ^ \
same thing. L h A,
cos \ a /
8. We come now to the Elliptic Functions. These arose from the consideration of
fRdx
the integral I /-^ > where R is a rational function of x, and X is the general rational
and integral quartic function
otf4 + /3a? + 7#2 -I- &c + e ;
a form arrived at was
dx _ d<f)
f dx _ I
1 Vl-aM-ArV ' ~ J VI -
k* sin2 0 '
on putting therein x = sin $, and this last integral was represented by F<j>, and called
the elliptic integral of the first kind. In the particular case k = 0, the integral is
Ct IT
= sin"1 x, and it thus appears that F<f> is of the nature of an inverse function ;
l-x*
for passing to the direct functions we write F<f> = u, and consider <£ as hereby determ
ined as a function of u, <f> = amplitude of u, or for shortness am u. And the functions
sin <f>, cos </>, and vl — kz sin2 <£ were then considered as functions of the amplitude, and
written sin am u, cos am u, A am u ; these were afterwards written sn u, en u, dn u, which
may be regarded either as mere abbreviations of the former functional symbols, or (in
a different point of view) as functions, no longer of &mu, but of u itself as the
argument of the functions ; sn is thus a function in some respects analogous to a sine,
and en and dn functions analogous to a cosine ; they have the corresponding property
that the three functions of u-\-v are expressible in terms of the functions of u and of v.
The following formulae may be mentioned :
en2 u = l — sn2 u, dn2 u = 1 — k2 sn2 u,
sri' u = en u dn u, en' u = — sn u dn u, dn' u = — T<? sn u en u,
c. xi. 67
530 FUNCTION. [787
where the accent denotes differentiation in regard to u ; and the addition-formulae :
sn (u + v) = sn u en v dn v + sn v en u dn u, (-T-),
en (u + v) = en u en v — sn u dn u sn v dn v, (-T-),
dn (u + v) = dn u dn w — k2 sn M en w sn v en v, (-4-),
each of the expressions on the right-hand side being the numerator of a fraction of
which
Denom. = 1 — k2 sn2 u sn2 v.
It may be remarked that any one of the fractional expressions, differentiated in regard
to u and to v respectively, gives the same result; such expression is therefore a
function of u + v, and the addition-formula? can be thus directly verified.
9. The existence of a denominator in the addition-formulae suggests that sn, en, dn
are not, like the sine and cosine, functions having zeros only, without infinities; they
are in fact functions, having each its own zeros, but having a common set of infinities;
moreover, the zeros and the infinities are simple zeros and infinities respectively. And
this further suggests, what in fact is the case, that the three functions are quotients
having each its own numerator but a common denominator, say they are the quotients
of four ^-functions, each of them having zeros only (and these simple zeros) but no
infinities.
The functions sn, en, dn, but not the ^-functions, are moreover doubly periodic;
that is, there exist values 2o>, 2v, =4>K and 4>(K + iK') in the ordinary notation, such
that the sn, en, or dn of u + 2«o, and the sn, en, and dn of u+2v are equal to the
sn, en, and dn respectively of u; or say that <f>(u + 2a)) = $ (u+ 2t>) = <f>u, where <f> is
any one of the three functions.
As regards this double periodicity, it is to be observed that the equations
<£ (u + 2<u) = <f>u, $ (u + 2w) = (f>u, imply $ (u + 2ma> + 2nv) = <f>u, and hence it easily follows
that if ft>, v were commensurable, say if they were multiples of some quantity a, we
should have <j)(u + 2a) = <f>u, an equation which would replace the original two equations
<f>(u + 2&>) = <}>u, <f>(u + 2v) = (J>u, or there would in this case be only the single period
a ; &) and v must therefore be incommensurable. And this being so, they cannot have
a real ratio, for if they had, the integer values ra, n could be taken such as to make
2ma> + 2nv = k times a given real or imaginary value, k as small as we please ; the
ratio &) : v must be therefore imaginary, as is in fact the case when the values are
4>K and
10. The function sn u has the zero 0 and the zeros ma> + nv, m and n any
positive or negative integers whatever ; and this suggests that the numerator of sn u is
equal to a doubly infinite product, (Cayley, "On the Inverse Elliptic Functions," Camb.
Math. Jour. t. iv., 1845, [24] ; and " Mdmoire sur les fonctions doublement periodiques,"
Liouville, t. x., 1845, [25]). The numerator is equal to
uUU (l +
* \
mca + nv) '
787] FUNCTION. 531
m and n having any positive or negative integer values whatever, including zero, except
that m, n must not be simultaneously = 0, these values being taken account of in the
factor u outside the product. But until further defined, such a product has no definite
value, and consequently no meaning whatever. Imagine m, n to be coordinates, and
suppose that we have, surrounding the origin, a closed curve having the origin for its
centre, i.e. the curve is such that, if a, ft be the coordinates of a point thereof, then
— a, — /3 are also the coordinates of a point thereof; suppose further that the form
of the curve is given, but that its magnitude depends upon a parameter h, and that
the curve is such that, when h is indefinitely large, each point of the curve is at an
indefinitely large distance from the origin; for instance, the curve might be a circle
or ellipse, or a parallelogram, the origin being in each case the centre. Then if in
the double product, taking the value of h as given, we first give to m, n all the
positive or negative integer values (the simultaneous values 0, 0 excluded) which corre
spond to points within the curve, and then make h indefinitely large, the product will
thus have a definite value ; but this value will still be dependent on the form of the curve.
Moreover, varying in any manner the form of the curve, the ratio of the two values
of the double product will be = exp (3u2, where /3 is a determinate value depending only
on the forms of the two curves; or, what is the same thing, if we first give to the
curve a certain form, say we take it to be a circle, and then give it any other form,
the product in the latter case is equal to its former value multiplied by exp /3i*2,
where ft depends only upon the form of the curve in the latter case.
Considering the form of the bounding curve as given, and writing the double
product in the form
nn (u + m" + ™
V mco + nv
the simultaneous values m = 0, n = 0 being now admitted in the numerator, although
still excluded from the denominator, then if we write for instance u 4- 2<o instead of u,
each factor in the numerator is changed into a contiguous factor, and the numerator
remains unaltered, except that we introduce certain factors which lie outside the
bounding curve, and omit certain factors which lie inside the bounding curve ; we, in
fact, affect the result by a singly infinite series of factors belonging to points adjacent
to the bounding curve ; and it appears on investigation that we thus introduce a con
stant factor exp7(w + o>). The final result thus is that the product
1+ "
mco + nv
does not remain unaltered when u is changed into u + 2a>, but that it becomes there
fore affected with a constant factor, exp7(w + a>). And similarly the function does not
remain unaltered when u is changed into u + 2v, but it becomes affected with a factor,
exp B (u + v). The bounding curve may however be taken such that the function is
unaltered when u is changed into u + 2o> : this will be the case if the curve is a
rectangle such that the length in the direction of the axis of m is infinitely great in
comparison of that in the direction of the axis of n ; or it may be taken such that
the function is unaltered when u is changed into u + 2v : this will be so if the curve
67—2
532 FUNCTION. [787
be a rectangle such that the length in the direction of the axis of n is indefinitely
great in comparison with that in the direction of the axis of m ; but the two con
ditions cannot be satisfied simultaneously.
11. We have three other like functions, viz. writing for shortness m, n to denote
m + %, n + ^ respectively, and (m, n) to denote mw + nv, then the four functions are
,
(m, n)
the bounding curve being in each case the same ; and, dividing the first three of these
each by the last, we have (except as to constant factors) the three functions sn u, en u, dn u ;
writing in each of the four functions u + 2&> or u+2v in place of u, the functions
acquire each of them the same exponential factor exp 7 (u + «), or exp 8 (u + v), and
the quotient of any two of them, and therefore each of the functions sn u, en u, dn u,
remains unaltered.
It is easily seen that, disregarding constant factors, the four ^-functions are in
fact one and the same function with different arguments, or they may be written
6u, #(w + £&>), 0(u + ^v), 6(u + ^a) + %v); by what precedes, the functions may be so
determined that they shall remain unaltered when u is changed into u + 2(o, that is,
be singly periodic, but that the change u into u + 2u shall affect them each with the
same exponential factor exp 8 (u + v).
12. Taking the last-mentioned property as a definition of the function 6, it
appears that 6u may be expressed as a sum of exponentials
TTt
0u = A^ exp — (vm? + urn},
60
where the summation extends to all positive and negative integer values of m,
including zero. In fact, if we first write herein u+2a) instead of u, then in each
term the index of the exponential is altered by — 2&>ra, = 2w7n', and the term itself
to
thus remains unaltered ; that is, 6 (u + 2<o) = 6u. But writing u + 2v in place of u, each
term is changed into the succeeding term, multiplied by the factor exp — (u + v); in fact,
making the change in question u into u + 2u, and writing also m — 1 in place of m,
vm? + um becomes v (m - I)2 + (u + 2v) (m - 1), = vm2 +um-u-v, and we thus have
6 (u + 2u) = exp •< -- (u + v)r.0tt. In order to the convergency of the series it is
TT? \)rfY) ^
necessary that exp - should vanish for indefinitely large values of m, and this will
lu
be so if — be a complex quantity of the form a + fti, a. negative; for instance, this will
be the case if o> be real and positive and v be =i multiplied by a real and positive
quantity. The original definition of 6 as a double product seems to put more clearly
in evidence the real nature of this function, but the new definition has the advantage
that it admits of extension to the ^-functions of two or more variables.
787] FUNCTION. 533
The elliptic functions sn u, en a, dn u, have thus been expressed each of them as
the quotient of two ^-functions, but the question arises to express conversely a ^-function
by means of the elliptic functions ; the form is found to be
/ r r \
6u = C exp Aii? + B I sn2 u du- ,
\ J QjQ 1
viz. du is expressible as an exponential, the index of which depends on the double integral
sn2 u du2.
The object has been to explain the general nature of the elliptic functions smt, cnw, dnu,
and of the ^-functions with which they are thus intimately connected ; it would be out
of place to go into the theories of the multiplication, division, and transformation of
the elliptic functions, or into the theory of the elliptic integrals, and of the applic
ation of the ^-functions to the representation of the elliptic integrals of the second
and third kinds.
13. The reasoning which shows that for a doubly periodic function the ratio of
the two periods 2eo, 2u is imaginary shows that we cannot have a function of a single
variable, which shall be triply periodic, or of any higher order of periodicity. For if
the periods of a triply periodic function <f> (u) were 2&>, 2u, 2^;, then m, n, p being any
positive or negative integer values, we should have <£ (u + 2rao> + 2?it> + 2p^) = $u ; o>, v, %
must be incommensurable, for if not, the three periods would really reduce themselves
to two periods, or to a single period ; and being incommensurable, it would be possible
to determine the integers m, n, p in such manner that the real part and also the
coefficient of i of the expression mco + nv + p% shall be each of them as small as
we please, say <£ (u + e) = (f>u, and thence <£ (u + ke) = <f>u (k an integer), and ke as
near as we please to any given real or imaginary value whatever. We have thus the
nugatory result <f)U = a constant, or at least the function if not a constant is a function
of an infinitely and perpetually discontinuous kind, a conception of which can hardly
be formed. But a function of two variables may be triply or quadruply periodic —
viz. we may have a function <f>(u, v) having for u, v the simultaneous periods 2o>, 2&>';
2u, 2 1/ ; 2;j£, 2^' ; 2i/r, 2-v/r' ; or, what is the same thing, it may be such that, m, n, p, q
being any integers whatever, we have
</> (u + 2ma) + 2nv + 2p% + 2q^r, v + Zmw + 2nv + 2p%' + 2g^') = <£ (u, v) ;
and similarly a function of 2?i variables may be 2n-tuply periodic.
It is, in fact, in this manner that we pass from the elliptic functions and the
single ^-functions to the hyperelliptic or Abelian functions and the multiple ^-functions ;
the case next succeeding the elliptic functions is when we have X, T the same rational
and integral sextic functions of x, y respectively, and then writing
534 FUNCTION. [787
we regard certain symmetrical functions of x, y, in fact, the ratios of (24 =) 16 such
symmetrical functions, as functions of (u, v) ; say we thus have 15 hyperelliptic
functions f (u, v), analogous to the 3 elliptic functions sn u, en u, dn u, and being
quadruply periodic. And these are the quotients of 16 double ^-functions 0 (u, v), the
general form being
0 (u, v) = A%2< exp {£ (a, h, b) (in, n)2 + mu + nv],
where the summations extend to all positive and negative integer values of (m, n) ;
and we thus see the form of the ^-function for any number of variables whatever.
The epithet " hyperelliptic " is used in the case where the differentials are of the form
doc
just mentioned —T=- , where X is a rational and integral function of x ; the epithet
V X
" Abelian " extends to the more general case where the differential involves the
irrational function of x, determined by any rational and integral equation <j> (x, y) = 0
whatever.
As regards the literature of the subject, it may be noticed that the various
memoirs by Riemann, 1851 — 1866, are republished in the collected edition of his works,
Leipsic, 1876 ; and shortly after his death we have the Theorie der Abel'schen Functionen,
by Clebsch and Gordan, Leipsic, 1866. Preceding this, we have by MM. Briot and
Bouquet, the Theorie des Fonctions doubletnent periodiques et en particulier des Fonctions
Elliptiques, Paris, 1859, the results of which are reproduced and developed in their larger
work, Theorie des Fonctions Elliptiques, 2nd ed., Paris, 1875.
14. It is proper to mention the gamma (F) or II function, F(n + l)= II?i, =1.2.3...?i,
when n is a positive integer. In the case just referred to, n a positive integer, this
presents itself almost everywhere in analysis, — for instance, the binomial coefficients,
and the coefficients of the exponential series are expressible by means of such functions
of a number n. The definition for any real positive value of n is taken to be
F?i = / xn~l e~x dx •
f*
= I xn~l e~x
Jo
it is then shown that, n being real and positive, T(n + 1) = nTn, and by assuming that
this equation holds good for positive or negative real values of n, the definition is
extended to real negative values ; the equation gives Fl = OFO, that is, FO = oo , and
similarly F (— n) = oo , where — n is any negative integer. The definition by the definite
integral has been or may be extended to imaginary values of n, but the theory is not
an established one. A definition extending to all values of n is that of Gauss
1 2 3 ... k
Tin = limit — — =— —f kn,
the ultimate value of k being = oo ; but the function is chiefly considered for real values
of the variable.
A formula for the calculation, when x has a large real and positive value, is
787] FUNCTION. 535
or as this may also be written, neglecting the negative powers of as,
Hx = \/2?r exp {(so + £) log x — x}.
7T
Another formula is TxT (1 — x) = -. : or, as this may also be written.
sin TTX
sin TTX
It is to be observed that the function II serves to express the product of a set
of factors in arithmetical progression ; we have
We can consequently express by means of it the product of any number of the factors
which present themselves in the factorial expression of sin u. Starting from the form
where II is here as before the sign of a product of factors corresponding to the
different integer values of s, this is thus converted into
/u \ f u \ fu\ ( u\
uTI ( — h wi I II ( h M ) -j- II ( — ] n ( Ilm Tin,
VTT / \ TT / VTT/ V 9T/
or as this may also be written,
*up + m]u(-^ + n} + iip-i)ii(-«} nmnn,
\7T J \ IT J \7T J \ TTj
which, in virtue of
smw
becomes
= sin wll (- + m } U ( h A] -*- Timlin.
VTT ) \ TT J
Here m and n are large and positive ; calculating the second factor by means of the
formula for Hx, in this case we have the before-mentioned formula
STTJ \ sir/ \m
The gamma or II function is the so-called second Eulerian integral ; the first Eulerian
integral
I xf~l (1 - x)i-1 dx, = TpTq -r T (p + q),
Jo
is at once expressible in terms of F, and is therefore not a new function to be con
sidered.
536 FUNCTION. [787
15. We have the function defined by its expression as a hypergeometric series
.
1.7 i.z. 7.7 + 1
i.e., this expression of the function serves as a definition, if the series be finite or
if, being infinite, it is convergent. The function may also be defined as a definite
integral; in other words, if, in the integral
I a-'-1 (1 - xY~l (1 - ux)~y' dx,
Jo
we expand the factor (1 — ux)~v' in powers of ux, and then integrate each term
separately by the formula for the second Eulerian integral, the result is
~P-/ "P/O' "P //«/' I 1\ "P/OX '
la. ip i (a + ij . i p 7
which is
_ Fa'.Fp" f a'. 7 a', a' + l. 7'. 7' + !
"H / ~' i O'\ 1 ~"~ „/ i Ql t ^ ' „/ i £)' .,/ _i /p' , 1 1 n
. v* ~r /J -f i . J- . ^i
or writing of, ft', 7' = a, 7 — a, ft respectively, this is
r«r(7 -<*)„. _
-^—JF(a, ft, 7, ti),
so that the new definition is
F(a, ft, 7, u) = TaT (? ~ ") P x«-> (1 - tsy-1 (1 - MB)-* dx •
1 7 Jo
but this is in like manner only a definition under the proper limitations as to the
values of a, ft, 7, u. It is not here considered how the definition is to be extended
so as to give a meaning to the function ^(a, ft, 7, u) for all values, say of the
parameters a, ft, 7, and of the variable u. There are included a large number of
special forms which are either algebraic or circular or exponential, for instance
F (a, ft, ft, u) = (1 — u)~a, &c. ; or which are special transcendents which have been
separately studied, for instance, Bessel's functions, the Legendrian functions Xn presently
referred to, series occurring in the development of the reciprocal of the distance
between two planets, &c.
16. There is a class of functions depending upon a variable or variables x, y, ...
and a parameter n, say the function for the parameter n is Xn such that the product
of two functions having the same variables, multiplied it may be by a given function
of the variables, and integrated between given limits, gives a result = 0 or not = 0,
f f
according as the parameters are unequal or equal ; I UXmXn dx = 0, but I UXn-dx not
= 0 ; the admissible values of the parameters being either any integer values, or
it may be the roots of a determinate algebraical or transcendental equation; and the
functions Xn may be either algebraical or transcendental. For instance, such a function
is cos nx ; m and n being integers, we have I cos mx . cos nx dx = 0, but I cos2 nx dx = £?r.
Jo Jo
787 J FUNCTION. 537
Assuming the existence of the expansion of a function fx, in a series of multiple
cosines, we thus obtain at once the well-known Fourier series, wherein the coefficient
[n
of cos mx is = £TT cos mx .fxdx. The question whether the process is applicable is-
.'o
elaborately discussed in Riemann's memoir (1854), Ueber die Darstellbarkeit einer
Function durch eine trigonometrische Reihe, No. XII. in the collected works. And again
we have the Legendrian functions, which present themselves as the coefficients of the
successive powers of a in the development of (1 — 2a#+a2)~i, X0=I, Xl=x, JT2=f(«2— ^), &c.:
P f1 2
here m, n being any positive integers, I Xm Xn dx = 0, but Xn2dx = = — — . And
J — i J — i An + I
we have also Laplace's functions, &c.
Functions in General.
17. In what precedes, a review has been given, not by any means an exhaustive
one, but embracing the most important kinds of known functions ; but there are
questions to be considered in regard to functions in general.
A function of x + iy has been built up by means of analytical operations performed
upon x + iy, (x + iy)2 = x2 — y2 + i . Zxy, &c., and the question next referred to has not
arisen. But observe that, knowing x + iy, we know x and y, and therefore any two
given functions <f> (x, y), *fy {x, y} of x and y : we therefore also know (f> (x, y) + i-\Jr (x, y),
and the question is, whether such a function of x, y (being known when x + iy is
known) is to be regarded as a function of x + iy ; and if not, what is the condition
to be satisfied in order that </> (x, y) + ity (x, y) may be a function of x + iy. Cauchy
at one time considered that the general form was to be regarded as a function of
x + iy, and he introduced the expression " fonction monogene," monogenous function, to
denote the more restricted form which is the proper function of x + iy.
Consider for a moment the above general form, say x' + iy' = </> (x, y) + ity (x, y),
where <j>, ty are any real functions of the real variables (x, y) ; or what is the same
thing, assume x = <£ (x, y), y' = -vjr (x, y} ; if these functions have each or either of them
more than one value, we attend only to one value of each of them. We may then
as before take x, y to be the coordinates of a point P in a plane II, and x, y' to
be the coordinates of a point P' in a plane II'. If, for any given values of x, y, the
increments of <£ (x, y), -fy (x, y) corresponding to the indefinitely small real increments
h, k of x, y be Ah + Bk, Ch + Dk, A, B, C, D being functions of x, y, then if the new
coordinates of P are x + h, y + k, the new coordinates of P' will be x' + Ah + Bk,
y' + Cfi + Dk ; or P, P' will respectively describe the indefinitely small straight paths
k Ch + Dk
at the inclinations tan"1 j- , tan"1 -j-? pT to the axes of x, x' respectively ; calling
ll A. fl -\- Jj/C
these angles 0, 6', we have therefore tan 6' = -j p- Q . Now in order that x + iy'
A. -|- -D tan u
may be = <f>(x + iy), a function of x + iy, the condition to be satisfied is that the
increment of x' + iy' shall be proportional to the increment h + ik of x + iy, or say
that it shall be = (A, + ip) (h + ik), \, ^ being functions of x, y, but independent of
C. XI. 68
538 FUNCTION. [787
A, k; we must therefore have Ah + Bk, Gh + Dk = \h — pk, ph + \k respectively, that is
A, B, C, D = X, — n, fjL, X respectively, and the equation for tan & thus becomes
tan 6' = -- 0 ; hence writing ^ = tan a, where a is a function of x, y, but inde-
X — IJL tan a A.
, f , , /,, tan a + tan 6
pendent of h, k, we have tan 0 = - - — -a , that is, 9 = a + 0 ; or for the given
1 — tan a tan 6
points (x, y), (x' , y'}, the path of P being at any inclination whatever 6 to the axis
of x, the path of P' is at the inclination 0 + constant angle a to the axis of x ;
also (\h-nk)* + (nh + \k)z = (\* + ij?)(hz + k*), i.e., the lengths of the paths are in a
constant ratio.
The condition may be written 8 (x + iy'} = (X + ip) (So; + iSy) ; or what is the same
thing
dx .
that is,
dx .dy' ,^ . . dx
-
consequently
dx' . dy' . ,'dx . dy'\
_ _1_ 2, '' — i I _ _ I i _«? l •
dy dy \da; dx)'
that is, •
dx' _ dy' dy' dx'
dy dx' dy dx'
as the analytical conditions in order that x +iy' may be a function of x + iy. They
obviously imply
ffirf ffiaf dty dy_
~d& dif ' da? "*" dy*
and if x' be a function of x, y, satisfying the first of these conditions, then
is a complete differential, and
dx , dx -,
— j— dx + v dy
dy dec '
C/ dv' rlv' \
i If Ubds i \4nJU -, \
y = If j-das + -.- dy}.
A dy dx v)
18. We have, in what just precedes, the ordinary behaviour of a function
in the neighbourhood of the value x + iy of the argument or point x + iy\ or say
the behaviour in regard to a point x + iy such that the function is in the neighbour
hood of this point a continuous function of x + iy (or that the point is not a point
of discontinuity) : the correlative definition of continuity will be that the function
<f> (x + iy), assumed to have at the given point x + iy a single finite value, is continuous
in the neighbourhood of this point, when the point x + iy describing continuously a
straight infinitesimal element h+ik, the point <f>(x + iy) describes continuously a straight
infinitesimal element (X + ip) (h + ik) ; or what is really the same thing, when the
function (x + iy) has at the point x + iy a differential coefficient.
787] FUNCTION.
19. It would doubtless be possible to give for the continuity of a function
$ (x + iy} a less stringent definition not implying the existence of a differential
coefficient; but we have this theory only in regard to the functions <f>x of a real
variable in memoirs by Riemann, Hankel, du Bois Reymond, Schwarz, Gilbert, Klein, arid
Darboux. The last- mentioned geometer, in his " Memoire sur les fonctions discontinues,"
Jour, de I'ficole Nurmale, t. iv. (1875), pp. 57 — 112, gives (after Bonnet) the following
definition of a continuous function (observe that we are now dealing with real
quantities only) : — the function f(x) is continuous for the value x = x0 when, h and e
being positive quantities as small as we please and 6 any positive quantity at pleasure
between 0 and 1, we have, for all the values of 9, f (x0 ± Oh) —f (x) less in absolute
magnitude than e; and moreover f(x} is continuous through the interval x0, xl (xl>x0r
that is, nearer + x ) when f (x) is continuous for every value of x between x0 and xl ,
and, h tending to zero through positive values, f(x0 + h) and f(x0 — h) tend to the limits
f(x0), /(#i) respectively. It is possible, consistently with this definition, to form con
tinuous functions not having in any proper sense a differential coefficient, and having
other anomalous properties; thus if al} a.,, a3, ... be an infinite series of real positive
or negative quantities, such that the series Saw is absolutely convergent (i.e. the sum
S + cin> each term being made positive, is convergent), then the function San (sin mrx)3
is a continuous function actually calculable for any assumed value of x ; but it is
shown in the memoir that, taking x = any commensurable value - whatever, and then
f)
writing x = - + h, h indefinitely small, the increment of the function is of the form
(k + e) h3, k finite, e an indefinitely small quantity vanishing with h ; there is thus no
term varying with h, nor consequently any differential coefficient. See also Riemann's
Memoir, Ueber die Darstellburkeit, &c. (No. xn. in the collected works), already referred to.
20. It was necessary to allude to the foregoing theory of (as they may be termed)
infinitely discontinuous functions ; but the ordinary and most important functions of
analysis are those which are continuous, except for a finite number (or it may be an
infinite number) of points of discontinuity. It is to be observed that a point at
which the function becomes infinite is ipso facto a point of discontinuity; a value of
the variable for which the function becomes infinite is, as already mentioned, said to
be an " infinity " (or a " pole ") of the function ; thus, in the case of a rational
function expressed as a fraction in its least terms, if the denominator contains a factor
(x — a)m, a a real or imaginary value, in a positive integer, then a is said to be an
infinity of the with order (and in the particular case m = 1 , it is said to be a simple
infinity). The circular functions tana;, sec*1 are instances of a function having an
infinite number of simple infinities.
A rational function is a one-valued function, and in regard to a rational function
the infinities are the only points of discontinuity ; but a one-valued function may have
points of discontinuity of a character quite distinct from an infinity : for instance, in the
exponential function exp f— — j where a is real or imaginary, the value u (= x + iy) = a,
is a point of discontinuity but not an infinity ; taking u = a + peai, where p is an
68—2
540 FUNCTION. [787
indefinitely small real positive quantity, the value of the function is exp (-e~ai
= exp - (cos a — i sin a), which is indefinitely large or indefinitely small according as cos a
is positive or negative, and in the separating case cos a = 0, and therefore sin a = + 1,
it is = cos - + i sin - which is indeterminate. If, instead of exp , we consider a
p p u— a
linear function
Ll +5 exp [ -=- j(7 + Dexp
u — a] * u — a
then writing as before u = a + peai, the value is = A -f- C, or = B -=- D, according as cos a
is negative or positive. As regards the theory of one-valued functions in general, the
memoir by Weierstrass, " Zur Theorie der eindeutigen analytischen Functionen," Berl.
Abh. 1876, pp. 11 — 60, may be referred to.
21. A one-valued function ex m termini cannot have a point of discontinuity of
the kind next referred to ; if the representative point P, moving in any manner
whatever, returns to its original position, the corresponding point P' cannot but return
to its original position. But consider a many-valued function, say an rc-valued function
oc' + iy', of x + iy ; to each position of P there correspond n positions, in general all
of them different, of P'. But the point P may be such that (to take the most simple
case) two of the corresponding points P' coincide with each other, say such a position
of P is at V, then (using for greater distinctness a different letter W instead of
V) corresponding thereto we have two coincident points (W), and n— 2 other points
W ; V is then a branch-point (Verzweigungspunkt). Taking for P any point which
is not a branch-point, then in the neighbourhood of this value each of the n functions
x' + iy' is a continuous function of x + iy, and by what precedes, if P describing an
infinitely small closed curve (or oval) return to its original position, then each of the
corresponding points P' describing a corresponding indefinitely small oval will return
to its original position. But imagine the oval described by P to be gradually enlarged,
so that it comes to pass through a branch-point V; the ovals described by two of
the corresponding points P' will gradually approach each other, and will come to unite
at the point (W), each oval then sharpening itself out so that the two form together
a figure of eight. And if we imagine the oval described by P to be still further
enlarged so as to include within it the point V, then the figure of eight, losing
the crossing, will be at first an hour-glass form, or twice-indented oval, and ultimately
in form an ordinary oval, but having the character of a twofold oval ; i.e. to the
oval described by P (and which surrounds the branch-point V] there will correspond
this twofold oval, and n — 2 onefold ovals, in such wise that to a given position of
P on its oval there correspond two points, say P/, P/, on the twofold oval, and
n — 2 points P3', . . . , Pn', each on its own onefold oval. And then as P describing its
oval returns to its original position, the point P/ describing a portion only of the
twofold oval, will pass to the original position of P.>', while the point P2' describing
the remaining portion of the twofold oval will pass to the original position of P/;
the other points P3', . . . , Pn', describing each of them its own onefold oval, will
787] FUNCTION. 541
return each of them to its original position. And it is easy to understand how,
when the oval described by P surrounds two or more of the branch-points V, the
corresponding curves for P' may be a system of manifold ovals, such that the sum of
the manifoldness is always = n, and to conceive in a general way the behaviour of
the corresponding points P and P'.
Writing for a moment x + iy = u, x' + iy' = v, the branch-points are the points of
contact of parallel tangents to the curve <f> (u, v) = Q, a line through a cusp (but
not a line through a node), being reckoned as a tangent ; that is, if this be a curve
of the order n and class m, with 8 nodes and K cusps, the number of branch-points
is =m + K, that is, it is =n*-n-28 — 2ic, or if p, = $ (n - 1) (n - 2) - 8 - K, be the
deficiency, then the number is = 2n — 2 + 2p.
To illustrate the theory of the ?i-valued algebraical function x' + iy of the complex
variable x + iy, Riemann introduces the notion of a surface composed of n coincident
planes or sheets, such that the transition from one sheet to another is made at the
branch-points, and that the n sheets form together a multiply-connected surface, which
can be by cross-cuts (Querschnitte) converted into a simply-connected surface ; the
n- valued function x' + iy' becomes thus a one-valued function of x + iy, considered as
belonging to a point on some determinate sheet of the surface : and upon such con
siderations he founds the whole theory of the functions which arise from the integration
of the differential expressions depending on the w-valued algebraical function (that is,
any irrational algebraical function whatever) of the independent variable, establishing
as part of the investigation the theory of the multiple ^-functions. But it would be
difficult to give a further account of these investigations.
The Calculus of Functions.
22. The so-called Calculus of Functions, as considered chiefly by Herschel and
Babbage and De Morgan, is not so much a theory of functions as a theory of the
solution of functional equations ; or, as perhaps should rather be said, the solution of
functional equations by means of known functions, or symbols, — the epithet known
being here used in reference to the actual state of analysis. Thus for a functional
equation <f>x + <f>y = </> (xy), taking the logarithm as a known function, the solution is
fa = c log as ; or if the logarithm is not taken to be a known function, then a solution
fdx
may be obtained by means of the sign of integration fa = c \ — ; but the establish-
J x
ment of the properties of the function logarithm (assumed to be previously unknown)
would not be considered as coming within the theory. A class of equations specially
considered is where aac, fix, ... being given functions of x, the unknown function <f> is
to be determined by means of a given relation between a;, fa, fyax, $>ftx, ...; in part
icular the given relation may be between x, fa, <j>ax; this can be at once reduced
to equations of finite differences; for writing x = un, aas = un+1, we have un+1 = aun,
giving un, and therefore also x, each of them as a function of n; and then writing
fa = vn, </><&» will be the same function of n+l, =vn+1, and the given relation is
again an equation of finite differences in vn+1, vn, and n ; we have thus vn, = fa,
542 FUNCTION. [787
as a function of n, that is, of x. As regards the equation un+l = a.un, considered in
itself apart from what precedes, observe that this is satisfied by writing un = an (x),
or the question of solving this equation of finite differences is, in fact, identical with
that of finding the nth function a." (a;), where a (x) is a given function of x. It of
course depends on the form of a.(x) whether this question admits of solution in any
proper sense; thus, for a function such as log x, the nth logarithm is expressible in
its original function log" x, ( = log log ... x), and not in any other form. But there
(1 I /)'/*
are forms, for instance ax — — — =- , where the nth function a.nx is a function of the
c + ax
like form a'l#= ~ — ^ , in which the actual value can be expressed as a function
U + Ux
of n ; if a be such a form, then <f>cuf>~\ whatever </> may be, is a like form, for we
obviously have (^a^r1)'1 = <f>a.n<f>~1. The determination of the nth function is, in fact, a
leading question in the calculus of functions.
It is to be observed that considering the case of two variables, if for instance
a. (x, y) denote a given function of x, y, the notation a2 (x, y) is altogether meaningless ;
in order to generalize the question, we require an extended notation wherein a single
functional symbol is used to denote two t functions of the two variables. Thus
$ (x, y} = a (x, y}, /3 (x, y), a and /3 given functions ; writing for shortness x^ = a (x, y),
2/j = /8 (x, y), then <f>2 (x, y) will denote <f> (sclt yj, that is, two functions a. (xlt y±), 0 (xlt y^,
say these are xz, y2: <f>3(x, y) will denote <£(#2> #2), and so on, so that <f>n (x, y) will
have a determinate meaning. And the like is obviously the case in regard to any
number of variables, the single functional symbol denoting in each case a set of
functions equal in number to the variables.
788] 543
788.
GALOIS.
[From the Encyclopaedia Britannica, Ninth Edition, vol. x. (1879), p. 48.]
GALOIS, EVARISTE (1811 — 1832), an eminently original and profound French mathe
matician, born 26th October 1811, killed in a duel May 1832. A necrological notice
by his friend M. Auguste Chevalier appeared in the Revue Encyclopedique, September
1832, p. 744; and his collected works are published, Liouville, t. XL (1846), pp. 381 —
444, about fifty of these pages being occupied by researches on the resolubility of
algebraic equations by radicals. But these researches, crowning as it were the previous
labours of Lagrange, Gauss, and Abel, have in a signal manner advanced the theory,
and it is not too much to say that they are the foundation of all that has since
been done, or is doing, in the subject. The fundamental notion consists in the
establishment of a group of permutations of the roots of an equation, such that every
function of the roots invariable by the substitutions of the group is rationally known,
and reciprocally that every rationally determinable function of the roots is invariable
by the substitutions of the group ; some further explanation of the theorem, and in
connexion with it an explanation of the notion of an adjoint radical, is given under
Equation, No. 32, [786]. As part of the theory (but the investigation has a very high
independent value as regards the Theory of Numbers, to which it properly belongs),
Galois introduces the notion of the imaginary roots of an irreducible congruence of a
degree superior to unity; i.e., such a congruence, F(x) = Q (mod. a prime number p),
has no integer root ; but what is done is to introduce a quantity i subjected to the
condition of verifying the congruence in question, F(i)=l (mod. p), which quantity
i is an imaginary of an entirely new kind, occupying in the theory of numbers a
position analogous to that of */ — 1 in algebra.
544 [789
789.
GAUSS.
[From the Encyclopaedia Britannica, Ninth Edition, vol. x. (1879), p. 116.]
GAUSS, CARL FRIEDRICH (1777 — 1855), an eminent German mathematician, was
born of humble parents at Brunswick, April 23, 1777, and was indebted for a liberal
education to the notice which his talents procured him from the reigning duke.
His name became widely known by the publication, in his twenty-fifth year (1801),
of the Disquisitiones Arithmetics. In 1807 he was appointed director of the Gottingen
observatory, an office which he retained to his death : it is said that he never slept
away from under the roof of his observatory, except on one occasion, when he accepted
an invitation from Humboldt to attend a meeting of natural philosophers at Berlin.
In 1809 he published at Hamburg his Theoria Motus Corporum Ccelestium, a work
which gave a powerful impulse to the true methods of astronomical observation ; and
his astronomical workings, observations, calculations of orbits of planets and comets, &c.,
are very numerous and valuable. He continued his labours in the theory of numbers
and other analytical subjects, and communicated a long series of memoirs to the
Royal Society of Sciences at Gottingen. His first memoir on the theory of magnetism,
Intensitas vis magneticce terrestris ad mensuram absolutam revocata, was published in 1833,
and he shortly afterwards proceeded, in conjunction with Professor Wilhelm Weber,
to invent new apparatus for observing the earth's magnetism and its changes ; the
instruments devised by them were the declination instrument and the bifilar mag
netometer. With Weber's assistance he erected in 1833 at Gottingen a magnetic
observatory free from iron (as Humboldt and Arago had previously done on a smaller
scale), where he made magnetic observations, and from this same observatory he sent
telegraphic signals to the neighbouring town, thus showing the practicability of an
electromagnetic telegraph. He further instituted an association (Magnetische Yerein),
composed at first almost entirely of Germans, whose continuous observations on fixed
term-days extended from Holland to Sicily. The volumes of their publication, Resultate
aus den Beobachtungen des magnetischen Vereins, extend from 1836 to 1839 ; and in
those for 1838 and 1839 are contained the two important memoirs by Gauss, A II-
gemeine Theorie des Erdmagnetismus, and the Allgemeine Lehrsatze — on the theory of
forces attracting according to the inverse square of the distance. The instruments
and methods thus due to him are substantially those employed in the magnetic
OF THE
UNIVERSITY
789] GAUSS. 545
observatories throughout the world. He co-operated in the Danish and Hanoverian
measurements of an arc and trigonometrical operations (1821 — 48), and wrote (1843,
1846) the two memoirs Ueber Gegenstdnde der hohern Geoddsie. Connected with
observations in general we have (1812 — 26) the memoir Theoria combinations observ-
ationum erroribus minimis obnoxia, with a second part and a supplement. Another
memoir of applied mathematics is the Dioptrische Untersuchungen, 1840. Gauss was
well versed in general literature and the chief languages of modern Europe, and was
a member of nearly all the leading scientific societies in Europe. He died at Gottingen
early in the spring of 1855. The centenary of his birth was celebrated (1877) at his
native place, Brunswick.
Gauss's collected works have been recently published by the Royal Society of
Gottingen, in 7 vols. 4to, Gott., 1863—71, edited by E. J. Sobering— (1) the Dis
quisitiones Arithmetics, (2) Theory of Numbers, (3) Analysis, (4) Geometry and Method
of Least Squares, (5) Mathematical Physics, (6) Astronomy, and (7) the Theoria Motus
Corporum Ccelestium. They include, besides his various works and memoirs, notices by
him of many of these, and of works of other authors in the Gottingisclie gelehrte
Anzeigen, and a considerable amount of previously unpublished matter, Nachlass. Of
the memoirs in pure mathematics, comprised for the most part in vols. II., ill., and
IV. (but to these must be added those on Attractions in vol. v.), it may be safely
said there is not one which has not signally contributed to the progress of the branch
of mathematics to which it belongs, or which would not require to be carefully
analysed in a history of the subject. Running through these volumes in order, we have
in the second the memoir, Summatio quarundam serierum singularium, the memoirs on
the theory of biquadratic residues, in which the notion of complex numbers of the
form a + bi was first introduced into the theory of numbers ; and included in the
Nachlass are some valuable tables. That for the conversion of a fraction into decimals
(giving the complete period for all the prime numbers up to 997) is a specimen of
the extraordinary love which Gauss had for long arithmetical calculations; and the
amount of work gone through in the construction of the table of the number of the
classes of binary quadratic forms must also have been tremendous. In vol. in. we have
memoirs relating to the proof of the theorem that every numerical equation has a
real or imaginary root, the memoirs on the Hyper geometric Series, that on Interpolation,
and the memoir Determinatio Attractionis — in which a planetary mass is considered
as distributed over its orbit according to the time in which each portion of the orbit
is described, and the question (having an implied reference to the theory of secular
perturbations) is to find the attraction of such a ring. In the solution the value of
an elliptic function is found by means of the arithmetico-geometrical mean. The
Nachlass contains further researches on this subject, and also researches (unfortunately
very fragmentary) on the lemniscate-function, &c., showing that Gauss was, even before
1800, in possession of many of the discoveries which have made the names of Abel
and Jacobi illustrious. In vol. iv. we have the memoir Allgemeine Auftosung..., on the
graphical representation of one surface upon another, and the Disquisitiones generates
circa superficies curvas. And in vol. V. we have a memoir On the Attraction of
Homogeneous Ellipsoids, and the already mentioned memoir Allgemeine Lehrsdtze . . . , on
the theory of forces attracting according to the inverse square of the distance.
c. xi. 69
546 [790
790.
GEOMETRY (ANALYTICAL).
[From the Encyclopedia Britannica, Ninth Edition, vol. x. (1879), pp. 408 — 420.]
THIS will be here treated as a method. The science is Geometry ; and it would
be possible, analytically, or by the method of coordinates, to develope the truths of
geometry in a systematic course. But it is proposed not in any way to attempt this,
but simply to explain the method, giving such examples, interesting (it may be) in
themselves, as are suitable for showing how the method is employed in the demon
stration and solution of theorems or problems.
Geometry is one-, two-, or three-dimensional, or, what is the same thing, it is lineal,
plane, or solid, according as the space dealt with is the line, the plane, or ordinary
(three-dimensional) space. No more general view of the subject need here be taken : —
but in a certain sense one-dimensional geometry does not exist, inasmuch as the
geometrical constructions for points in a line can only be performed by travelling out
of the line into other parts of a plane which contains it, and conformably to the usual
practice Analytical Geometry will be treated under the two divisions, Plane and Solid.
It is proposed to consider Cartesian coordinates almost exclusively ; for the proper
development of the science homogeneous coordinates (three and four in plane and solid
geometry respectively) are required ; and it is moreover necessary to have the correlative
line- and plane-coordinates ; and in solid geometry to have the six coordinates of the
line. The most comprehensive English works are those of Dr Salmon, Conic Sections
(5th edition, 1869), Higher Plane Curves (2nd edition, 1873), and Geometry of Three
Dimensions (3rd edition, 1874) ; we have also, on plane geometry, Clebsch's Vorlesungen
ilber Geometrie, posthumous, edited by Dr F. Lindemann, Leipsic, 1875, not yet complete.
I. PLANE ANALYTICAL GEOMETRY (§§ 1 — 25).
1. It is assumed that the points, lines, and figures considered exist in one and
the same plane, which plane, therefore, need not be in any way referred to. The
position of a point is determined by means of its (Cartesian) coordinates ; i.e. as
790]
GEOMETRY.
547
explained under the article Curve, [785], we take the two lines x'Ox and y'Oy, called the
axes of x and y respectively, intersecting in a point 0 called the origin, and determine
the position of any other point P by means of its coordinates x= OM (or NP}, and
X'
Fig. i.
y
N
M
y=MP (or ON). The two axes are usually (as in fig. 1) at right angles to each
other, and the lines PM, PN are then at right angles to the axes of x and y
respectively. Assuming a scale at pleasure, the coordinates x, y of a point have
numerical values.
It is necessary to attend to the signs : x has opposite signs according as the
point is on one side or the other of the axis of y, and similarly y has opposite signs
according as the point is on the one side or the other of the axis of x. Using the
letters N., E., S., W. as in a map, and considering the plane as divided into four
quadrants by the axes, the signs are usually taken to be —
x y for quadt.
+ + N. E.
+ S. E.
+ N. W.
S. W.
A point is said to have the coordinates (a, b), and is referred to as the point (a, 6),
when its coordinates are x = a, y=b; the coordinates x, y of a variable point, or of
a point which is for the time being regarded as variable, are said to be current
coordinates.
2. It is sometimes convenient to use oblique coordinates; the only difference is
that the axes are not at right angles to each other ; the lines PM, PN are drawn
parallel to the axes of y and x respectively, and the figure OMPN is thus a
parallelogram. But in all that follows, the Cartesian coordinates are taken to be
rectangular; polar coordinates and other systems will be briefly referred to in the sequel.
3. If the coordinates (x, y) of a point are not given, but only a relation between
them f(x, 2/) = 0, then we have a curve. For, if we consider # as a real quantity
varying continuously from — x to +00, then, for any given value of x, y has a value
69—2
548 GEOMETRY. [790
or values. If these are all imaginary, there is not any real point ; but if one or more
of them be real, we have a real point or points, which (as the assumed value of x
varies continuously) varies or vary continuously therewith ; and the locus of all these
real points is a curve. The equation completely defines the curve ; to trace the curve
directly from the equation, nothing else being known, we obtain as above a series of
points sufficiently near to each other, and draw the curve through them. For instance,
let this be done in a simple case. Suppose y = 2x — 1 ; it is quite easy to obtain and
lay down a series of points as near to each other as we please, and the application of
a ruler would show that these were in a line ; that the curve is a line depends upon
something more than the equation itself, viz. the theorem that every equation of the
form y = ax + b represents a line ; supposing this known, it will be at once understood
how the process of tracing the curve may be abbreviated; we have x=0, y = — \, and
x=\, y = 0; the curve is thus the line passing through these two points. But in
the foregoing example the notion of a line is taken to be a known one, and such
notion of a line does in fact precede the consideration of any equation of a curve
whatever, since the notion of the coordinates themselves rests upon that of a line. In
other cases it may very well be that the equation is the definition of the curve ; the
points laid down, although (as finite in number) they do not actually determine the
curve, determine it to any degree of accuracy ; and the equation thus enables us to
construct the curve.
A curve may be determined in another way ; viz. the coordinates x, y may be
given each of them as a function of the same variable parameter 6 ; x, y =f(0), <f> (6}
respectively. Here, giving to 0 any number of values in succession, these equations
determine the values of x, y, that is, the positions of a series of points on the curve.
The ordinary form y = (f> (x}, where y is given explicitly as a function of x, is a
particular case of each of the other two forms : we have f(x, y}, = y—<J> (x\ = 0 ;
and x = 0, y — $ (0).
4. As remarked under Curve, [785], it is a useful exercise to trace a considerable
number of curves, first taking equations which are purely numerical, and then equations
which contain literal constants (representing numbers); the equations most easily dealt
with are those wherein one coordinate is given as an explicit function of the other,
say y = $ (x) as above. A few examples are here given, with such explanations as
seem proper.
(i) y = 2x — 1, as before ; it is at once seen that this is a line ; and taking it
to be so, any two points, for instance, (0, — 1) and (£, 0), determine the line.
(ii) y = x2. The equation shows that x may be positive or negative, but that y
is always positive, and has the same values for equal positive and negative values of
x: the curve passes through the origin, and through the points (+1,1). It is already
known that the curve lies wholly above the axis of x. To find its form in the
neighbourhood of the origin, give x a small value, x = ± O'l or + O'Ol, then y is very
much smaller, = O'Ol and O'OOOl in the two cases respectively ; this shows that the
curve touches the axis of x at the origin. Moreover, x may be as large as we please,
790] GEOMETRY. 549
but when it is large, y is much larger; for instance, #=10, ?/=100. The curve is a
parabola (fig. 2).
0
(iii) y •= a?. Here x being positive y is positive, but x being negative y is also
negative : the curve passes through the origin, and also through the points (1, 1) and
(— 1, — 1). Moreover, when x is small, = 0'1 for example, then not only is y, =0*001,
very much smaller than x, but it is also very much smaller than y was for the last-
mentioned curve y = x2, that is, in the neighbourhood of the origin the present curve
approaches more closely the axis of x. The axis of x is a tangent at the origin, but
it is a tangent of a peculiar kind (a stationary or inflexional tangent), cutting the
curve at the origin, which is an inflexion. The curve is the cubical parabola (fig. 3).
Fig. 3.
y
(iv) y* = x — l.x — 3.# — 4. Here y = 0 for x = 1, =3, =4. Whenever x—\.x— 3. #—4
is positive, y has two equal and opposite values ; but when x — I . x — 3.x — 4 is
negative, then y is imaginary. In particular, for x less than 1, or between 3 and 4,
y is imaginary, but for x between 1 and 3, or greater than 4, y has two values. It
Fig. 4.
y
is clear that for x somewhere between 1 and 3, y will attain a maximum : the values
of x and y may be found approximately by trial. The curve will consist of an oval
and infinite branch, and it is easy to see that, as shown in fig. 4, the curve where
it cuts the axis of x cuts it at right angles. It may be further remarked that, as
550
GEOMETRY.
[790
x increases from 4, the value of y will increase more and more rapidly ; for instance,
a;=5, y2 = 8, # = 10, 2/2=378, &c., and it is easy to see that this implies that the curve
has on the infinite branch two inflexions as shown.
(v) 7/2 = x — c . x — b . x — a, where a > b > c (that is, a nearer to +00, c to — oc ).
The curve has the same general form as in the last figure, the oval extending between
the limits x = c, x = b, the infinite branch commencing at the point x = a.
(vi) y<* = (x — c)z(x — a). Suppose that in the last-mentioned curve, y1=x— c.x— b.x— a,
b gradually diminishes, and becomes ultimately = c. The infinite branch (see fig. 5)
changes its form, but not in a very marked manner, and it retains the two inflexions.
Fig. 5.
CD
The oval lies always between the values x = c, x = b, and therefore its length con
tinually diminishes ; it is easy to see that its breadth will also continually diminish ;
ultimately it shrinks up into a mere point. The curve has thus a conjugate or isolated
point, or acnode. For a direct verification observe that x = c, y = 0, so that (c, 0) is
a point of the curve, but if x is either less than c, or between c and a, y2 is negative,
and y is imaginary.
(vii) y~ = (x — c) (x — of. If in the same curve b gradually increases and becomes
ultimately = a, the oval and the infinite branch change each of them its form, the
oval extending always between the values x = c, x=b, and thus continually approaching
the infinite branch, which begins at x = a. The consideration of a few numerical
examples, with careful drawing, would show that the oval and the infinite branch as
they approach sharpen out each towards the other, the two inflexions on the infinite
branch coming always nearer to the point (a, 0), — so that finally, when b becomes
Fig. 6.
= a, the curve has the form shown in fig. 6, there being now a double point or node
(crunode) at A, and the inflexions on the infinite branch having disappeared.
790]
GEOMETRY.
551
In the last four examples, the curve is one of the cubical curves called the
divergent parabolas: (iv) is a mere numerical example of (v), and (vi), (vii), (viii) are in
Newton's language the parabola cum ovali, punctata> and nodata respectively. When
a, b, c are all equal, or the form is y2 = (x — c)3, we have a cuspidal form, Newton's
parabola cuspidata, otherwise the semicubical parabola.
(viii) As an example of a curve given by an implicit equation, suppose the
equation is
a? + yz - %xy = 0 ;
this is a nodal cubic curve, the node at the origin, and the axes touching the two
branches respectively (fig. 7). An easy mode of tracing it is to express x, y each of them
Fig. 7.
y
in terms of a variable 8, x =
directly from its equation.
3(9
; but it is instructive to trace the curve
5. It may be remarked that the purely algebraical process, which is, in fact, that
employed in finding a differential coefficient -J*- , if applied directly to the equation of
(JLCC
the curve, determines the point consecutive to any given point of the curve, that is,
the direction of the curve at such given point, or, what is the same thing, the
direction of the tangent at that point. In fact, if a, /3 are the coordinates of any
point on a curve f(x, y) = 0, then writing in the equation of the curve x=a + h,
y = ft + k, and in the resulting equation /(a + h, (3 + k) = 0, developed in powers of h
and k, omitting the term f(a, /3), which vanishes, and the terms containing the second
and higher powers of h, k, we have a linear equation Ah + Bk = 0, which determines
the ratio of the increments h, k. Of course, in the analytical development of the theory,
we translate this into the notation of the differential calculus; but the question pre
sents itself, and is thus seen to be solvable, as soon as it is attempted to trace a
curve from its equation.
552 GEOMETRY. [790
Geometry is Descriptive, or Metrical.
6. A geometrical proposition is either descriptive or metrical : in the former case
it is altogether independent of the idea of magnitude (length, inclination, &c.) ; in the
latter case it has reference to this idea. It is to be noticed that, although the method
of coordinates seems to be by its inception essentially metrical, and we can hardly,
except by metrical considerations, connect an equation with the curve which it repre
sents (for instance, even assuming it to be known that an equation Ax + By + C=0
represents a line, yet if it be asked what line, the only form of answer is, that it
is the line cutting the axes at distances from the origin — C+ A, — C + B respectively),
yet in dealing by this method with descriptive propositions, we are, in fact, eminently
free from all metrical considerations.
7. It is worth while to illustrate this by the instance of the well-known theorem
of the radical centre of three circles. The theorem is that, given any three circles
A, B, G (fig. 8), the common chords aa', /3/3', 77' of the three pairs of circles meet in
a point.
The geometrical proof is metrical throughout : —
Take 0 the point of intersection of oca', @/3', and joining this with 7', suppose
that y'O does not pass through 7, but that it meets the circles A, B in two distinct
points 71( 72 respectively. We have then the known metrical property of intersecting
chords of a circle ; viz. in the circle C, where a a', /3/3' are chords meeting at a point 0,
OOL . OOL' = 0/3 . 0&,
where, as well as in what immediately follows Oa, &c., denote, of course, lengths or
distances.
Similarly in the circle A,
and in the circle B,
Oa . Oa' = 0% . Oy.
Consequently 0% . Oy = 0% . Oy', that is, Oyl=0y^, or the points ^ and 7, coincide;
that is, they each coincide with 7.
790] GEOMETRY. 553
We contrast this with the analytical method.
Here it only requires to be known that an equation Ax + By+C=0 represents
a line, and an equation a? + y2 + Ax + By + C—0 represents a circle. A, B, C have, in
the two cases respectively, metrical significations ; but these we are not concerned with.
Using 8 to denote the function #2 4- y* + Ax + By + C, the equation of a circle is 8=0,
where 8 stands for its value; more briefly, we say the equation is S, =x2+y* + Ax+By+C, =0.
Let the equation of any other circle be S', = x'2 + y* + A'x + B'y + C' = 0 ; the equation
S — S' = 0 is a linear equation: 8 — 8' is, in fact, = (A — A'}x + (B — B')y + C— C': and
it thus represents a line ; this equation is satisfied by the coordinates of each of the
points of intersection of the two circles (for at each of these points 8=0 and S' = 0,
therefore also S— S' = 0); hence the equation S — S' = 0 is that of the line joining the
two points of intersection of the two circles, or say it is the equation of the common
chord of the two circles. Considering then a third circle S", =x2+y2+A"x+B"y + C"=0,
the equations of the common chords are S — 8' = 0, S — S" = 0, S' — S" = 0 (each of
these a linear equation); at the intersection of the first and second of these lines 8=8'
and 8=8", therefore also S' = S", or the equation of the third line is satisfied by the
coordinates of the point in question ; that is, the three chords intersect in a point 0,
the coordinates of which are determined by the equations S = S' = S".
It further appears that, if the two circles 8 = 0, S' =0 do not intersect in any real
points, they must be regarded as intersecting in two imaginary points, such that the
line joining them is the real line represented by the equation S — S' = 0 ; or that two
circles, whether their intersections be real or imaginary, have always a real common
chord (or radical axis), and that for any three circles the common chords intersect in
a point (of course real) which is the radical centre. And by this very theorem, given
two circles with imaginary intersections, we can, by drawing circles which meet each
of them in real points, construct the radical axis of the first-mentioned two circles.
8. The principle employed in showing that the equation of the common chord of
two circles is S — S' = 0 is one of very extensive application, and some more illustrations
of it may be given.
Suppose 8=0, S' = 0 are lines, that is, let S, S' now denote linear functions
Ax+By + C, A'x + B'y + C', then 8 - kS' = 0 (k an arbitrary constant) is the equation
of any line passing through the point of intersection of the two given lines. Such a
line may be made to pass through any given point, say the point (XQ, yQ) ; i.e., if S0, S0'
are what S, S' respectively become on writing for (x, y) the values (#„, y0), then the
value of k is k = S0 + S0'. The equation in fact is SS0' — S0S' = 0; and starting from
this equation we at once verify it a posteriori; the equation is a linear equation
satisfied by the values of (x, y} which make 8 = 0, S' = 0; and satisfied also by the
values (x0, y0) ; and it is thus the equation of the line in question.
If, as before, 8=0, S' = 0 represent circles, then (k being arbitrary) S-kS' = 0 is
the equation of any circle passing through the two points of intersection of the two
circles; and to make this pass through a given point (x0, y0) we have again k = S0-r-S0\
In the particular case k=l, the circle becomes the common chord; more accurately,
c. xi. 70
554
GEOMETRY.
[790
it becomes the common chord together with the line infinity, but this is a question
which is not here gone into.
If S denote the general quadric function,
S = ax* + 2hxy + by2 + 2fy + 2gx + c, = (a, b, c, f, g, h) (x, y, I)2,
then the equation S = 0 represents a conic ; assuming this, then, if S' = 0 represents
another conic, the equation S — kS' = 0 represents any conic through the four points
of intersection of the two conies.
Returning to the equation Ax + By + C = 0 of a line, if this pass through two given
points (xlt 2/i), (#2, 2/aX then we must have Ax^ + By^ + C = 0, Ax2 + By2 + C = 0, equations
which determine the ratios A : B : C, and it thus appears that the equation of the
line through the two given points is
or, what is the same thing,
«, y ,
=0.
2, 2/2 >
9. The object still being to illustrate the mode of working with coordinates, we
consider the theorem of the polar of a point in regard to a circle. Given a circle and
a point 0 (fig. 9), we draw through 0 any two lines meeting the circle in the points
Fig. 9.
B'
A, A' and B, B' respectively, and then taking Q as the intersection of the lines AR
and A'B, the theorem is that the locus of the point Q is a right line depending only
upon 0 arid the circle, but independent of the particular lines OAA and OBB'.
Taking 0 as the origin, and for the axes any two lines through 0 at right angles
to each other, the equation of the circle will be
x2 + y2 + 2Aas + 2By + (7 = 0;
and if the equation of the line OAA' is taken to be y = mx, then the points A, A'
are found as the intersections of the straight line with the circle ; or to determine x we
have
«2 (1 + m2) + 2x (A + Em) + C= 0.
790] GEOMETRY. 555
If (a?!, 2/1) are the coordinates of A, and (#2, ?/2) of A', then the roots of this equation
are xlt #2, whence easily
_ g
-- 1 -- — — i
x-i #2
And similarly, if the equation of the line OBB' is taken to be y = m'x, and the
coordinates of B, B' to be (ac3, y3) and (#4, yt) respectively, then
11 A + Bm'
I -- - - 2t - -~ .
a?3 a?4
We have then
- ^y2 = o,
as the equations of the lines AB' and yl'5 respectively ; for the first of these equations,
being satisfied if we write therein (xlf y,) or (a?4, yt) for (a?, y), is the equation of
the line AB' : and similarly the second equation is that of the line A'B. Reducing by
means of the relations yl — mxl = 0, y2 — mx^ = 0, ys — m'x3 = 0, y^ — m'xt — 0, the two
equations become
x (maci — m'xt) — y(xl — #4) + (m — m) x^xt = 0,
' — m) x2x3 = 0 ;
and if we divide the first of these equations by m1m4, and the second by m2m3, and
then add, we obtain
or, what is the same thing,
1 , , x .
— i (y — m ^) - — + — (y - m*) + 2w - 2m = o,
v
which by what precedes is the equation of a line through the point Q. Substituting
herein for — 1 — , — I — their foregoing values, the equation becomes
/¥• /y» /y» /y> *•
t«/j *x/2 w3 **/4
- (A + Bm) (y - m'x) + ( A + Bm') (y - mx) + m' - m = 0 ;
that is,
(m - m) (Ax + By+C) = Q;
or finally it is Ax + By + C= 0, showing that the point Q lies in a line the position
of which is independent of the particular lines OAA, OBB' used in the construction.
It is proper to notice that there is no correspondence to each other of the points
A, A' and B, B' ; the grouping might as well have been A, A and B', B; and it
thence appears that the line Ax + By + G = 0 just obtained is in fact the line joining
the point Q with the point R which is the intersection of AB and A'B'.
10. The equation Ax + By + C = 0 of a line contains in appearance 3, but really
only 2 constants (for one of the constants can be divided out), and a line depends
70—2
556 GEOMETRY. [790
accordingly upon 2 parameters, or can be made to satisfy 2 conditions. Similarly, the
equation (a, b, c, f. g, h\x, y, I)2 = 0 of a conic contains really 5 constants, and the
equation (*$#, y, 1)3 = 0 of a cubic contains really 9 constants. It thus appears that
a cubic can be made to pass through 9 given points, and that the cubic so passing
through 9 given points is completely determined. There is, however, a remarkable
exception. Considering two given cubic curves 8=0, S' = 0, these intersect in 9 points,
and through these 9 points we have the whole series of cubics S — kS' = 0, where k
is an arbitrary constant : k may be determined so that the cubic shall pass through a
given tenth point, viz. k = SQ-i-S0', if the coordinates are (x0, y0), and 80, S0' denote the
corresponding values of S, S'. The resulting curve SS0'-S'S0=Q may be regarded as
the cubic determined by the conditions of passing through 8 of the 9 points and
through the given point (#„, y0); and from the equation it thence appears that the
curve passes through the remaining one of the 9 points. In other words, we thus
have the theorem, any cubic curve which passes through 8 of the 9 intersections of
two given cubic curves passes through the 9th intersection.
The applications of this theorem are very numerous ; for instance, we derive from
it Pascal's theorem of the inscribed hexagon. Consider a hexagon inscribed in a conic.
The three alternate sides constitute a cubic, and the other three alternate sides
another cubic. The cubics intersect in 9 points, being the 6 vertices of the hexagon,
and the 3 Pascalian points, or intersections of the pairs of opposite sides of the
hexagon. Drawing a line through two of the Pascalian points, the conic and this
line constitute a cubic passing through 8 of the 9 points of intersection, and it
therefore passes through the remaining point of intersection — that is, the third Pascalian
point; and since obviously this does not lie on the conic, it must lie on the line
that is, we have the theorem that the three Pascalian points (or points of intersection
of the pairs of opposite sides) lie on a line.
Metrical Theory.
11. The foundation of the metrical theory consists in the simple theorem that if
a finite line PQ (fig. 10) be projected upon any other line 00' by lines perpendicular
Fig. 10.
'0'
to 00', then the length of the projection P'Qf is equal to the length of PQ multiplied
by the cosine of its inclination to P'Q' ; or, what is the same thing, that the perpen
dicular distance P'Q' of any two parallel lines is equal to the inclined distance PQ
790]
GEOMETRY.
557
multiplied by the cosine of the inclination. It at once follows that the algebraical sum
of the projections of the sides of a closed polygon upon any line is = 0 ; or, reversing
the signs of certain sides, and considering the polygon as consisting of two broken lines,
each extending from the same initial to the same terminal point, the sum of the
projections of the lines of the first set upon any line is equal to the sum of the
projections of the lines of the second set. Observe that, if any line be perpendicular
to the line on which the projection is made, then its projection is =0.
Thus, if we have a right-angled triangle PQR (fig. 11), where QR, RP, QP are
Fig. 11.
R
= £, 7], p respectively, and whereof the base-angle is = a, then projecting successively
on the three sides, we have
£ = p cos a, 77 = p sin a, p = £ cos a + 77 sin a ;
and we thence obtain
/a2 = f 2 + rf ; cos2 a + sin2 a = 1.
And again, by projecting on a line Q^, inclined at the angle a' to QR, we have
p cos (a - a') = £ cos a' + 77 sin of ;
and by substituting for £, 77 their foregoing values,
cos (a — a') = cos a cos a' + sin a sin a'.
It is to be remarked that, assuming only the theory of similar triangles, we have
herein a proof of Euclid, Book I., Prop. 47 ; in fact, the same as is given Book VI.,
Prop. 31; and also a proof of the trigonometrical formula for cos (a — a'). The formulae
for cos (a -fa') and sin (a + a') could be obtained in the same manner.
Draw FT at right angles to Qx^, and suppose QT, TP = £, % respectively, so
that we have now the quadrilateral QRPTQ, or, what is the same thing, the two
broken lines QRP and QTP, each extending from Q to P. Projecting on the four sides
successively, we have
£ = £1 cos a/ — *7i sin a',
77 = £x sin a' + r]! cos a',
& = £ cos «' + V sin a',
??! = — £ sin a' + 77 cos a',
where the third equation is that previously written
p cos (a — a') = | cos a + 1] sin a.
558 GEOMETRY. [790
Equations of Right Line and Circle. — Transformation of Coordinates.
12. The required formulae are really contained in the foregoing results. For, in
fig. 11, supposing that the axis of x is parallel to QR, and taking a, b for the
coordinates of Q, and (x, y) for those of P, then we have f, 77 = x — a, y — b respectively ;
and therefore
x — a = p cos a, y — b = p sin a,
Writing the first two of these in the form
r
x-a^y-
cos a sin a
we may regard Q as a fixed point, but P as a point moving in the direction Q to P,
so that a remains constant, and then, omitting the equation (= p), we have a relation
between the coordinates x, y of the point P thus moving in a right line, — that is,
we have the equation of the line through the given point (a, 6) at a given inclination
a to the axis of x. And, moreover, if, using this equation (= p), we write x = a + p cos a,
y =. b + p sin a, then we have expressions for the coordinates x, y of & point of this
line, in terms of the variable parameter p.
Again, take the point T to be fixed, but consider the point P as moving in the
line TP at right angles to QT. If instead of (fx we take p for the distance QT,
then the equation £ = £ cos a' + 97 sin a' will be
(x — a) cos a +(y — b) sin a' =p ;
that is, this will be the equation of a line such that its perpendicular distance from
the point (a, b) is = p, and that the inclination of this distance to the axis of x is = a.
From either form it appears that the equation of a line is, in fact, a linear
equation of the form Ax + By + (7 = 0. It is important to notice that, starting from
this equation, we can determine conversely the a but not the (a, b) of the form of
equation which contains these quantities ; and in like manner the a' but not the
(a, 6) or p of the other form of equation. The reason is obvious. In each case (a, b)
denote the coordinates of a point, fixed indeed, but which is in the first form any
point of the line, and in the second form any point whatever. Thus, in the second
form the point from which the perpendicular is let fall may be the origin. Here
(a, 6) = (0, 0), and the equation is x cos a' + y sin a — p = 0. Comparing this with
Ax+ By + C = 0, we have the values of cos a', sin a', and p.
13. The equation
p* = (x-ay + (y-by~
is an expression for the squared distance of the two points (a, 6) and (x, y). Taking
as before the point Q, coordinates (a, 6), as a fixed point, and writing c in the place
of p, the equation
790] GEOMETRY. 559
expresses that the point (x, y) is always at a given distance c from the given point
(a, b) ; viz. this is the equation of a circle, having (a, b) for the coordinates of its
centre, and c for its radius.
The equation is of the form
#2 + f + 2Ax + 2% + (7=0,
and here, the number of constants being the same, we can identify the two equations ;
we find a = -A, b = — B, c2 = A2 + B2 — C, or the last equation is that of a circle
having - A, —B for the coordinates of its centre, and ^A- + J52 — C for its radius.
14. Drawing (fig. 11) Q?/i at right angles at Qx1, and taking Qx1} Q^ as a new
set of rectangular axes, if instead of £, % we write xly ylt we have x^, yl as the
new coordinates of the point P ; and writing also a in place of of, a now denoting
the inclination of the axes Qx-i and Ox, we have the formulae for transformation
between two sets of rectangular axes. These are
x — a = x-i cos a — yl sin a,
y — b = #j sin a + yt cos a,
and
x-i = (x — a) cos a + (y — b) sin a,
y1 = — (x — a) sin a + (y — b) cos a,
each set being obviously at once deducible from the other one. In these formulae
(a, b) are the ^-coordinates of the new origin Qlt and a is the inclination of Qxl
to Ox. It is to be noticed that Qxly Qyl are so placed that, by moving 0 to Q, and
then turning the axes Ox1} Oy-i round Q (through an angle a measured in the sense
Ox to Oy\ the original axes Ox, Oy will come to coincide with Qxlt Q^ respectively.
This could not have been done if Qyx had been drawn (at right angles always to
Q#i) in the reverse direction : we should then have had in the formulae — y± instead
of 2/j. The new formulae which would be thus obtained are of an essentially distinct
form : the analytical test is that in the formulae as written down we can, by giving
to a a proper value (in fact, a. = 0), make the (x — a) and (y — b) equal to xl and yl
respectively; in the other system we could only make them equal to x1} —y1} or
— #1, 2/1 respectively. But for the very reason that the second system can be so easily
derived from the first, it is proper to attend exclusively to the first system, — that is,
always to take the new axes so that the two sets admit of being brought into
coincidence.
In the foregoing system of two pairs of equations, the first pair give the original
coordinates x, y in terms of the new coordinates xl} yl\ the second pair the new
coordinates x1} y^ in terms of the original coordinates x, y. The formulae involve (a, b),
the original coordinates of the new origin ; it would be easy, instead of these, to
introduce (ax, frj), the new coordinates of the origin. Writing (a, b) = (0, 0), we have,
of course, the formulae for transformation between two sets of rectangular axes having
the same origin, and it is as well to write the formulae in this more simple form ;
the subsequent transformation to a new origin, but with axes parallel to the original
axes, can then be effected without any difficulty.
560 GEOMETRY. [790
15. All questions in regard to the line may be solved by means of one or other
of the foregoing forms —
Ax + By + C = 0,
y = Ax + B,
x— a y — b
cos a sin a '
(x — a) cos a' + (y — b) sin a' — p = 0 ;
or it may be by a comparison of these different forms: thus, using the first form, it
has been already shown that the equation of the line through two given points (x1} y^,
(x2, y2) is
x (2/1 ~ 2/2) - y (X - #8) + #i2/2 - x^ = 0,
or, as this may be written,
A particular case is the equation
representing the line through the points (a, 0) and (0, 6), or, what is the same thing,
the line meeting the axes of x and y at the distances from the origin a and b
respectively. It may be noticed that, in the form Ax + By + 0 = 0, -= denotes the
n
tangent of the inclination to the axis of x, or we may say that B -4- V J.2 4- B2 and
- A -T- ^/A2 + B2 denote respectively the cosine and the sine of the inclination to the
axis of x. A better form is this : A + */A2 + B2 and B -=- f^A2+B2 denote respectively
the cosine and the sine of inclination to the axis of x of the perpendicular upon
the line. So, of course, in regard to the form y = Ax + B, A is here the tangent of
the inclination to the axis of x ; 1 -f- VZ* +T and A -=- \/A2 + 1 are the cosine and
sine of this inclination, &c. It thus appears that the condition, in order that the
lines Ax + By + C = 0 and A'x+B'y-C' = 0 may meet at right angles, is AA' + BB' = 0;
so when the equations are y = Ax+B, y=A'x + B', the condition is ^' + 1 = 0, or
say the value of A' is = — 1 -=- A.
The perpendicular distance of the point (a, b) from the line Ax + By + C = 0 is
( Aa + Bb + C) H- \/A2 + B2. In all the formulae involving ^A^+B2 or \/A2 + 1, the radical
should be written with the sign ±, which is essentially indeterminate: the like
indeterminateness of sign presents itself in the expression for the distance of two
points p = ± \/(<c-a)2 + (y-b)2; if, as before, the points are Q, P, and the indefinite
line through these is z'QPz, then it is the same thing whether we measure off from
Q along this line, considered as drawn from z towards z, a positive distance k, or
along the line considered as drawn reversely from z towards z' , the equal negative
distance — k, and the expression for the distance p is thus properly of the form + k.
It is interesting to compare expressions which do not involve a radical : thus, in
790] GEOMETRY. 561
seeking for the expression for the perpendicular distance of the point (a, 6) from a given
line, let the equation of the given line be taken in the form, a; cos a + ys'ma — p = 0,
p being the perpendicular distance from the origin, a its inclination to the axis
of x: the equation of the line may also be written (x — a) cos a + (y- 6) sin a— pl = 0,
and we have thence pi=p — a cos a — b sin a, the required expression for the distance
P! : it is here assumed that pl is drawn from (a, b) in the same sense as p is drawn
from the origin, and the indeterminateness of sign is thus removed.
16. As an instance of the mode of using the formulae, take the problem of
finding the locus of a point such that its distance from a given point is in a given
ratio to its distance from a given line.
We take (a, 6) as the coordinates of the given point, and it is convenient to
take (x, y) as the coordinates of the variable point, the locus of which is required:
it thus becomes necessary to use other letters, say (X, Y), for current coordinates in
the equation of the given line. Suppose this is a line such that its perpendicular
distance from the origin is =p, and that the inclination of p to the axis of x is
= a; the equation is X cos a+ 7 sin a -p = Q. In the result obtained in § 15, writing
(x, y) in place of (a, b), it appears that the perpendicular distance of this line from
the point (x, y} is
=p — x cos a — y sin a ;
hence the equation of the locus is
V(ar - a)2 + (y - 6)2 = e (p - x cos a - y sin a),
or say
(x — a)2 + (y- b)2 - e2 (x cosa + y sin a -p)2 = 0,
an equation of the second order.
The Conies (Parabola, Ellipse, Hyperbola).
17. The conies or, as they were called, conic sections were originally defined as
the sections of a right circular cone; but Apollonius substituted a definition, which
is, in fact, that of the last example : the curve is the locus of a point such that its
Fig. 12.
y
D
distance from a given point (called the focus) is in a given ratio to its distance
from a given line (called the directrix); and taking the ratio as e : 1, then e is
called the eccentricity.
Take FJ) for the perpendicular from the focus F upon the directrix, and the
given ratio being that of e : 1 (e >, =, or < 1, but positive), and let the distance FD
c. XL 71
562 GEOMETRY. [790
be divided at 0 in the given ratio, say we have OD = m, OF=em, where m is
positive; — then the origin may be taken at 0, the axis Ox being in the direction OF
(that is, from 0 to F), and the axis Oy at right angles to it. The distance of the
point (x, y) from F is = *J(x — em)2 + y2, its distance from the directrix is = x + m ;
.the equation therefore is
(x — eirif + y2 = eP(a; + m)2 ;
or, what is the same thing, it is
(1 - e2) x2 - 2me (l+e)x + y- = 0.
If e'-=l, or, since e is taken to be positive, if e=l, this is
y- — 4moc = 0,
which is the parabola.
If e" not =1, then the equation may be written
me V m2
Supposing e positive and < 1, then, writing m= , the equation becomes
(1 - e2) (x - a)2 + y2=a2(l- e2),
that is,
a2 + a2(l-e?} =
, changing the origin and writing b2 = a2 (1 — e2), this is
or
which is the ellipse.
An<
becomes
(1 - e2) (x + a)2 + y2 = a2 (1 - e-),
that is,
(a + «)2 , _f_ ,
And similarly if e be positive and > 1, then writing m= — ', the equation
or changing the origin and writing 62 = a2(e2 — 1), this is
which is the hyperbola.
18. The general equation ax2 + 2hxy + by2 -f 2fy + 2gx + c = 0, or as it is written
(a, 6, c, f, g, h)(x, y, 1)2 = 0, may be such that the quadric function breaks up into
factors, = (ax + @y + 7) (OLX -f fly + 7') ; and in this case the equation represents a pair
of lines, or (it may be) two coincident lines. When it does not so break up, the
function can be put in the form X {(x — a')2 + (y — b')2 — & (x cos a + y sin a — p)2}, or,
equating the two expressions, there will be six equations for the determination of
X, a', b', e, p, a. ; and by what precedes, if a, b', e, p, a are real, the curve is either
790]
GEOMETRY.
563
a parabola, ellipse, or hyperbola. The original coefficients (a, b, c, f, g, h) may be such
as not to give any system of real values for a, b', e, p, a; but when this is so the
equation (a, b, c, f, g, ti)(x, y, I)2 = 0 does not represent a real curve*; the imaginary
curve which it represents is, however, regarded as a conic. Disregarding the special
cases of the pair of lines and the twice repeated line, it thus appears that the only
real curves represented by the general equation (a, b, c, f, g, h}(x, y, 1)2 = 0 are the
parabola, the ellipse, and the hyperbola. The circle is considered as a particular case
of the ellipse.
The same result is obtained by transforming the equation (a, b, c,f,g, ti)(x, y, I)2 = 0
to new axes. If in the first place the origin be unaltered, then the directions of the
new (rectangular) axes Oac1} Oy1 can be found so that h^ (the coefficient of the term
#i2/i) shall be =0; when this is done, then either one of the coefficients of x*, y* is
= 0, and the curve is then a parabola, or neither of these coefficients is = 0, and the
curve is then an ellipse or hyperbola, according as the two coefficients are of the
same sign or of opposite signs.
19. The curves can be at once traced from their equations : —
2/2 = 4m#, for the parabola (fig. 13),
p, = 1, for the ellipse (fig. 14),
_
a1
x2 ij-
— — j- = 1, for the hyperbola (fig. 15) ;
Ct \j"
Fig. 13.
* It is proper to remark that, when (a, b, c, f, g, h) (x, y, 1)2 = 0 does represent a real curve, there are,
in fact, four systems of values of a', b', e, p, a, two real, the other two imaginary ; we have thus two real
equations and two imaginary equations, each of them of the form (x-a')2 + (y -6')3=c2 (xcosa + ycoa @-p)2,
representing each of them one and the same real curve. This is consistent with the assertion of the text
that the real curve is in every case represented by a real equation of this form.
71—2
564 GEOMETRY. [790
and it will be noticed how the form of the last equation puts in evidence the two
/>> ijt
asymptotes - = + f of the hyperbola. Referred to the asymptotes (as a set of oblique
a o
axes) the equation of the hyperbola takes the form xy = c ; and in particular, if in
this equation the axes are at right angles, then the equation represents the rectangular
hyperbola referred to its asymptotes as axes.
Tangent, Normal, Circle and Radius of Curvature, &c.
20. There is great convenience in using the language and notation of the
infinitesimal analysis ; thus we consider on a curve a point with coordinates (x, y),
and a consecutive point the coordinates of which are (x+dx, y + dy), or again a second
consecutive point with coordinates (x + dx + \ d2x, y + dy + ^d'2y), &c. ; and in the final
results the ratios of the infinitesimals must be replaced by differential coefficients in
the proper manner; thus, if x, y are considered as given functions of a parameter 0,
dec dii
then dx, dy have in fact the values ,.. dO, -^ dd, and (only the ratio being really
dec du
material) they may in the result be replaced by -v^, -j^. This includes the case where
Ciu atr
the equation of the curve is given in the form y = <f) (x) ; 9 is here = x, and the
increments dx, dy are in the result to be replaced by 1, ,- . So also with the
doc
infinitesimals of the higher orders d?x, &c.
21. The tangent at the point (x, y} is the line through this point and the
consecutive point (x + dx, y + dy) : hence, taking £, 77 as current coordinates, the
equation is
dx dy
an equation which is satisfied on writing therein £, 77 = (x, y) or = (x + dx, y + dy).
The equation may be written
*-y=Tx(t-^
dii
—- being now the differential coefficient of y in regard to x; and this form is applicable
whether y is given directly as a function of x, or in whatever way y is in effect
given as a function of x: if as before x, y are given each of them as a function
of 6, then the value of ,- is = -^ -r- -rn . which is the result obtained from the original
dx dd dd
form on writing therein jg» 3JL f°r ^ % respectively.
So again, when the curve is given by an equation u = 0 between the coordinates
(x, y), then -/ is obtained from the equation - j- + -y ~- = 0. But here it is more
dx dx dy dx
790] GEOMETRY. 565
fi t/ fill
elegant, using the original form, to eliminate dx, dy by the formula -7- dx + -, -- dy ;
we thus obtain the equation of the tangent in the form
du , „ . du .
For example, in the case of the ellipse —^ + j^=^, the equation is
CL 0"
or reducing by means of the equation of the curve the equation of the tangent is
The normal is a line through the point at right angles to the tangent ; the
equation therefore is
(£ - x) dx + (i)-y)dy = 0,
where dx, dy are to be replaced by their proportional values as before.
22. The circle of curvature is the circle through the point and two consecutive
points of the curve. Taking the equation to be
the values of a, /3 are given by
dx d2y - dy d2x ' dx d?y - dy d?x '
and we then have
In the case where y is given directly as a function of x, then, writing for shortness
dy d*u .. (i+p2\3 n+rtOl
P~dx' 3- = dx2 ' 1S ^ = - • ' Or' as equation is usually written, 7 = v ^ ' ,
the radius of curvature, considered to be positive or negative according as the curve
is concave or convex to the axis of x.
It may be added that the centre of curvature is the intersection of the normal
by the consecutive normal.
The locus of the centre of curvature is the evolute. If from the expressions of
a, $ regarded as functions of x we eliminate x, we have thus an equation between
{a, /3), which is the equation of the evolute.
566 GEOMETRY. [790
Polar Coordinates.
23. The position of a point may be determined by means of its distance from a
fixed point and the inclination of this distance to a fixed line through the fixed
point. Say we have r the distance from the origin, and 6 the inclination of r to the
axis of x] r and 6 are then the polar coordinates of the point, r the radius vector,
and 6 the inclination. These are immediately connected with the Cartesian coordinates
x, y by the formulas x — r cos 0, y = r sin 0 ; and the transition from either set of
coordinates to the other can thus be made without difficulty. But the use of polar
coordinates is very convenient, as well in reference to certain classes of questions
relating to curves of any kind — for instance, in the dynamics of central forces — as in
relation to curves having in regard to the origin the symmetry of the regular polygon
(curves such as that represented by the equation r = cos md), and also in regard to
the class of curves called spirals, where the radius vector r is given as an algebraical
or exponential function of the inclination 6.
Trilinear Coordinates.
24. Consider a fixed triangle ABC, and (regarding the sides as indefinite lines)
suppose for a moment that p, q, r denote the distances of a point P from the sides
BC, CA, AB respectively, — these distances being measured either perpendicularly to the
several sides, or each of them in a given direction. To fix the ideas each distance
may be considered as positive for a point inside the triangle, and the sign is thus
fixed for any point whatever. There is then an identical relation between p, q, r: if
a, b, c are the lengths of the sides, and the distances are measured perpendicularly
thereto, the relation is ap + bq + cr = twice the area of triangle. But taking x, y, z
proportional to p, q, r, or if we please proportional to given multiples of p, q, r, then
only the ratios of x, y, z are determined ; their absolute values remain arbitrary. But
the ratios of p, q, r, and consequently also the ratios of x, y, z determine, and that
uniquely, the point ; and it being understood that only the ratios are attended to, we
say that (x, y, z} are the coordinates of the point. The equation of a line has thus
the form ax + by + cz = 0, and generally that of a curve of the nth order is a homo
geneous equation of this order between the coordinates, (*$#, y, z)n = Q. The advantage
over Cartesian coordinates is in the greater symmetry of the analytical forms, and in
the more convenient treatment of the line infinity and of points at infinity. The method
includes that of Cartesian coordinates, the homogeneous equation in x, y, z is, in fact, an
/>> /iy
equation in - , - , which two quantities may be regarded as denoting Cartesian coordi-
Z 2
nates; or, what is the same thing, we may in the equation write z—\. It may be
added that, if the trilinear coordinates (x, y, z} are regarded as the Cartesian coordi
nates of a point of space, then the equation is that of a cone having the origin for
its vertex ; and conversely that such equation of a cone may be regarded as the equation
in trilinear coordinates of a plane curve.
General Point-Coordinates. — Line-Coordinates.
25. All the coordinates considered thus far are point-coordinates. More generally,
any two quantities (or the ratios of three quantities) serving to determine the position
790]
GEOMETRY.
567
of a point in the plane may be regarded as the coordinates of the point ; or, if
instead of a single point they determine a system of two or more points, then as the
coordinates of the system of points. But, as noticed under Curve, [785], there are also
line- coordinates serving to determine the position of a line ; the ordinary case is when
the line is determined by means of the ratios of three quantities £, 77, £ (correlative to
the trilinear coordinates x, y, z). A linear equation a£ + br) + c%=Q represents then the
system of lines such that the coordinates of each of them satisfy this relation, in fact,
all the lines which pass through a given point ; and it is thus regarded as the line-
equation of this point ; and generally a homogeneous equation (*$£, rj, f)n = 0 represents
the curve which is the envelope of all the lines the coordinates of which satisfy this
equation, and it is thus regarded as the line-equation of this curve.
II. SOLID ANALYTICAL GEOMETRY (§§ 26 — 40).
26. We are here concerned with points in space, — the position of a point being
determined by its three coordinates x, y, z. We consider three coordinate planes, at
right angles to each other, dividing the whole of space into eight portions called
octants, the coordinates of a point being the perpendicular distances of the point from
the three planes respectively, each distance being considered as positive or negative
according as it lies on the one or the other side of the plane. Thus the coordinates
in the eight octants have respectively the signs
Fig. 16.
The positive parts of the axes are usually drawn as in fig. 16, which represents
a point P, the coordinates of which have the positive values OM, MN, NP.
568
GEOMETRY.
[790
27. It may be remarked, as regards the delineation of such solid figures, that if
we have in space three lines at right angles to each other, say Oa, Ob, Oc, of equal
lengths, then it is possible to project these by parallel lines upon a plane in such
wise that the projections Oa', Ob', Oc' shall be at given inclinations to each other, and
that these lengths shall be to each other in given ratios : in particular, the two lines
Oa', Oc may be at right angles to each other, and their lengths equal, the direction
of Ob', and its proportion to the two equal lengths Oa', Oc', being arbitrary. It thus
appears that we may as in the figure draw Ox, Oz at right angles to each other, and
Oy in an arbitrary direction ; and moreover represent the coordinates x, z on equal
scales, and the remaining coordinate y on an arbitrary scale (which may be that of
the other two coordinates x, z, but is in practice usually smaller). The advantage, of
course, is that a figure in one of the coordinate planes xz is represented in its proper
form without distortion ; but it may be in some cases preferable to employ the
isometrical projection, wherein the three axes are represented by lines inclined to each
other at angles of 120°, and the scales for the coordinates are equal (fig. 17).
Fig. 17.
For the delineation of a surface of a tolerably simple form, it is frequentlv
sufficient to draw (according to the foregoing projection) the sections by the coordi
nate planes ; and in particular, when the surface is symmetrical in regard to the
coordinate planes, it is sufficient to draw the quarter-sections belonging to a single
octant of the surface ; thus fig. 18 is a convenient representation of an octant of the
790] GEOMETRY. 569
wave surface. Or a surface may be delineated by means of a series of parallel sections,
or (taking these to be the sections by a series of horizontal planes) say by a series
of contour lines. Of course, other sections may be drawn or indicated, if necessary.
For the delineation of a curve, a convenient method is to represent, as above, a series
of the points P thereof, each point P being accompanied by the ordinate PN, which
serves to refer the point to the plane of xy ; this is in effect a representation of each
point P of the curve, by means of two points P, N such that the line PN has a
fixed direction. Both as regards curves and surfaces, the employment of stereographic
representations is very interesting.
28. In plane geometry, reckoning the line as a curve of the first order, we have
only the point and the curve. In solid geometry, reckoning a line as a curve of the
first order, and the plane as a surface of the first order, we have the point, the curve,
and the surface ; but the increase of complexity is far greater than would hence at
first sight appear. In plane geometry a curve is considered in connexion with lines
(its tangents); but in solid geometry the curve is considered in connexion with lines
and planes (its tangents and osculating planes), and the surface also in connexion with
lines and planes (its tangent lines and tangent planes) ; there are surfaces arising out
of the line — cones, skew surfaces, developables, doubly and triply infinite systems of
lines, and whole classes of theories which have nothing analogous to them in plane
geometry : it is thus a very small part indeed of the subject which can be even
referred to in the present article.
In the case of a surface, we have between the coordinates (a, y, z) a single, or
say a onefold relation, which can be represented by a single relation f(x, y, z} = 0 ;
or we may consider the coordinates expressed each of them as a given function of
two variable parameters p, q ; the form z=f(x, y) is a particular case of each of these
modes of representation; in other words, we have in the first mode f(x, y, z)=z — f(x, y),
and in the second mode x=p, y = q for the expression of two of the coordinates in
terms of the parameters.
In the case of a curve, we have between the coordinates (x, y, z) a twofold relation :
two equations f(x, y, z) = Q, <j> (x, y, z) = Q give such a relation ; i.e., the curve is here
considered as the intersection of two surfaces (but the curve is not always the com
plete intersection of two surfaces, and there are hence difficulties) ; or, again, the
coordinates may be given each of them as a function of a single variable parameter.
The form y = <f>x, z = tyx, where two of the coordinates are given in terms of the third,
is a particular case of each of these modes of representation.
29. The remarks under plane geometry as to descriptive and metrical propositions,
and as to the non-metrical character of the method of coordinates when used for the
proof of a descriptive proposition, apply also to solid geometry ; and they might be
illustrated in like manner by the instance of the theorem of the radical centre of four
spheres. The proof is obtained from the consideration that S and S' being each of
them a function of the form #2 + y1 + z- + ax + by + cz + d, the difference S— S' is a
mere linear function of the coordinates, and consequently that S—S'=0 is the equation
of the plane containing the circle of intersection of the two spheres S = 0 and 8' = 0.
c. xr. 72
570 GEOMETRY. [790
Metrical Theory.
30. The foundation in solid geometry of the metrical theory is, in fact, the before-
mentioned theorem that, if a finite right line PQ be projected upon any other line
00' by lines perpendicular to 00', then the length of the projection P'Q' is equal to the
length of PQ multiplied by the cosine of its inclination to PQ' — or (in the form in
which it is now convenient to state the theorem) the perpendicular distance P'Q' of two
parallel planes is equal to the inclined distance PQ into the cosine of the inclination.
Hence also the algebraical sum of the projections of the sides of a closed polygon
upon any line is = 0 ; or, reversing the signs of certain sides and considering the
polygon as made up of two broken lines each extending from the same initial to the
same terminal point, the sum of the projections of the one set of lines upon any
line is equal to the sum of the projections of the other set of lines upon the same
line. When any of the lines are at right angles to the given line (or, what is the
same thing, in a plane at right angles to the given line), the projections of these
lines severally vanish.
31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being
respectively parallel to the three rectangular axes Ox, Oy, Oz\ let the lengths of these
sides be f , 77, £, and that of the side QP be = p ; and let the cosines of the inclin
ations (or say the cosine-inclinations) of p to the three axes be a, /3, 7 ; then pro
jecting successively on the three sides and on QP, we have
£ 77, £=pa, p&, py,
and
p- of +£17 +76
whence /a2 = £2 + if + £2, which is the relation between a distance p and its projections
f, 77, £ upon three rectangular axes. And from the same equations we obtain a2 + ^ + <f — 1,
which is a relation connecting the cosine-inclinations of a line to three rectangular axes.
Suppose we have through Q any other line QT, and let the cosine-inclinations of
this to the axes be a', /3', 7', and S be its cosine-inclination to QP; also let p be the
length of the projection of QP upon QT; then projecting on QT, we have
And in the last equation substituting for f, 77, £ their values pa, pft, py, we find
8 = aaf + ft/3' + 77',
which is an expression for the mutual cosine-inclination of two lines, the cosine-
inclinations of which to the axes are a, /3, 7 and a', £', 7' respectively. We have of
course o2 + /32+72 = l, and a/2 + /3'2 + y'2 = 1, and hence also
- S2 = (a2 + £»
(7*' ~ 7'
so that the sine of the inclination can only be expressed as a square root. These
formulae are the foundation of spherical trigonometry.
790] GEOMETRY. 571
The Line, Plane, and Sphere.
32. The foregoing formulse give at once the equations of these loci.
For first, taking Q to be a fixed point, coordinates (a, b, c) and the cosine-
inclinations (a, /3, 7) to be constant, then P will be a point in the line through Q
in the direction thus determined ; or, taking (x, y, z) for its coordinates, these will be
the current coordinates of a point in the line. The values of £, 77, £ then are x — a,
y — b, z — c, and we thus have
x — a y — b z — c , N
^T=p ~T~(=pl
which (omitting the last equation, = p) are the equations of the line through the
point (a, b, c), the cosine-inclinations to the axes being a, /3, 7, and these quantities
being connected by the relation a2 + /32 + 72 = 1. This equation may be omitted, and
then a, /3, 7, instead of being equal, will only be proportional to the cosine-inclinations.
Using the last equation, and writing
x, y, z = a + ap, b + /3p, c + yp,
these are expressions for the current coordinates in terms of a parameter p, which is in
fact the distance from the fixed point (a, b, c).
It is easy to see that, if the coordinates (x, y, z} are connected by any two
linear equations, these equations can always be brought into the foregoing form, and
hence that the two linear equations represent a line.
Secondly, taking for greater simplicity the point Q to be coincident with the origin,
and a', /3', 7', p to be constant, then p is the perpendicular distance of a plane from
the origin, and a, ft', 7' are the cosine-inclinations of this distance to the axes
(a'2 + /3'2 + 7/2 = 1). Now P is any point in this plane; taking its coordinates to be
(x, y, z}, then (£, 77, f) are =(x, y, z}, and the foregoing equation £> = cc'£+ /3V
becomes
which is the equation of the plane in question.
If, more generally, Q is not coincident with the origin, then, taking its coordinates
to be (a, b, c), and writing pl instead of p, the equation is
a' (x - a) + p (y - b) + 7 (z - c) = p, ;
and we thence have p^ =p - (aaf + bfi' + cy'), which is an expression for the perpendicular
distance of the point (a, b, c) from the plane in question.
It is obvious that any linear equation Ax +By + Cz+D = Q between the coordinates
can always be brought into the foregoing form, and hence that such equation
represents a plane.
72—2
572 GEOMETRY. [790
Thirdly, supposing Q to be a fixed point, coordinates (a, b, c) and the distance
QP, = p, to be constant, say this is =d, then, as before, the values of f, 17, f are
x — a, y — b, z — c, and the equation |2 + rf + £2 = p2 becomes
(x - a)2 + (y - 6)2 + (z - c)2 = d2,
which is the equation of the sphere, coordinates of the centre = (a, b, c) and radius = d.
A quadric equation wherein the terms of the second order are a? + y2 4- z*y viz.
an equation
tf + y* + z* + Ax + By + Gz + D = 0,
can always, it is clear, be brought into the foregoing form; and it thus appears that
this is the equation of a sphere, coordinates of the centre — ^A, — ^B, — \G, and
squared radius = \ ( A2 + B2 + (72) - D.
Cylinders, Cones, Ruled Surfaces.
33. A singly infinite system of lines, or a system of lines depending upon one
variable parameter, forms a surface ; and the equation of the surface is obtained by
eliminating the parameter between the two equations of the line.
If the lines all pass through a given point, then the surface is a cone; and, in
particular, if the lines are all parallel to a given line, then the surface is a cylinder.
Beginning with this last case, suppose the lines are parallel to the line x = mz,
y = nz, the equations of a line of the system are x = mz + a, y = nz + b, — where a, b
are supposed to be functions of the variable parameter, or, what is the same thing,
there is between them a relation /(a, 6)=0: we have a = x — mz, b = y-nz, and the
result of the elimination of the parameter therefore is f(x — mz, y — nz) = 0, which is
thus the general equation of the cylinder the generating lines whereof are parallel to
the line x = mz, y = nz. The equation of the section by the plane z = 0 is f(x, y) = 0,
and conversely if the cylinder be determined by means of its curve of intersection
with the plane z = 0, then, taking the equation of this curve to be f(x, y} = 0, the
equation of the cylinder is f(x -mz, y- nz} = 0. Thus, if the curve of intersection
be the circle (x - a)2 + (y - /3)2 = 72, we have (x - mz - a)2 + (y - nz- /3)2 = y* as the
equation of an oblique cylinder on this base, and thus also (x - a)2 + (y - /3)2 = y as
the equation of the right cylinder.
If the lines all pass through a given point (a, b, c), then the equations of a line
are x-a = a.(z-c}, y-b=@(z-c), where a, /3 are functions of the variable parameter,
or, what is the same thing, there exists between them an equation /(a, /3)=0; the
elimination of the parameter gives, therefore, / (— " , £_*)«<); and this equation, or,
\ Z ~~ C Z — 0 /
what is the same thing, any homogeneous equation f(x -a, y-b, z-c) = 0, or, taking
/ to be a rational and integral function of the order n, say (*)(«- a, y-b, z-c)n = 0,
is the general equation of the cone having the point (a, b, c) for its vertex. Taking the
vertex to be at the origin, the equation is (*)(#, y, z)n = Q- and, in particular,
(*)(#. y> z)2=0 is the equation of a cone of the second order, or quadricone, having the
origin for its vertex.
790]
GEOMETRY.
573
34. In the general case of a singly infinite system of lines, the locus is a ruled
surface (or regulus}. If the system be such that a line does not intersect the con
secutive line, then the surface is a skew surface, or scroll ; but if it be such that
each line intersects the consecutive line, then it is a developable, or torse.
Suppose, for instance, that the equations of a line (depending on the variable
parameter 6) are
® z -/, y\ x z 1 /., y\
- + -=6 1 + | , = TJ 1-f ,
a c \ bj a c 0 \ bj
then, eliminating 6, we have — — - = 1 — ^ -, or say —2 + j-z — 1~^-> *ne equation of a
quadric surface, afterwards called the hyperboloid of one sheet; this surface is con
sequently a scroll. It is to be remarked that we have upon the surface a second
singly infinite series of lines ; the equations of a line of this second system (depending
on the variable parameter <£) are
x z — A (~\ y\ x z — ^ 1 1 y
a c \ bj ' a c <f>\ b
It is easily shown that any line of the one system intersects every line of the other
system.
Considering any curve (of double curvature) whatever, the tangent lines of the
curve form a singly infinite system of lines, each line intersecting the consecutive line
of the system, — that is, they form a developable, or torse ; the curve and torse are
thus inseparably connected together, forming a single geometrical figure. A plane
through three consecutive points of the curve (or osculating plane of the curve)
contains two consecutive tangents, that is, two consecutive lines of the torse, and is
thus a tangent plane of the torse along a generating line.
Transformation of Coordinates.
35. There is no difficulty in changing the origin, and it is for brevity assumed
that the origin remains unaltered. We have, then, two sets of rectangular axes,
Ox, Oy, Oz, and Ox1} Oy1} Oz1} the mutual cosine-inclinations being shown by the
diagram —
that is, a, ft,
Oyi, &c.
are the cosine -inclinations of 0^ to Ox, Oy, Oz\ a, ft', y those of
574
GEOMETRY.
[790
And this diagram gives also the linear expressions of the coordinates (xlt ylt
or (x, y, z) of either set in terms of those of the other set ; we thus have
xl = a. as + /3 y + 7 z,
x = dxl + cfyi + OL"ZI ,
y = fa + ffyi + /3"zlt
which are obtained by projection, as above explained. Each of these equations is, in
fact, nothing else than the before-mentioned equation p = a?j~ + p^ + y'£ adapted to the
problem in hand.
But we have to consider the relations between the nine coefficients. By what
precedes, or by the consideration that we must have identically x? + yz + z2 = x? + y? + zf,
it appears that these satisfy the relations —
a"2 =
/32 + /3'2 + /3"2 = 1,
72 +7/2 + 7"' =1,
+ 7 = 1,
//a 1
+ 7 =1,
a." a. + P'& + y"y = 0, ya + y'a! + y"a" = 0,
a a' + P p + y y' = 0, a/3 4- a'/3' + a"/3" = 0,
either set of six equations being implied in the other set.
It follows that the square of the determinant
a , /3 , y
is = 1 ; and hence that the determinant itself is = + 1. The distinction of the two
cases is an important one: if the determinant is = + 1, then the axes Oxlt Oylf Ozl
are such that they can by a rotation about 0 be brought to coincide with Ox, Oy, Oz
respectively; if it is =-1, then they cannot. But in the latter case, by measuring
#!, ylt Sj in the opposite directions we change the signs of all the coefficients and so
make the determinant to be = + 1 ; hence this case need alone be considered, and it
is accordingly assumed that the determinant is = + 1. This being so, it is found that
we have a further set of nine equations, a = j3'y" - & '7', &c.; that is, the coefficients
arranged as in the diagram have the values
/3V'-yS'V
y'a" — y"a'
a'P" - a" ft'
P"y - Py"
y a — ya"
a"(3 - a/3"
/¥ -Py
ya — y'a
a/3' -a(3
GEOMETRY.
575
790]
36. It is important to express the nine coefficients in terms of three independent
quantities. A solution which, although unsymmetrical, is very convenient in Astronomy
and Dynamics is to use for the purpose the three angles 6, </>, T of fig. 19; say
6 = longitude of the node ; <f> = inclination ; and T = longitude of xl from node.
Fig. 19.
The diagram of transformation then is
X
y
2
«a
cos T cos 0 — sin T sin 6 cos <£
cos T sin 0 + sin T cos 0 cos <j>
sin T sin <£
2/i
— sin T cos 0 — cos T sin 0 cos <£
— sin T sin 6 + cos T cos 6 cos </>
cos T sin <£
«i
sin 6 sin <£
— cos 6 sin <£
cos <f>
But a more elegant solution (due to Rodrigues) is that contained in the diagram
X
y
z
«1
\ + \2 _ ^ _ V2
2(V-*)
2 (Xv + /*)
2/i
2 (X/x + v)
1 _ \2 + ^2 _ V2
2 (/xv - X)
*1
2 (vX - p.)
2 (/xv + A)
1 - X2 - p* + v2
The nine coefficients of transformation are the nine functions of the diagram, each
divided by 1 + X2 + p? + i/2 ; the expressions contain as they should do the three arbitrary
576 GEOMETRY. [790
quantities \, /A, v ; and the identity x? + yf -f- z? = a? + y2 + zs can be at once verified.
It may be added that the transformation can be expressed in the quaternion form
ix-L +jyi + kzl = (1 + A) (ix +jy + kz) (1 + A)"1,
where A denotes the vector i\ -\-j^ + kv.
Quadric Surfaces (Paraboloids, Ellipsoid, Hyperboloids).
37. It appears, by a discussion of the general equation of the second order
(a,... $#, y, z, 1)2 = 0, that the proper quadric surfaces* represented by such an equation
are the following five surfaces (a and b positive) : —
(1) z = = — h ^r , elliptic paraboloid.
yS. yV
(2) z = = ~r , hyperbolic paraboloid.
(3) -2 + -jr, + -a = 1, ellipsoid.
(4) — + j- — -= 1, hyperboloid of one sheet.
Ct 0 C
y& 7/2 Z^
(5) -+r = — 1, hyperboloid of two sheets.
a2 o2 c2
It is at once seen that these are distinct surfaces ; and the equations also show
very readily the general form and mode of generation of the several surfaces.
Fig. 20.
In the elliptic paraboloid (fig. 20), the sections by the planes of zx and zy are
the parabolas
z-^- z-y-
~2a' ~26'
* The improper quadric surfaces represented by the general equation of the second order are (1) the pair
of planes or plane-pair, including as a special case the twice repeated plane, and (2) the cone, including as
a special case the cylinder. There is but one form of cone ; but the cylinder may be parabolic, elliptic, or
hyperbolic.
790]
GEOMETRY.
577
having the common axis Oz ; and the section by any plane z=y parallel to that of
xy is the ellipse
x" y-
7=2« + 26;
so that the surface is generated by a variable ellipse moving parallel to itself along
the parabolas as directrices.
In the hyperbolic paraboloid (fig. 21), the sections by the planes of zx, zy are
the parabolas
z-
~
— --
2a ' 26 '
Fig. 21.
having the opposite axes Oz, Oz' ; and the section by a plane z = 7 parallel to that
of xy is the hyperbola
#2 y-
which has its transverse axis parallel to Ox or Oy according as 7 is positive or
negative. The surface is thus generated by a variable hyperbola moving parallel to
Fig. 22.
itself along the parabolas as directrices. The form is best seen from fig. 22, which
represents the sections by planes parallel to the plane of xy, or say the contour lines ;
C. XL 73
578
GEOMETRY.
[790
the continuous lines are the sections above the plane of xy, and the dotted lines the
sections below this plane. The form is, in fact, that of a saddle.
In the ellipsoid (fig. 23), the sections by the planes of zx, zy, and xy are each
•of them an ellipse, and the section by any parallel plane is also an ellipse. The
Fig. 23.
surface may be considered as generated by an ellipse moving parallel to itself along
two ellipses as directrices.
In the hyperboloid of one sheet (fig. 24), the sections by the planes of zx, zy
are the hyperbolas
- - - = 1 y' - - - l
a2 c2 ' 62 c2 '
having a common conjugate axis zOz' \ the section by the plane of xy, and that by
any parallel plane, is an ellipse ; and the surface may be considered as generated by
a variable ellipse moving parallel to itself along the two hyperbolas as directrices.
In the hyperboloid of two sheets (fig. 25), the sections by the planes of zx and zy
are the hyperbolas
zl
c2
b-
790]
GEOMETRY.
having the common transverse axis zOz ; the section by any plane z = + 7 parallel to
that of ocy, 7 being in absolute magnitude > c, is the ellipse
__ L. y = L — i •
•> I 7 o - o A)
a- 6- c-
and the surface, consisting of two distinct portions or sheets, may be considered as
Fig. 2.5.
3
generated by a variable ellipse moving parallel to itself along the hyperbolas as
directrices.
The hyperbolic paraboloid is such (and it is easy from the figure to understand
how this may be the case) that there exist upon it two singly infinite series of right
lines. The same is the case with the hyperboloid of one sheet (ruled or skew hyper-
boloid, as with reference to this property it is termed). If we imagine two equal
and parallel circular disks, their points connected by strings of equal length, so that
these are the generating lines of a right circular cylinder, then by turning one of
the disks about its centre through the same angle in one or the other direction,
the strings will in each case generate one and the same hyperboloid, and will in
regard to it be the two systems of lines on the surface, or say the two systems of
generating lines ; and the general configuration is the same when instead of circles we
/v»2
have ellipses. It has been already shown analytically that the equation — + ~-2 — - = 1
CL 0 C
is satisfied by each of two pairs of linear relations between the coordinates.
Curves; Tangent, Osculating Plane, Curvature, &c.
38. It will be convenient to consider the coordinates (x, y, z} of the point on
the curve as given in terms of a parameter 6, so that dec, dy, dz, S?x, &c., will be
dec dij dz d^cc
proportional to -j^ , ,^ , ,-a , -j^-t , &c. But only a part of the analytical formulaB will
at/ do du du-
be given ; in them £, 97, £ are used as current coordinates.
73—2
580 GEOMETRY. [790
The tangent is the line through the point (x, y, z) and the consecutive point
(x + dx, y + dy, z + dz) ; its equations therefore are
j_-as = rj-y = £-_z
dx dy dz
The osculating plane is the plane through the point and two consecutive points,
and contains therefore the tangent ; its equation is
77 -y,
= 0,
dx , dy , dz
dj~3C Cb II CvZ
or, what is the same thing,
(£ ~ x) (dyd2z — dzd-y) + (77 — y) (dzd-x — dxd-z) + (% — z) (dxdzy — dyd-x) = 0.
The normal plane is the plane through the point at right angles to the tangent.
It meets the osculating plane in a line called the principal normal; and drawing
through the point a line at right angles to the osculating plane, this is called the
biriormal. We have thus at the point a set of three rectangular axes — the tangent,
the principal normal, and the binormal.
We have through the point and three consecutive points a sphere of spherical
curvature, — the centre and radius thereof being the centre, and radius, of spherical
curvature. The sphere is met by the osculating plane in the circle of absolute
curvature, — the centre and radius thereof being the centre, and radius, of absolute
curvature. The centre of absolute curvature is also the intersection of the principal
normal by the normal plane at the consecutive point.
Surfaces; Tangent Lines and Plane, Curvature, &c.
39. It will be convenient to consider the surface as given by an equation
f(x, y, z)=0 between the coordinates; taking (x, y, z) for the coordinates of a given
point, and (x + dx, y -f dy, z -t- dz) for those of a consecutive point, the increments
dx, dy, dz satisfy the condition
dx dy dz
but the ratio of two of the increments, suppose dx : dy, may be regarded as arbitrary.
Only a part of the analytical formulas will be given; in them f, 17, £ are used as
current coordinates.
We have through the point a singly infinite series of right lines, each meeting
the surface in a consecutive point, or say having each of them two-point intersection
with the surface. These lines lie all of them in a plane which is the tangent plane ;
its equation is
as is at once verified by observing that this equation is satisfied (irrespectively of
the value of dx : dy) on writing therein £, ??, £ = x + dx, y + dy, z + dz.
790] GEOMETRY. 581
The line through the point at right angles to the tangent plane is called the
normal ; its equations are
L~L? _ 5J1? — %~z
df df df •
dx dy dz
In the series of tangent lines there are in general two (real or imaginary) lines,
each of which meets the surface in a second consecutive point, or say it has three-
point intersection with the surface ; these are called the chief-tangents (Haupt-
tangenten). The tangent-plane cuts the surface in a curve, having at the point of
contact a node (double point), the tangents to the two branches being the chief-tangents.
In the case of a quadric surface the curve of intersection, qua curve of the
second order, can only have a node by breaking up into a pair of lines ; that is,
every tangent-plane meets the surface in a pair of lines, or we have on the surface
two singly infinite systems of lines; these are real for the hyperbolic paraboloid and
the hyperboloid of one sheet, imaginary in other cases.
At each point of a surface the chief-tangents determine two directions; and passing
along one of them to a consecutive point, and thence (without abrupt change of
direction) along the new chief-tangent to a consecutive point, and so on, we have on
the surface a chief-tangent curve; and there are, it is clear, two singly infinite series
of such curves. In the case of a quadric surface, the curves are the right lines on the
surface.
40. If at the point we draw in the tangent-plane two lines bisecting the angles
between the chief-tangents, these lines (which are at right angles to each other) are
called the principal tangents*. We have thus at each point of the surface a set of
rectangular axes, the normal and the two principal tangents.
Proceeding from the point along a principal tangent to a consecutive point on
the surface, and thence (without abrupt change of direction) along the new principal
tangent to a consecutive point, and so on, we have on the surface a curve of curvature ;
there are, it is clear, two singly infinite series of such curves, cutting each other at
right angles at each point of the surface.
Passing from the given point in an arbitrary direction to a consecutive point on
the surface, the normal at the given point is not intersected by the normal at the
consecutive point : but passing to the consecutive point along a curve of curvature
(or, what is the same thing, along a principal tangent) the normal at the given point
is intersected by the normal at the consecutive point ; we have thus on the normal
two centres of curvature, and the distances of these from the point on the surface are
the two principal radii of curvature of the surface at that point ; these are also the
radii of curvature of the sections of the surface by planes through the normal and the
two principal tangents respectively ; or say they are the radii of curvature of the
* The point on the surface may be such that the directions of the principal tangents become arbitrary ;
the point is then an umbilicus. It is in the text assumed that the point on the surface is not an
umbilicus.
582 GEOMETRY. [790
normal sections through the two principal tangents respectively. Take at the point
the axis of z in the direction of the normal, and those of x and y in the directions
of the principal tangents respectively, then, if the radii of curvature be a, b (the signs
being such that the coordinates of the two centres of curvature are z = a and z = b
respectively), the surface has in the neighbourhood of the point the form of the
paraboloid
x2 y2
= 2a + Zb'
g£ ,y2
and the chief-tangents are determined by the equation 0 = — + —- . The two centres
2.1 1 —i>
of curvature may be on the same side of the point or on opposite sides ; in the
former case a and b have the same sign, the paraboloid is elliptic, and the chief-
tangents are imaginary ; in the latter case a and 6 have opposite signs, the para
boloid is hyperbolic, and the chief-tangents are real.
The normal sections of the surface and the paraboloid by the same plane have
the same radius of curvature ; and it thence readily follows that the radius of curvature
of a normal section of the surface by a plane inclined at an angle 0 to that of zx
is given by the equation
1 cos2 6 sin2 6
— i
pa b
The section in question is that by a plane through the normal and a line in
the tangent plane inclined at an angle 6 to the principal tangent along the axis
of x. To complete the theory, consider the section by a plane having the same trace
upon the tangent plane, but inclined to the normal at an angle <f>; then it is shown
without difficulty (Meunier's theorem) that the radius of curvature of this inclined
section of the surface is = p cos <£.
583
791.
LANDEN.
[From the Encyclopaedia Britannica, Ninth Edition, vol. xiv. (1882), p. 271.]
LANDEN, JOHN, a distinguished mathematician of the 18th century, was born at
Peakirk near Peterborough in Northamptonshire in 1719, and died 15th January 1790
at Milton in the same county. Most of his time was spent in the pursuits of active life,
but he early showed a strong talent for mathematical study, which he eagerly cultivated
in his leisure hours. In 1762 he was appointed agent to the Earl Fitzwilliam, and
held that office to within two years of his death. He lived a very retired life', and
saw little or nothing of society ; when he did mingle in it, his dogmatism and
pugnacity caused him to be generally shunned. He was first known as a mathematician
by his essays in the Ladies Diary for 1744. In 1766 he was elected a Fellow of the
Royal Society. He was well acquainted and au courant with the works of the mathe
maticians of his own time, and has been called the English DAlembert. In his
Discourse on the "Residual Analysis," in which he proposes to substitute for the method
of fluxions a purely algebraical method, he says, "It is by means of the following
theorem, viz.
™n mil m -
\JU ^^ V — — J
X— V
V /V\2
x 1 +-+-+... (m terms)
X \XJ
(where m and n are integers), that we are able to perform all the principal operations
in our said analysis ; and I am not a little surprised that a theorem so obvious, and
of such vast use, should so long escape the notice of algebraists." The idea 'is of
course a perfectly legitimate one, and may be compared with that of Lagrange's Calcul
des Fonctions. His memoir (1775) on the rotatory motion of a body contains (as the
author was aware) conclusions at variance with those arrived at by DAlembert and
584 LANDEN. [791
Euler in their researches on the same subject. He reproduces and further develops
and defends his own views in his Mathematical Memoirs, and in his paper in the
Philosophical Transactions for 1785. But Landen's capital discovery is that of the
theorem known by his name (obtained in its complete form in the memoir of 1775,
and reproduced in the first volume of the Mathematical Memoirs} for the expression
of the arc of an hyperbola in terms of two elliptic arcs. To find this, he integrates
a differential equation derived from the equation
/rav-o?
t = qx \ / — ,
y V m- — ax-
gx*
interpreting geometrically in an ingenious and elegant manner three integrals which
present themselves. If in the foregoing equation we write m = 1, g = k2, and instead
of t consider the new variable y = t -r- (1 — k'}, then
/ 1 -a?
^(l + tf)^——,
which is the form known as Landen's transformation in the theory of elliptic functions:
but his investigation does not lead him to obtain the equivalent of the resulting
differential equation
where X = , -,, ,
due it would appear to Legendre, and which (over and above Landen's own beautiful
result) gives importance to the theorem as leading directly to the quadric transformation
of an elliptic integral in regard to the modulus.
The list of his writings is as follows : — Ladies' Diary, various communications,
1744—1760; papers in the Phil. Trans., 1754, 1760, 1768, 1771, 1775, 1777, 1785;
Mathematical Lucubrations, 1755; A Discourse concerning the Residual Analysis, 1758;
The Residual Analysis, book I., 1764; Animadversions on Dr Stewart's Method of com
puting the Suns Distance from the Earth, 1771 ; Mathematical Memoirs, 1780, 1789.
792] 585
792.
LOCUS.
[From the Encyclopaedia Britannica, Ninth Edition, vol. xiv. (1882), pp. 764, 765.]
Locus, in Greek TOTTO?, a geometrical term, the invention of the notion of which
is attributed to Plato. It occurs in such statements as these: — the locus of the points
which are at the same distance from a fixed point, or of a point which moves so as
to be always at the same distance from a fixed point, is a circle ; conversely a circle
is the locus of the points at the same distance from a fixed point, or of a point
moving so as to be always at the same distance from a fixed point ; and so, in general,
a curve of any given kind is the locus of the points which satisfy, or of a point
moving so as always to satisfy, a given condition. The theory of loci is thus identical
with that of curves ; and it is in fact in this very point of view that a curve is
considered in the article Curve, [785] ; see that article, and also Geometry (Analytical),
[790]. It is only necessary to add that the notion of a locus is useful as regards deter
minate problems or theorems : thus, to find the centre of the circle circumscribed about
a given triangle ABC, we see that the circumscribed circle must pass through the two
vertices A, B, and the locus of the centres of the circles which pass through these two
points is the straight line at right angles to the side AB at its mid-point ; similarly the
circumscribed circle must pass through A, C, and the locus of the centres of the circles
through these two points is the line at right angles to the side AC at its mid -point ;
thus we get the ordinary construction, and also the theorem that the lines at right
angles to the sides, at their mid-points respectively, meet in a point. The notion of
a locus applies, of course, not only to plane but also to solid geometry. Here the locus
of the points satisfying a single (or onefold) condition is a surface ; the locus of the
points satisfying two conditions (or a twofold condition) is a curve in space, which is
in general a twisted curve or curve of double curvature.
C. XL 74
586 [793
793.
MONGE.
[From the Encyclopaedia Britannica, Ninth Edition, vol. XVI. (1883), pp. 738, 739.]
MONGE, GASPARD (1746 — 1818), a French mathematician, the inventor of descriptive
geometry, was born at Beaune on the 10th May 1746. He was educated first at the
college of the Oratorians at Beaune, and then in their college at Lyons, — where, at
sixteen, the year after he had been learning physics, he was made a teacher of it.
Returning to Beaune for a vacation, he made, on a large scale, a plan of the town,
inventing the methods of observation and constructing the necessary instruments ; the
plan was presented to the town, and preserved in their library. An officer of engineers
seeing it wrote to recommend Monge to the commandant of the military school at
Mezieres, and he was received as draftsman and pupil in the practical school attached
to that institution ; the school itself was of too aristocratic a character to allow of
his admission to it. His manual skill was duly appreciated : " I was a thousand times
tempted," he said long afterwards, " to tear up my drawings in disgust at the esteem
in which they were held, as if I had been good for nothing better." An opportunity,
however, presented itself: being required to work out from data supplied to him the
" defilement " of a proposed fortress (an operation then only performed by a long
arithmetical process), Monge, substituting for this a geometrical method, obtained the
result so quickly that the commandant at first refused to receive it — the time necessary
for the work had not been taken ; but upon examination the value of the discovery
was recognized, and the method was adopted. And Monge, continuing his researches,
arrived at that general method of the application of geometry to the arts of con
struction which is now called descriptive geometry. But such was the system in France
before the Revolution that the officers instructed in the method were strictly forbidden
to communicate it even to those engaged in other branches of the public service ;
and it was not until many years afterwards that an account of it was published. The
method consists, as is well known, in the use of the two halves of a sheet of paper
to represent say the planes of xy and xz at right angles to each other, and the
793] MONGE. 587
consequent representation of points, lines, and figures in space by means of their plan
and elevation, placed in a determinate relative position.
In 1768 Monge became professor of mathematics, and in 1771 professor of physics,
at Mdzieres ; in 1778 he married Madame Horbon, a young widow whom he had
previously defended in a very spirited manner from an unfounded charge; in 1780 he
was appointed to a chair of hydraulics at the Lyceum in Paris (held by him together
with his appointments at Mezieres), and was received as a member of the Academy ;
his intimate friendship with Berthollet began at this time. In 1783, quitting Mezieres,
he was, on the death of Bezout, appointed examiner of naval candidates. Although
pressed by the minister to prepare for them a complete course of mathematics, he
declined to do so, on the ground that it would deprive Madame Bezout of her only
income, arising from the sale of the works of her late husband ; he wrote, however
(1786), his Traite elementaire de la Statique.
Monge contributed (1770 — 1790) to the Memoirs of the Academy of Turin, the
Mdmoires des Savants Strangers of the Academy of Paris, the Mdmoires of the same
Academy, and the Annales de Chimie, various mathematical and physical papers. Among
these may be noticed the memoir " Sur la the'orie des deblais et des remblais "
(Me'm. de I'Acad. de Paris, 1781), which, while giving a remarkably elegant investi
gation in regard to the problem of earthwork referred to in the title, establishes in
connexion with it his capital discovery of the curves of curvature of a surface. Euler,
in his paper on curvature in the Berlin Memoirs for 1760, had considered, not the
normals of the surface, but the normals of the plane sections through a particular
normal, so that the question of the intersection of successive normals of the surface
had never presented itself to him. Monge's memoir just referred to gives the ordinary
differential equation of the curves of curvature, and establishes the general theory in
a very satisfactory manner; but the application to the interesting particular case of
the ellipsoid was first made by him in a later paper in 1795. A memoir in the
volume for 1783 relates to the production of water by the combustion of hydrogen;
but Monge's results in this matter had been anticipated by Watts and Cavendish.
In 1792, on the creation by the Legislative Assembly of an executive council,
Monge accepted the office of minister of the marine, but retained it only until April
1793. When the Committee of Public Safety made an appeal to the savants to assist
in producing the materiel required for the defence of the republic, he applied himself
wholly to these operations, and distinguished himself by his indefatigable activity
therein ; he wrote at this time his Description de I'art de fabriquer les canons, and
his Avis aux ouvriers en fer sur la fabrication de I'acier. He took a very active
part in the measures for the establishment of the Normal School (which existed only
during the first four months of the year 1795), and of the School for Public Works,
afterwards the Polytechnic School, and was at each of them professor of descriptive
geometry ; his methods in that science were first published in the form in which the
shorthand writers took down his lessons given at the Normal School in 1795, and
again in 1798—99. In 1796 Monge was sent into Italy with Berthollet and some
artists to receive the pictures and statues levied from several Italian towns, and made
74—2
588 MONGE. [793
there the acquaintance of General Bonaparte. Two years afterwards he was sent to
Rome on a political mission, which terminated in the establishment, under Massena, of
the shortlived Roman republic; and he thence joined the expedition to Egypt, taking
part with his friend Berthollet as well in various operations of the war as in the
scientific labours of the Egyptian Institute of Sciences and Arts; they accompanied
Bonaparte to Syria, and returned with him in 1798 to France. Monge was appointed
president of the Egyptian commission, and he resumed his connexion with the Poly
technic School. His later mathematical papers are published (1794 — 1816) in the
Journal and the Correspondence of the Polytechnic School. On the formation of the
Senate he was appointed a member of that body, with an ample provision and the
title of count of Pelusium ; but on the fall of Napoleon he was deprived of all his
honours, and even excluded from the list of members of the reconstituted Institute.
He died at Paris on the 28th July 1818.
For further information see B. Brisson, Notice historique sur Gaspard Monge;
Dupin, Essai historique sur les services et les travaux scientiftques de Gaspard Monge,
Paris, 1819, which contains (pp. 162 — 166) a list of Monge's memoirs and works; and
the biography by Arago (GEuvres, t. II., 1854).
Monge's various mathematical papers are to a considerable extent reproduced in
the Application de V Analyse a la Georndtrie, 4th edition (last revised by the author),
Paris, 1819 — the pure text of this is reproduced in the 5th edition (revue, corrige'e et
annotee par M. Liouville), Paris, 1850, which contains also Gauss's Memoir, " Disquisitiones
generales circa superficies curvas," and some valuable notes by the editor. The other
principal separate works are Traite" ddmentaire de la Statique, 8e edition, conformee d
la prtcedente, par M. Hachette, et suivie d'une Note etc., par M. Gauchy, Paris, 1846 ;
and the Gdometrie Descriptive (originating, as mentioned above, in the lessons given
at the Normal School). The 4th edition, published shortly after the author's death,
seems to have been substantially the same as the 7th (Geometrie Descriptive, par
G. Monge, suivie d'une thdorie des Ombres et de la Perspective, extraite des papiers de
I'auteur, par M. Brisson, Paris, 1847).
794] 589
794.
NUMBERS, PARTITION OF.
[From the Encyclopaedia Britannica, Ninth Edition, vol. xvil. (1884), p. 614.]
THIS subject, created by Euler, though relating essentially to positive integer
numbers, is scarcely regarded as a part of the Theory of Numbers. We consider in
it a number as made up by the addition of other numbers: thus the partitions of
the successive numbers 1, 2, 3, 4, 5, 6, &c., are as follows : —
i;
2, 11;
3, 21, 111;
4, 31, 22, 211, 1111 ;
5, 41, 32, 311, 221, 2111, 11111;
6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These are formed each from the preceding ones; thus, to form the partitions of 6 we
take first 6; secondly, 5 prefixed to each of the partitions of 1 (that is, 51); thirdly,
4 prefixed to each of the partitions of 2 (that is, 42, 411); fourthly, 3 prefixed to
each of the partitions of 3 (that is, 33, 321, 3111); fifthly, 2 prefixed, not to each
of the partitions of 4, but only to those partitions which begin with a number not
exceeding 2 (that is, 222, 2211, 21111); and lastly, 1 prefixed to all the partitions of
5 which begin with a number not exceeding 1 (that is, 111111); and so in other cases.
The method gives all the partitions of a number, but we may consider different
classes of partitions: the partitions into a given number of parts, or into not more
than a given number of parts ; or the partitions into given parts, either with
repetitions or without repetitions, &c. It is possible, for any particular class of parti
tions, to obtain methods more or less easy for the formation of the partitions either
of a given number or of the successive numbers 1, 2, 3, &c. And of course in any
case, having obtained the partitions, we can count them and so obtain the number
of partitions.
590 NUMBERS. [794
Another method is by Arbogast's rule of the last and the last but one; in fact,
taking the value of a to be unity, and, understanding this letter in each term, the
rule gives 6; c, 62; d, be, b3; e, bd, c2, b'2c, b\ &c., which, if b, c, d, e, &c., denote
1, 2, 3, 4, &c., respectively, are the partitions of 1, 2, 3, 4, &c., respectively.
An important notion is that of conjugate partitions. Thus a partition of 6 is 42;
writing this in the form -L , and summing the columns instead of the lines, we
obtain the conjugate partition 2211; evidently, starting from 2211, the conjugate
partition is 42. If we form all the partitions of 6 into not more than three parts,
these are
6, 51, 42, 33, 411, 321, 222,
and the conjugates are
111111, 21111, 2211, 222, 3111, 321, 33,
where no part is greater than 3 ; and so, in general, we have the theorem, the number
of partitions of n into not more than k parts is equal to the number of partitions
of n with no part greater than k.
We have for the number of partitions an analytical theory depending on generating
functions; thus for the partitions of a number n with the parts 1, 2, 3, 4, 5, &c.,
without repetitions, writing down the product
it is clear that, if #*, a£, #?, ... are terms of the series x, a?, x3, ... for which
a + /3 + 7 + . . . = n, then we have in the development of the product a term xn, and
hence that, in the term Nxn of the product, the coefficient N is equal to the number
of partitions of n with the parts 1, 2, 3,..., without repetitions; or say that the
product is the generating function (G. F.) for the number of such partitions. And so
in other cases we obtain a generating function.
Thus for the function
1
observing that any factor 1/1— x1 is = 1 + x1 + a?1 + ..., we see that, in the term Nxn,
the coefficient is equal to the number of partitions of n, with the parts 1, 2, 3,...,
with repetitions.
Introducing another letter z, and considering the function
II rip n> \ ,1 , /yi2 y | I ,y>3 ty . — . | I ty ( /yi I /y>2 I \ I I /V /y»?l <yfC I
l^ w& • A \^ \AJ & • -L ^^ \Aj & • • . , —"• L i^ & I w |^ w ^^ • • • i (^ • • • ^\ JL" i*/ A ^\ . • . ,
we see that, in the term Nxnzk of the development, the coefficient N is equal to the
number of partitions of n into k parts, with the parts 1, 2, 3, 4, ..., without repetitions.
And similarly, considering the function
i
= 1 +
1 — xz . 1 — a?z . 1 — x3z . . . '
we see that, in the term Nxnzk of the development, the coefficient N is equal to the
number of partitions of n into k parts, with the parts 1, 2, 3, 4, ..., with repetitions.
794] NUMBERS. 591
We have such analytical formulae as
1 _ zx z*xz
— 1 4-
1 — XZ . 1 — X*Z . 1 — X3Z ... 1 — X 1 — (C.I — X2
which lead to theorems in the Partition of Numbers. A remarkable theorem is
where the only terms are those with an exponent ^ (3n? + ri), and for each such pair
of terms the coefficient is (— )n 1. The formula shows that, except for numbers of the
form £ (3n2 + n), the number of partitions without repetitions into an odd number of
parts is equal to the number of partitions without repetitions into an even number
of parts, whereas for the excepted numbers these numbers differ by unity. Thus for
the number 11, which is not an excepted number, the two sets of partitions are
11, 821, 731, 641, 632, 542,
10.1, 92, 83, 74, 65, 5321,
in each set 6.
We have
1-«.1+*.1 + «2.
or, as this may be written,
. z ,
I — x
showing that a number n can always be made up, and in one way only, with the
parts 1, 2, 4, 8, — The product on the left-hand side may be taken to k terms
only: thus if k = 4>, we have
,
1 — x
a?
that is, any number from 1 to 15 can be made up, and in one way only, with the
parts 1, 2, 4, 8 ; and similarly any number from 1 to 2* — 1 can be made up, and in
one way only, with the parts 1, 2, 4, ... , 2fc-1. A like formula is
1-a? 1-a? \-aF 1 -Xs1 1 - a?1
x.l-x' a? ,\-a?' x».l -x9' a? .l-x* x*> . 1 - x '
that is,
ar1 + 1 + x . x~3 + 1 + a? . x-9 + 1 + of> . ar27 -f 1 + x* = x~to + x~™ +... + !+«+.
showing that any number from —40 to +40 can be made up, and that in one way only,
with the parts 1, 3, 9, 27 taken positively or negatively ; and so in general any number
from — 1(3* — 1) to +-|(3fc — 1) can be made up, and that in one way only, with the
parts 1, 3, 9,..., S*"1 taken positively or negatively.
592 [795
795.
NUMBERS, THEORY OF.
[From the Encyclopaedia Britannica, Ninth Edition, vol. xvil. (1884), pp. 614 — 624.]
THE Theory of Numbers, or higher arithmetic, otherwise arithmology, is a subject
which, originating with Euclid, has in modern times, in the hands of Legendre, Gauss,
Lejeune-Dirichlet, Kummer, Kronecker, and others, been developed into a most extensive
and interesting branch of mathematics. We distinguish between the ordinary (or say
the simplex) theory and the various complex theories.
In the ordinary theory we have, in the first instance, positive integer numbers,
the unit or unity 1, and the other numbers 2, 3, 4, 5, &c. We introduce the zero 0,
which is a number sui generis, and the negative numbers — 1, — 2, — 3, — 4, &c., and
we have thus the more general notion of integer numbers, 0, +1, +2, +3, &c. ; +1
and — 1 are units or unities. The sum of any two or more numbers is a number ;
conversely, any number is a sum of two or more parts; but even when the parts are
positive a number cannot be, in a determinate manner, represented as a sum of parts.
The product of two or more numbers is a number; but (disregarding the unities + 1,
— 1, which may be introduced as factors at pleasure) it is not conversely true that
every number is a product of numbers. A number such as 2, 3, 5, 7, 11, &c., which
is not a product of numbers, is said to be a prime number; and a number which
is not prime is said to be composite. A number other than zero is thus either
prime or composite ; and we have the theorem that every composite number is, in a
determinate way, a product of prime factors.
We have complex theories in which all the foregoing notions (integer, unity, zero,
prime, composite) occur; that which first presented itself was the theory with the unit i
(i2 = — 1) ; we have here complex numbers, a + bi, where a and b are in the before-
mentioned (ordinary) sense positive or negative integers, not excluding zero ; we have
the zero 0, = 0 + Qi, and the four units I, —I, i, —i. A number other than zero is
here either prime or else composite; for instance, 3, 7, 11, are prime numbers, and
5, = (2 + i) (2 - i), 9, =3.3, 13, = (3 + 2i) (3 — 2i), are composite numbers (generally any
795] NUMBERS. 593
positive real prime of the form 4>n + 3 is prime, but any positive real prime of the
form 4>n + 1 is a sum of two squares, and is thus composite). And disregarding unit
factors we have, as in the ordinary theory, the theorem that every composite number
is, in a determinate way, a product of prime factors.
There is, in like manner, a complex theory involving the cube roots of unity — if
a be an imaginary cube root of unity (a2 + a + 1 = 0), then the integers of this theory
are a + ba. (a and b real positive or negative integers, including zero) ; a complex
theory with the fifth roots of unity — if a be an imaginary fifth root of unity
(a4 + as + a2 + a + 1 = 0), then the integers of the theory are a + ba. + ca? + da? (a, b, c, d,
real positive or negative integers, including zero); and so on for the roots of the
orders 7, 11, 13, 17, 19. In all these theories, or at any rate for the orders 3, 5, 7
(see No. 37, post), we have the foregoing theorem : disregarding unit factors, a number
other than zero is either prime or composite, and every composite number is, in a
determinate way, a product of prime factors. But coming to the 23rd roots of unity
the theorem ceases to be true. Observe that it is a particular case of the theorem
that, if N be a prime number, any integer power of N has for factors only the lower
powers of N, — for instance, N3 = N . N2 ; there is no other decomposition N3 = AB.
This is obviously true in the ordinary theory, and it is true in the complex theories
preceding those for the 3rd, 5th, and 7th roots of unity, and probably in those for
the other roots preceding the 23rd roots ; but it is not true in the theory for the
23rd roots of unity. We have, for instance, 47, a number not decomposable into factors,
but 473, =AB, is a product of two numbers each of the form a + ba. + ... + &a21 (a a
23rd root). The theorem recovers its validity by the introduction into the theory of
Rummer's notion of an ideal number.
The complex theories above referred to would be more accurately described as
theories for the complex numbers involving the periods of the roots of unity : the
units are the roots either of the equation xp~l + xp~2 +...+# + 1 = 0 (p a prime number)
P-I
or of any equation x e +... + 1=0 belonging to a factor of the function of the
order p — 1 : in particular, this may be the quadric equation for the periods each of
\ (p — 1) roots ; they are the theories which were first and have been most completely
considered, and which led to the notion of an ideal number. But a yet higher
generalization which has been made is to consider the complex theory, the units
whereof are the roots of any given irreducible equation which has integer numbers
for its coefficients.
There is another complex theory the relation of which to the foregoing is not
very obvious, viz. Galois's theory of the numbers composed with the imaginary roots
of an irreducible congruence, F(x) = 0 (modulus a prime number p) ; the nature of
this will be indicated in the sequel.
In any theory, ordinary or complex, we have a first part, which has been termed
(but the name seems hardly wide enough) the theory of congruences ; a second part,
the theory of homogeneous forms : this includes in particular the theory of the binary
quadratic forms (a, b, c)(x, 7/)2; and a third part, comprising those miscellaneous
investigations which do not come properly under either of the foregoing heads.
c. XL 75
594 NUMBERS. [795
Ordinary Theory, First Part.
1. We are concerned with the integer numbers 0, +1, + 2, + 3, &c., or in the
first place with the positive integer numbers 1, 2, 3, 4, 5, 6, &c. Some of these,
1, 2, 3, 5, 7, &c., are prime, others, 4, = 22, 6, = 2 . 3, &c., are composite ; and we have
the fundamental theorem that a composite number is expressible, and that in one way
only, as a product of prime factors, N = aa63cy . . . (a, b, c, ... primes other than 1;
a, 0, 7, ... positive integers).
Gauss makes the proof to depend on the following steps : (i) the product of two
numbers each smaller than a given prime number is not divisible by this number :
(ii) if neither of two numbers is divisible by a given prime number the product is
not so divisible ; (iii) the like as regards three or more numbers ; (iv) a composite
number cannot be resolved into factors in more than one way.
2. Proofs will in general be only indicated or be altogether omitted, but, as a
specimen of the reasoning in regard to whole numbers, the proofs of these funda
mental propositions are given at length, (i) Let p be the prime number, a a number
less than p, and if possible let there be a number b less than p, and such that ab
is divisible by p ; it is further assumed that b is the only number, or, if there is
more than one, then that b is the least number having the property in question;
b is greater than 1, for a being less than p is not divisible by p. Now p as a
prime number is not divisible by b, but must lie between two consecutive multiples
mb and (m + l)6 of b. Hence, ab being divisible by p, mab is also divisible by p;
moreover, ap is divisible by p, and hence the difference of these numbers, = a (p — mb),
must also be divisible by p, or, writing p — mb = b', we have ab' divisible by p, where
b' is less than b ; so that b is not the least number for which ab is divisible by p.
(ii) If a and 6 are neither of them divisible by p, then a divided by p leaves a
remainder a which is less than p, say we have a = mp + a ; and similarly b divided
by p leaves a remainder /? which is less than p, say we have b = np + B ; then
ab = (mp + a) (np + 0), = (mnp + no. + m/3) p + a/3,
and a.0 is not divisible by p, therefore ab is not divisible by p. (iii) The like proof
applies to the product of three or more factors a, b, c, ... (iv) Suppose that the
number N, = aabficv ... (a, b, c, ... prime numbers other than 1), is decomposable in
some other way into prime factors ; we can have no prime factor p, other than
a, b, c,..., for no such number can divide a^c?...; and we must have each of the
numbers a, b, c, ..., for if any one of them, suppose a, were wanting, the number N
would not be divisible by a. Hence the new decomposition if it exists must be a
decomposition N = aa'bP'c?' ... ; and here, if any two corresponding indices, say a, a/, are
different from each other, then one of them, suppose a', is the greater, and we have
N + pa = bl*cv ... =aa'-a We"*' ... That is, we have the number N+pa expressed in two
different ways as a product, the number a being a factor in the one case, but not a
factor in the other case. Thus the two exponents cannot be unequal, that is, we
must have a = a', and similarly we have 0 = 0', 7 = 7',...; that is, there is only the
original decomposition N = a^
795] NUMBERS. 595
3. The only numbers divisible by a number N = aab^c^ . . . are the numbers
orb^'cy' . . . , where each exponent a! is equal to or greater than the corresponding
exponent or. And conversely the only numbers which divide N are those of the form
a*'Wc*' . . . , where each index a! is at most equal to the corresponding index a ; and
in particular each or any of the indices a! may be = 0. Again, the least common
multiple of two numbers N = aab^c^ . . . and N' = arWtf' ... is aa"¥"c^" ... , where each
index a" is equal to the largest of the corresponding indices a, a ; — observe that any
one or more of the indices ct, fi, 7, . . . , a.', @', 7', . . . , may be = 0, so that the theorem
extends to the case where either of the numbers N, N', has prime factors which are
not factors of the other number. And so the greatest common measure of two
numbers N = &*¥&... and N' = aa'6^'c>' . . . is aa"&3"cy" ..., where each index a" is equal
to the least of the corresponding indices a and a'.
4. The divisors of N = aa^c? . . . are the several terms of the product
where unity and the number N itself are reckoned each of them as a divisor. Hence
the number of divisors is = (a + 1) (/8 + 1) (7+ 1) ..., and the sum of the divisors is
(aa+1 -
5. In N = a^cv . . . the number of integers less than N and prime to it is
To find the numbers in question write down the series of numbers 1, 2, 3,..., jV;
strike out all the numbers divisible by a, then those divisible by b, then those divisible
by c, and so on ; there will remain only the numbers prime to N. For actually
finding the numbers we may of course in striking out those divisible by b disregard
the numbers already struck out as divisible by a, and in striking out with respect to
c disregard the numbers already struck out as divisible by a or 6, and so on; but
in order to count the remaining numbers it is more convenient to ignore the previous
strikings out. Suppose, for a moment, there are only two prime factors a and b, then
the number of terms struck out as divisible by a is = N . - , and the number of
a
terms struck out as divisible by b is =N.j-; but then each term divisible by ab will
have been twice struck out; the number of these is =N.—r, and thus the number
ab
of the remaining terms is Nil --- r + -Tli which is =N(l -- ) ( 1 — r ) . By
V a b ab] \ a)\ b)
treating in like manner the case of three or more prime factors a, 6, c, . . . we arrive at the
general theorem. The formula gives <£ (1) = 1 viz. when N = 1, there is no factor 1 -- ;
Gb
and it is necessary to consider <£ (1) as being = 1. The explanation is that <j>(N)
75—2
596 NUMBERS. [795
properly denotes the number of integers not greater than N and prime to it; so that,
when N = l, we have 1 an integer not greater than N and prime to it; but in every
other case the two definitions agree.
6. If N, N', are numbers prime to each other, then <£ (NNf) = </> (N) <j> (N'\ and
so also for any number of numbers having no common divisor; in particular,
and the theorem is at once verified. We have N = ^<f> (N'\ where the summation
extends to all the divisors N' of N, unity and the number N itself being included ; thus
15 = <f> (15) + <f> (5) + <f> (3) + <£> (1), =8 + 4+2 + 1.
7. The prime factor of the binomial function xN—l is
l)(a^/» -1)...'
a rational and integral function of the degree <j>(N)', sav tnig ^s ca^ed [a^— 1], and
we have OCN - 1 = II [aF - 1], where the product extends to all the divisors N' of J\T,
unity and the number N included. For instance
r\»
— 1)
and we have
tf15 - 1 = |>15 - 1] [a* - 1] [a* - 1] 0 - 1],
8. Congruence to a given modulus. A number so is congruent to 0, to the
modulus N, # = 0(mod. N), when as is divisible by N\ two numbers x, y are congruent
to the modulus N, x = y (mod. N), when their difference x — y divides by N, or, what is
the same thing, if x - y = 0 (mod. N). Observe that, if xy = 0 (mod. N), and x be prime
to N, then y = 0 (mod. N).
9. Residues to a given modulus. For a given modulus N we can always find,
and that in an infinity of ways, a set of N numbers, say N residues, such that every
number whatever is, to the modulus N, congruent to one and only one of these
residues. For instance, the residues may be 0, 1, 2, 3,..., N— 1 (the residue of a
given number is here simply the positive remainder of the number when divided by
N) ; or, ^V being odd, the system may be
0, ±1, ±2,..., ±i(JV-l),
and N even,
0, ±1, ±2,..., +1(^-2), +$N.
10. Prime residues to a given modulus. Considering only the numbers which are
prime to a given modulus N, we have here a set of <j> (N) numbers, say <f>(N) prime
residues, such that every number prime to N is, to the modulus N, congruent to
795] NUMBERS. 597
one and only one of these prime residues. For instance, the prime residues may be
the numbers less than N and prime to it. In particular, if N is a prime number p,
then the residues may be the p — I numbers, 1, 2, 3, ...,p — 1.
In all that follows, the letter p, in the absence of any statement to the contrary,
will be used to denote an odd prime other than unity. A theorem for p may hold
good for the even prime 2, but it is in general easy to see whether this is so or not.
11. Fermat's theorem, 0P-1— 1 = 0 (mod.^). The generalized theorem is «*W — 1 = 0
(mod. N). The proof of the generalized theorem is as easy as that of the original
theorem. Consider the series of the <f>(N) numbers a, b, c, ..., each less than N and
prime to it ; let x be any number prime to N, then each of the numbers xa, xb, xc, . . . ,
is prime to N, and no two of them are congruent to the modulus N, that is, we
cannot have x (a — b) = 0 (mod. N) ; in fact, x is prime to N, and the difference a — b
of two positive numbers each less than N will be less than N. Hence the numbers
xa, xb, xc, ..., are in a different order congruent to the numbers a, b, c, ... ; and
multiplying together the numbers of each set we have a&W abc ... = abc ... (mod. N), or,
since a, b, c, ..., are each prime to N, and therefore also the product abc... is prime
to N, we have «* W = 1, or say gfi W - 1 = 0 (mod. N).
In particular, if N be a prime number =p, then (f>(N) is =p — l, and the
theorem is x?~l — 1 = 0 (mod. p), x being now any number not divisible by p.
12. The general congruence f(x) = 0 (mod. p). f(x) is written to denote a rational
and integral function with integer coefficients which may without loss of generality
be taken to be each of them less than p ; it is assumed that the coefficient A of the
highest power of x is not = 0. If there is for x an integer value a such that
f(a) = 0 (mod. p, throughout), then a is said to be a root of the congruence f (x} = 0 ;
we may, it is clear, for a substitute any value whatever a' = a + kp, or say any value
a which is = a, but such value a' is considered not as a different root but as the same
root of the congruence. We have thus /(a) = 0 ; and therefore f(x) =f(x) —f(a},
= (x — a)yi (x), where /j (x) is a function of like form with f(x), that is, with integer
coefficients, but of the next inferior order n—1. Suppose there is another root b of the
congruence, that is, an integer value b such that f(b) = 0 ; we have then (b — a)/i (b) = 0,
and b — a is not = 0 (for then b would be the same root as a). Hence /j (b) = 0, and
f(x) = (x — a) {fi(®)—fi(b)}, =(x — a)(x — b}fz(x\ where /2(#) is an integral function
such as f(x), but of the order n — 2 ; and so on, that is, if there exist n different
(non-congruent) roots of the congruence f(x) = 0, then f(x) = A (x — a)(x—b) ... (x—k),
and the congruence may be written A(x — a}(x — b) ...(x — k) = 0. And this cannot be
satisfied by any other value I ; for if so we should have A (I — a) (I — b) ...(l—k) = Q,
that is, some one of the congruences (I — a) = 0, &c., would have to be satisfied, and
I would be the same as one of the roots a, b, c, ..., k. That is, a congruence of the
order n cannot have more than n roots, and if it have precisely n roots a, b, c, ..., k,
then the form is / (x) = A (x — a) (x — b) . . . (x — k), = 0.
Observe that a congruence may have equal roots, viz. if the form be
598 NUMBERS. [795
then the roots a, b, ... are to be counted a times, /3 times, . . . respectively ; but clearly
the whole number of roots a + /3 + . . . is at most = n.
It is hardly necessary to remark that this theory of a congruence of the order
n is precisely analogous to that of an equation of the order n, when only real roots
are attended to. The theory of the imaginary roots of a congruence will be considered
further on (see No. 41).
13. The linear congruence ax = c (mod. b). This is equivalent to the indeterminate
equation ax+by = c; if a and b are not prime to each other, but have a greatest
common measure q, this must also divide c ; supposing the division performed, the
equation becomes a'x + b'y = c', where a and b' are prime to each other, or, what is the
same thing, we have the congruence a'x = c' (mod. b'). This can always be solved, for,
if we consider the b' numbers 0, 1, 2, ..., b' — I, one and only one of these will be
= c'(mod. b'). Multiplying these by any number a' prime to b', and taking the remainders
in regard to b', we reproduce in a different order the same series of numbers
0, 1, 2, ..., b' — 1 ; that is, in the series a', 2a', ..., (b' — I) a' there will be one and
only one term = c' (mod. 'b'), or, calling the term in question o, we have x = a as
the solution of the congruence a'x = c' (mod. b') ; a'a — c' is then a multiple of b', say
it is = — Z//3, and the corresponding value of y is y = /3. We may for a. write a. + mb',
m being any positive or negative integer, not excluding zero (but, as already remarked,
this is not considered as a distinct solution of the congruence); the corresponding value
of y is clearly = ft — ma'.
The value of x can be found by a process similar to that for finding the greatest
common measure of the two numbers a' and c'; this is what is really done in the
apparently tentative process which at once presents itself for small numbers, thus
6# = 9 (mod. 35), we have 36«= 54, or, rejecting multiples of 35, x=l9, or, if we please,
19.
In particular, we can always find a number £ such that a'g = 1 (mod. b') ; and we
have then x = c'% as the solution of the congruence a'x = c'. The value of £ may be
written £ = — (mod. V), where - stands for that integer value £ which satisfies the
original congruence a'f=l (mod. b'); and the value of x may then be written x=C— (mod. b').
C—
Lv
Another solution of the linear congruence is given in No. 21.
14. Wilson's theorem, 1 . 2 . 3 ...p- 1 + 1 = 0(mod.j9). It has been seen that, for
any prime number p, the congruence a^-1 -1 = 0 (mod. p) of the order p - 1 has the
p — 1 roots 1, 2, . . . , p — 1 ; we have therefore
xP-i - 1 = (x - 1) (x- 2) . . . (x -p~-l),
or, comparing the terms independent of x, it appears that 1. 2. 3. ..£>-! = - 1, that is,
1 . 2. 3 ... p — 1 + 1 = 0(mod.p), — the required theorem. For instance, where p = 5, then
1.2.3.4 + 1 = 0 (mod. 5), and where p = 7, then 1.2.3.4.5.6+1 = 0 (mod. 7).
15. A proof on wholly different principles may be given. Suppose, to fix the
ideas, p = 7 ; consider on a circle 7 points, the summits of a regular heptagon, and join
795] NUMBERS. 599
these in any manner so as to form a heptagon ; the whole number of heptagons is
£.1.2.3.4.5.6. Now of these we have £(7 — 1), =3, which are regular heptagons
(convex or stellated); the number of remaining heptagons must be divisible by 7, for
with any one such heptagon we connect the 6 heptagons which can be obtained from it by
making it rotate through successive angles of f360°. That is, £. 1.2. 3.4.5. 6-£ (7 — 1) = 0
(mod. 7), whence 1.2.3.4.5.6-7 + 1 = 0, or finally 1.2.3.4.5.6 + 1 = 0 (mod. 7). It
is clear that the proof applies without alteration to the case of any prime number p.
If p is not a prime number, then 1.2.3...j)-l=0 (mod. p) ; hence the theorem
shows directly whether a number p is or is not a prime number; but it is not of
any practical utility for this purpose.
16. Prime roots of a prime number — application to the binomial equation acP— 1=0.
Take, for instance, p = 7. By what precedes, we have
of- 1 = [Xs - 1] [a? - 1] |>2 - 1] [>-l], = («2 - a>+ 1) (#2 + x + 1) (ff+ 1) (x - 1) ;
and we have
of- 1 = (x- l)(x- 2) (a; - 3) (a?- 4) (x- 5) (x-6) (mod. 7) ;
whence also
(x^-x + l}(x-+x + l}(x + l}(x-l} = (x-l}(x-'2l)(x-^}(x-^)(x-5)(x-Q}.
These two decompositions must agree together, and we in fact have
x* -a:+l=(tK- 3) (#-5), xz + x + 1 = (as - 2) (as- 3), x + l = os-6, x-l=x-l.
In particular, we thus have 3, 5, as the roots of the congruence x2 — x + 1 = 0, that is,
[x6— 1] = 0, and these roots 3, 5, are not the roots of any other of the congruences
[y-l] = 0, [>2-l] = 0, [« — 1] = 0; that is, writing a = 3 or 5 in the series of numbers
a, a2, a3, a4, a5, a6, we have a6 as the first term which is = 1 (mod. 7) ; the series in
fact are
3, 9 , 27 , 81 , 243 , 729 =3, 2, 6, 4, 5, 1,
5, 25, 125, 625, 3125, 15625 = 5, 4, 6, 2, 3, 1.
And so, in general, the congruence xP~l -1=0 (mod. p) has the p - 1 real roots
1, 2, 3, ...,p-l ; hence the congruence [x?-1 - 1] = 0, which is of the order tf>(p — l), has
this number <f)(p — I) of real roots; and, calling any one of these g, then in the series
of powers g, g2, g3, ..., <JP~I, the first term which is = 1 (mod. p) is ff^~\ that is, we have
g, g\ g3, ..., gp~l = l, 2, 3, ..., p-l in a different order. Any such number g is said
to be a prime root of p, and the number of prime roots is </> (p — 1), the number of
integers less than and prime to p — 1.
The notion of a prime root was applied by Gauss to the solution of the binomial
equation ^ — 1=0, or, what is the same thing, to the question of the division of
the circle (Kreistheilung), see Equation, Nos. 30 and 31, [786] ; and, as remarked in
the introduction to the present article, the roots or periods of roots of this equation
present themselves as the units of a complex theory in the Theory of Numbers.
600 NUMBERS. [795
17. Any number x less than p is = gm, and, if m is not prime to p—l, but
has with it a greatest common measure e, suppose m = ke, p — l=ef, then
x = gke, x* = gkef = gk (p~1} = 1 ,
that is, x? = 1 ; and it is easily seen that in the series of powers x, x*, ..., xf, we have
xf as the first term which is = 1 (mod. p}. A number = gm, where m is not prime to
p — l, is thus not a prime root; and it further appears that, g being any particular
prime root, the <j> (p — 1) prime roots are = the numbers gm, where m is any number
less than p — l and prime to it. Thus in the foregoing example p = 7, where the
prime roots were 3 and 5, the integers less than 6 and prime to it are 1, 5 ; and
we, in fact, have 5 = 35 and 3 = 55 (mod. 7).
18. Integers belonging to a given exponent ; index of a number. If, as before,
p — l=ef, that is, if f be a submultiple of p — l, then any integer x such that xf
is the lowest power of x which is = 1 (mod. p) is said to belong to the exponent f.
The number of residues, or terms of the series 1, 2, 3, .... p — 1, which belong to the
exponent / is <j> (/), the number of integers less than / and prime to it ; these are
the roots of the congruence \acf- 1] = 0 of the order </>(/). It is hardly necessary to
remark that the prime roots belong to the exponent p — 1.
A number x = gm is said to have the index m; observe the distinction between
the two terms exponent and index ; and, further, that the index is dependent on the
selected prime root g.
19. Special forms of composite modulus. If instead of a prime modulus p we
have a modulus pm which is the power of an odd prime, or a modulus 2p or 2pm
which is twice an odd prime or a power of an odd prime, then there is a theory
analogous to that of prime roots, viz. the numbers less than the modulus and prime
to it are congruent to successive powers of a prime root g] thus,
if p™ = S2, we have
2, 4, 8, 16, 32, 64 = 2, 4, 8, 7, 5, 1 (mod. 9),
and if 2pm= 2 . 32, we have
5, 25, 125, 625, 3125, 15625 = 5, 7, 11, 13, 17, 1 (mod. 18).
As regards the even prime 2 and its powers — for the modulus 2 or 4 the theory of
prime roots does not come into existence, and for the higher powers it is not
applicable ; thus with modulus = 8 the numbers less than 8 and prime to it are
1, 3, 5, 7 ; and we have 32 = 52=72 = ] (mod. 8).
20. Composite modulus N= aab^cy... — no prime roots — irregularity. In the general
case of a composite modulus it has been seen that, if x is any number less than N
and prime to it, then #* W — 1 = 0 (mod. N). But, except in the above-mentioned cases
pm, 2pm, 2 or 4, there is not any number a such that a*^ is the first power of a
which is = 1 ; there is always some submultiple i = ^(j) (N) such that a* is the first
795] NUMBERS. 601
power which is = 1. For instance, say N = 24, <f> (N) = 8, then the numbers less than
24 and prime to it are 1, 5, 7, 11, 13, 17, 19, 23; and we have
I1 = 1, 52 = 72 = 132 = 172 = 192 = 23a = 1 (mod. 24),
that is, 1 has the exponent 1, but all the other numbers have the exponent 2. So-
again where A" = 48, the 16 numbers less than 48 and prime to it have, 1 the
exponent 1, and 7, 13, 17, 23, 25, 31, 35, 41, 47 each the exponent 2, and the
remaining numbers 5, 11, 19, 29, 37, 43 each the exponent 4. We cannot in this case
by means of any single root or of any two roots express all the numbers, but we
can by means of three roots, for instance, 5, 7, 13, express all the numbers less than
48 and prime to it ; the numbers are in fact = 5a7*13*, where a = 0, 1, 2, or 3, and @
and 7 each =0 or 1.
Comparing with the theorem for a prime number p, where the several numbers
1, 2, 3, ..., p — 1, are expressed by means of a single prime root, =ga, where a=0, 1, 2, ...,p— 1,
we have the analogue of a case presenting itself in the theory of quadratic forms, —
the "irregularity" of a determinant (post, No. 31); the difference is that here (the
law being known, N= a composite number) the case is not regarded as an irregular
one, while the irregular determinants do not present themselves according to any apparent
law.
21. Maximum indicator — application to solution of a linear congruence. In the
case JV=48 it was seen that the exponents were 1, 2, 4, the largest exponent 4 being
divisible by each of the others, and this property is a general one, viz. if N=aabPcv...
in the series of exponents (or, as Cauchy calls them, indicators) of the numbers
less than N and prime to it, the largest exponent / is a multiple of each of the
other exponents, and this largest exponent Cauchy calls the maximum indicator; the
maximum indicator / is thus a submultiple of </>(JV), and it is the smallest number
such that for every number *• less than N and prime to it we have oc1 - 1 = 0 (mod. N}.
The values of / have been tabulated from N=2 to 1000.
Reverting to the linear congruence ax = c (mod. 6), where a and b are prime to
each other, then, if / is the maximum indicator for the modulus 6, we have az=l,
and hence it at once appears that the solution of the congruence is x = caf~l.
22. Residues of powers for an odd prime modulus. For the modulus p, if g be
a prime root, then every number not divisible by p is = one of the series of numbers
ff> 92> •••> SrP~1 5 ancl if k be any positive number prime to p - 1, then raising each of
these to the power k we reproduce in a different order the same series of numbers
g, g'\ ..., gp~l, which numbers are in a different order =1, 2, ..., p— 1, that is, the
residue of a kih power may be any number whatever of the series 1, 2, ..., p — 1.
But, if A; is not prime to p — 1, say their greatest common measure is e, and
that we have p — 1 = ef, k = me, then for any number not divisible by p the kth power
is = one of the series of/ numbers ge, g*e, ..., gfl; there are thus only / = -(p-l)f
C/
out of the p— 1 numbers 1, 2, 3, ..., p-l, which are residues of a kth power.
C. XI. 76
602 NUMBERS. [795
23. Quadratic residues for an odd prime modulus. In particular, if k = 2, then
e= 2, f= ^(p — 1), and the square of every number not divisible by p is = one of the
^(p — 1) numbers g2, g4, . . . , g'p~l ; that is, there are only $(p— 1) numbers out of the
series 1, 2, 3, ...,p — 1 which are residues of a square number, or say quadratic residues,
and the remaining ^(p — 1) numbers are said to be quadratic non-residues of the
modulus p, — we may say simply, residues and non-residues. But this result can be
obtained more easily without the aid of the theory of prime roots. Every number not
divisible by p is, to the modulus p, = one of the series of numbers +1, +2, ±3, ..., ±^(p— 1);
hence every square number is = one of the series of numbers I2, 22, 32, ..., £ (p — I)2 ;
and thus the p — I numbers 1, 2, 3,...,p—I, are one-half of them residues and the
other half non-residues of p. Thus, in the case p — Il, every number not divisible by
11 is, to this modulus, = one of the series ±1, +2, +3, +4, ± 5 ; whence the square
of any such number is = one of the series 1, 4, 9, 16, 25, or say the series 1, 4, 9, 5, 3;
that is, we have
residues 1, . 3, 4, 5, . . . 9, .
non-residues . 2, . . . 6, 7, 8, . 10
Calling as usual the residues a and the non-residues b, we have in this case
TV (26-2(1) = ^ (33 -22), =1,
a, positive integer ; this is a property true for any prime number of the form 4?i + 3,
but for a prime number of the form 4w + 1 we have 26 — 2a = 0 ; the demonstration
belongs to a higher part of the theory.
It is easily shown that the product of two residues or of two non-residues is a
residue ; but the product of a residue and a non-residue is a non- residue.
24. The law of reciprocity — Legendre's symbol. The question presents itself, given
that P is a residue or a non-residue of Q, can we thence infer whether Q is a
residue or a non-residue of P ? In particular, if P, Q, are the odd primes p, q, for
instance, given that 13 = jR(17), can we thence infer that 17 = _R (13), or that
17 = NR(13) ? The answer is contained in the following theorem: If p, q, are odd
primes each or one of them of the form 4w + l, then p, q, are each of them a residue
or each of them a non-residue of the other ; but, if p, q, are each of them of the
form 4n + 3, then, according as p is a residue or a non-residue of q, we have q a non-
residue or a residue of p.
The theorem is conveniently expressed by means of Legendre's symbol, viz. p being
a positive odd prime, and Q any positive or negative number not divisible by p, then
(-] denotes +1 or —1, according as Q is or is not a residue of p ; if, as before, q
is (as p) a positive odd prime, then the foregoing theorem is
qj \p
795] NUMBERS. 603
The denominator symbol may be negative, say it is — p, we then have as a
definition ( -—] = (--] — observe that ( — ~\ is not =(-- ) — and we have further the
\-p/ \pJ \ p I \—pJ
theorems
'=!).(- 1)10^, (?) = (_ i)^-!),
viz. — 1 is a residue or a non-residue of p according as p = 1 or =3 (mod. 4), and 2
is a residue or a non-residue of p according as p = 1 or 7, or =3 or 5 (mod. 8). If,
/• V \ fl>\
as definitions, - -) = + ! and (^)= + l, these may be written
V — I/ \^/
/_ l\ f n \ , /2\ fn\
I —\ " — ^_1^(J>— i) finrl l — lft-l — ( 1 M (P'~ i)
II Tf \ 1/" ' cillu I I loJ — \ A;
\ p J V— I/ \|>/ \2/
We have also, what is in fact a theorem given at the end of No. 23,
= (Q
\ P J \p \p
The further definition is sometimes convenient —
* = 0, when p divides 0.
\pf
The law of reciprocity, as contained in the theorem
P\
/
q/ \p
is a fundamental theorem in the whole theory; it was enunciated by Legendre, but
first proved by Gauss, who gave no less than six demonstrations of it.
25. Jacobi's generalized symbol Jacobi defined this as follows: The symbol
( -- -* ---- ] where p, p', p", . . . are positive odd primes equal or unequal, and Q is any
\±pp'p' ...J
positive or negative odd number prime to pp'p"..., denotes +1 or —1 according to the
definition
/
±PP'p"..~p\p'p
the symbols on the right-hand side being Legendre's symbols. But the definition may
be regarded as extending to the case where Q is not prime to pp'p"-.- : then we have
Q divisible by some factor p, and by the definition of Legendre's symbol in this case
we have ( -1 = 0 : hence in the case in question of Q not being prime to pp'p"...,
\pJ
the value of Jacobi's symbol is = 0.
We may further extend the definition of the symbol to the case where the
numerator and the denominator of the symbol are both or one of them even, and
present the definition in the most general form, as follows : suppose that p, p, p", . . .
76—2
604 NUMBERS. [795
being positive or negative even or odd primes, equal or unequal, and similarly
<l> 9'> <l"> • • • being positive or negative even or odd primes, equal or unequal, we have
P=pp'p"... and Q = qq'q"..., then the symbol f will denote + 1, -1, or 0, according
to the definition
(9\-(t\(sh(£\ /
\f) (P) (P') \P") '"(P P
the symbols on the right-hand being Legendre's symbols. If P and Q are not prime
to each other, then for some pair of factors p and q we have p — ± q, and the corre
sponding Legendrian symbol f -J is =0, whence in this case (~\ = 0.
It is important to remark that ( = + 1 is not a sufficient condition in order
that Q may be a residue of P ; if P = 2app'p"... , p, p', p",... being positive odd
primes, then, in order that Q may be a residue of P, it must be a residue of each
of the prime factors p, p', p", ..., that is, we must have
Q\_ , T Q\ , , Q
- *'
as many equations as there are unequal factors p, p', p", ... of the modulus P.
Ordinary Theory, Second Part, — Theory of Forms.
26. Binary quadratic (or quadric) forms — transformation and equivalence. We
consider a form
ax- + 2bxy + cy2, = (a, b, c) (x, y}~,
or when, as usual, only the coefficients are attended to, = (a, b, c). The coefficients
(a, b, c) and the variables (x, y) are taken to be positive or negative integers, not
excluding zero. The discriminant ac - 62 taken negatively, that is, b2 - ac, is said to be
the determinant of the form: and we thus distinguish between forms of a positive
and of a negative determinant.
Considering new variables, ax + fiy, yx + By, where a, 0, y, 8, are positive or negative
integers, not excluding zero, we have identically
(a, b, c)((uc + @y, yx + fy)2 = (a, b', c'}(x, y)2,
where
a' = (a, b, c) (a, y)-, = aar + 2bay + cy2,
b' = (a, b, c) (a, 7) (£, B), =aa@ + b (aS + £7) + cy8,
c = (a, b, c) (0, S)2, = a/32 + 26/3S + cS2 ;
and thence
6/a - a'c = (aS - /37)2 (b- - ac).
The form (a', b', c') is in this case said to be contained in the form (a, b, c);
and a condition for this is obviously that the determinant D' of the contained form
795] NUMBERS. 605
shall be equal to the determinant D of the containing form multiplied by a square
number; in particular, the determinants must be of the same sign. If the determ
inants are equal, then (aS — /fy)2 = 1, that is, a8 — j3y=±l. Assuming in this case
that the transformation exists, and writing aS — fty = e, and writing also
then conversely
x = CM; + fty,
By,
:=( Sx'-/3y'), = a'x'
suppose, where a', /3', 7', 8' are integers ; and we have, moreover,
~a-/7, = , = e,
that is, «'&' — /3Y = + 1 or — 1, according as a& — /3y is = + 1 or — 1. The two forms
(a, 6, c), (a', 6', c') are in this case said to be equivalent, and to be, in regard to the
particular transformation, equivalent properly or improperly according as aS—/3y(=a.'S'—/3'y)
is = + 1 or = — 1. We have, therefore, as a condition for the equivalence of two forms,
that their determinants shall be equal ; but this is not a sufficient condition. It is
to be remarked also that two forms of the same determinant may be equivalent
properly and also improperly ; there may exist a transformation for which a.8 — @y is
= +1, and also a transformation for which otS — (3y is =— 1. But this is only the
case when each of the forms is improperly equivalent to itself; for instance, a form
x2 — Dy2, which remains unaltered by the change x, y, into x, —y (that is, a, @, 7, 5 = 1,
0, 0, — 1, and therefore otS — $7 = — 1), is a form improperly equivalent to itself. A
form improperly equivalent to itself is said to be an ambiguous form. In what follows,
equivalent means always properly equivalent.
27. Forms for a given determinant — classes, &c. In the case where D, = b- — ac,
is a square, the form (a, b, c) (x, y)2 is a product of two rational factors ; this case
may be excluded from consideration, and we thus assume that the determinant D is
either negative, or, being positive, that it is not a square. The forms (a, b, c) of a
given positive or negative determinant are each of them equivalent to some one out
of a finite number of non-equivalent forms which may be considered as representing
so many distinct classes. For instance, every form of the determinant — 1 is equivalent to
(1, 0, 1), that is, given any form (a, b, c) for which 62 — ac = — 1, it is possible to find integer
values a, @, 7, S, such that «S - /3y = + 1, and (a, b, c) (oar+ 0y, yx + Sy)° = (1, 0, 1) (as, yj*,
that is, = x2 + y-. Or, to take a less simple example, every form of the determinant
— 35 is equivalent to one of the following forms: (1, 0, 35), (5, 0, 7), (3, + 1, 12),
(4, ±1, 18), — (2, 1, 8), (6, 1, 6); for the first six forms, the numbers a, 26, c have no
common factor, and these are said to be properly primitive forms, or to belong to the
properly primitive order; for the last two forms, the numbers a, b, c have no common
factor, but, a and c being each even, the numbers a, 26, c have a common factor 2,
606 NUMBERS. [795
and these are said to be improperly primitive forms, or to belong to the improperly
primitive order. The properly primitive forms are thus the six forms (1, 0, 35),
(5, 0, 7), (3, ±1, 12), (4, ±1, 18); or we may say that there are represented hereby
six properly primitive classes. Derived forms, or forms which belong to a derived
order, present themselves in the case of a determinant D having a square factor or
factors, and it is not necessary to consider them here.
It is not proposed to give here the rules for the determination of the system
of non-equivalent forms ; it will be enough to state that this depends on the determ
ination in the first instance of a system of reduced forms, that is, forms for which
the coefficients a, b, c, taken positively satisfy certain numerical inequalities admitting
only of a finite number of solutions. In the case of a negative determinant, the
reduced forms are no two of them equivalent, and we thus have the required system
of non-equivalent forms ; in the case of a positive determinant, the reduced forms
group themselves together in periods in such wise that the forms belonging to a
period are equivalent to each other, and the required system of non-equivalent forms
is obtained by selecting one form out of each such period. The principal difference in
the theory of the two cases of a positive and a negative determinant consists in these
periods ; the system of non-equivalent forms once arrived at, the two theories are nearly
identical.
28. Characters of a form or class — division into genera. Attending only to the
properly primitive forms : for instance, those mentioned above for the determinant
— 35: the form (1, 0, 35) represents only numbers f which are residues of 5, and also
residues of 7; we have, in fact, f=<v2 + %oy'2, = #2 (mod. 5), and also ^^(mod. 7).
Using the Legendrian symbols (4) and (4 ) » we sa.v that the form (1, 0, 35) has the
\o / \7 /
characters (•- ) , (•£• ] = + + . Each of the other forms has in like manner a determinate
\9/ \TJ
character -f or — in regard to r— ] and also in regard to ( =• ) ; and it is found that
\5/ \7 /
for each of them the characters are + + or else — — (that is, they are never + —
or — -f). We, in fact, have
(A
WV7
(1, 0, 35) + +
(4, ±1, 9)
(5, 0, 7)
(3, ± 1, 12)
and we thus arrange the six forms into genera, viz. we have three forms belonging
to the genus [4-j, (<4) = + +, and three to the genus r4), ( "L } = , these characters
v* / \ • / V 5 / \ 7 /
+ + and of genera being one-half of all the combinations ++, , H — , — K
The like theory applies to any other negative or positive determinant ; the several
characters have reference in some cases not only to the odd prime factors of D but
795] NUMBERS. 607
also to the numbers 4 and 8, that is, there is occasion to consider also the Legendrian
symbols f--i =(-l)i(/~1', and (-., = (- 1)*'/3-1', and there are various cases to be
considered according to the form of D in regard to its simple and squared factors
respectively; but in every case there are certain combinations of characters (in number
one-half of all the combinations) which correspond to genera, and the properly primitive
forms belong to different genera accordingly, the number of forms being the same in
each genus.
The form (1, 0, — D) has the characters all + , and this is said to be the principal
form, and the genus containing it the principal genus. For a given determinant, the
characters of two genera may be compounded together according to the ordinary rule
of signs, giving the characters of a new genus; in particular, if the characters of a
genus are compounded with themselves, then we have the characters of the principal
genus.
29. Composition of quadratic forms. Considering X, Y, as given lineo-linear
functions of (x, y), (x, y'}, defined by the equations
X = p»xx'
Y = q0xx + q,xy' + q.yx + q3yy'
the coefficients p0, plt p.,, p3, q0, qlt q.2j q3) may be so connected with the coefficients
(A, B, C), (a, b, c), (of, b', c'), of three quadratic forms as to give rise to the identity
(A, B, C)(X, F)> = («, b, c)(x, y?.(a', b', c}(x', yj-
and, this being so, the form (A, B, C} is said to be compounded of the two forms
(«, b, c) and (a, b', c'), the order of composition being indifferent.
The necessary and sufficient condition, in order that it may be possible to com
pound together two given forms (a, b, c), (a', b', c'), is that their determinants shall
be to each other in the proportion of two square numbers ; in particular, the two
forms may have the same determinant D ; and when this is so the compound form
(A, B, C) will also have the same determinant D. The rules for this composition of
two forms of the same determinant have been (as part of the general theory) investig
ated and established. The forms compounded of equivalent forms are equivalent to
each other ; we thus in effect compound classes, viz. considering any two classes, the
composition of their representative forms gives a form which is the representative of
a new class, and the composition of any two forms belonging to the two classes
respectively gives a form belonging to the new class. But, this once understood, it is
more simple to speak of the composition of forms, that is, of the forms belonging to
the finite system of representative forms for a given determinant : and it will be
enough to consider the properly primitive forms.
30. The principal form (1, 0, D), compounded with any other form (a, b, c), gives
rise to this same form (a, b, c) ; the principal form is on this account denoted by 1,
viz. denoting the other form by <£, and expressing composition in like manner with
608 NUMBERS. [795
multiplication, we have 1 . <£ = <f>. The form (f> may be compounded with itself, giving
a form denoted by </>- ; compounding this again with <£, we have a form denoted
by </>3 ; and so on. Since the whole number of forms is finite, we must in this
manner arrive at the principal form, say we have </>7l = l, n being the least exponent
for which this equation is satisfied. In particular, if the form </> belong to the
principal genus, then the forms </>2, <£3, . . . , (f>il~l will all belong to the principal genus,
or the principal genus will include the forms 1, <f>, </>2, ..., <f>n~\ the powers of a form
<£ having the exponent n.
31. Regular and irregular determinants. The principal genus may consist of such
a series of forms, and the determinant is then said to be regular; in particular, for
a negative determinant D, = — 1 to —1000, the determinant is always regular except
in the thirteen cases - D = 243, 307, 339, 459, 576, 580, 675, 755, 820, 884, 891, 900,
974 (and, Perott, in Crelle, vol. xcv., 1883, except also for -D = 468, 931); the
determinant is here said to be irregular. Thus for each of the values — D = 576, 580,
820, 900, the principal genus consists of four forms, not 1, <f>, </>2, <£3, where (/>4=1, but
1, </>, </>!, <£</>i, where <fr=l, </>i2=l, and therefore also (<j><f>1}-=l.
Compounding together any two forms, we have a form with the characters com
pounded of the characters of the two forms ; and in particular, combining a form
with itself, we have a form with the characters of the principal form. Or, what is
the same thing, any two genera compounded together give rise to a determinate genus,
viz. the genus having the characters compounded of the characters of the two genera ;
and any genus compounded with itself gives rise to the principal genus.
Considering any regular determinant, suppose that there is more than one genus,
and that the number of forms in each genus is —n; then, except in the case n = 2,
it can, be shown that there are always forms having the exponent 2w. For instance,
in the case D = — 35, we have two genera each of three forms ; there will be a form
g having the exponent 6, or cf° = I; and the forms are 1, g, g2, gs, g4, g5, where 1, g2, g4,
belong to the principal genus, and g, gs, g5, to the other genus. The characters refer
t° ( r I 1 H } > and the forms are
\5/ \7/
+ +, (1, 0, 35) 1 - -, (3, - 1, 12) g
(4, 1, 9)
(4, -1, 9)
(5, 0, 7) g*
(3, 1, 12) g5.
An instance of the case n = 1 is D = — 21, there are here four genera each of a single
form 1, c, d, cci, where c2 = l, Cf = 1 ; an instance of the case n = 2 is D=-88, there
are here two genera each of two forms 1, c, and c1} ccl} where c2=l, c^ = l, thus
there is here no form having the exponent 2?i. (See Cayley, Tables, &c., in Crelle,
t. LX., 1862, pp. 357—372, [335].) We may have 2*+1 genera, each of n forms, viz. such
a system may be represented by (1, </>2, ..., <£2'l~2; <£, <£3,..., ^^(l, c) (1, c,)...(l, c^),
where </>>2n = 1, c2 = l, cr = 1, ..., Cst_i=l; there is no peculiarity in the form <j>: we may
instead of it take any form such as c<f>, cc^, &c., for each of these is like <£, a form
belonging to the exponent 2?i, and such that the even powers give the principal genus.
795] NUMBERS. 609
32. Ternary and higher quadratic forms — cubic forms, &c. The theory of the
ternary quadratic forms
(a, b, c, a', b', c') (x, y, z)2, = ax1- + by2 + cz- + 2a'yz + 2b'zx + 2c'xy,
or when only the coefficients are attended to, ( ' / . ) , has been studied in a very
\a , o , c I J
complete manner; and those of the quaternary and higher quadratic forms have also
been studied ; in particular, the forms x2 + y2 + z2, a? + y2 + z2 + w2 composed of three or
four squares; and the like forms with five, six, seven, and eight squares. The binary
cubic forms (a, b, c, d) (x, y)3, = ax3 + 3bx2y + Sexy" + dy3, or when only the coefficients
are attended to, (a, b, c, d), have also been considered, though the higher oinary forms
have been scarcely considered at all. The special ternary cubic forms ax3+by3+cz3+6lxyz
have been considered. Special forms of the degree n with n variables, the products
of linear factors, present themselves in the theory of the division of the circle (the
Kreistheilung) and of the complex numbers connected therewith ; but it can hardly be
said that these have been studied as a part of the general theory of forms.
Complex Theories.
33. The complex theory which first presented itself is that of the numbers a + bi
composed with the imaginary i, = V — 1 ; here if a and b are ordinary, or say simplex
positive or negative integers, including zero, we regard a + bi as an integer number,
or say simply as a number in this complex theory. We have here a zero 0 (a = 0,
6 = 0) and the units 1, i, -I, -i, or as these may be written, 1, i, i2, i3 (^=1); the
numbers a + bi, a - bi, are said to be conjugate numbers, and their product (a + bi)(a - bi),
= a2 + b2, is the norm of each of them. And so the norm of the real number a is
= a2, and that of the pure imaginary number bi is = b-. Denoting the norm by the
letter JV, JV(a + bi) = a2 + fr.
Any simplex prime number, = 1 (mod. 4), is the sum of two squares a? + b2, for
instance 13 = 9 + 4, and it is thus a product (a + bi) (a — bi), that is, it is not a prime
number in the present theory, but each of these factors (or say any number a + bi,
where d- + b2 is a prime number in the simplex theory) is a prime; and any simplex
prime number, = 3 (mod. 4), is also a prime in the present theory. The number
2, =(l+i)(l—i), is not a prime, but the factors I+i, l—i are each of them prime;
these last differ only by a unit factor i — l=i(l+i)— so that 2, = -i(l+i)2, contains
a square factor.
In the simplex theory we have numbers, for instance 5, — 5, differing from each
other only by a unit factor, but we can out of these select one, say the positive
number, and attend by preference to this number of the pair. It is in this way —
viz. by restricting a, b, c, ... to denote terms of the series 2, 3, 5, 7, ... of positive
primes other than unity — that we are enabled to make the definite statement, a
positive number N is, and that in one way only, = a^tf . . . ; if N be a positive or
negative number, then the theorem of course is, N is, and that in one way only,
= (- l)m aWc? . . . , where m = 0, or 1, and a, b, c,..., a, /3, 7,... are as before. To
C. XI. 77
610 NUMBERS. [795
obtain a like definite statement in the present theory, we require to distinguish
between the four numbers a+bi, -a-bi, -b + ai, b - ai, which differ from each other
only by a unit factor - 1, ± i. Consider a number a + bi where a and b are the one
of them odd and the other even (a and b may be either of them =0, the other is
then odd), every prime number a + bi other than ± 1 ± i is necessarily of this form :
for if a and b were both even, the number would be divisible by 2, or say by (1 + 1)2,
and if a and b were both odd, it would be divisible by l + i; then of the four
associated numbers a + bi, -a-bi, -b + ai, b-ai, there is one and only one, a + bi,
such that b is even and a + b - I is evenly even ; or say one and only one which
is = 1 (mod. 2 (l+i)). We distinguish such one of the four numbers from the other
three and call it a primary number ; the units + 1, ± i, and the numbers + 1 ± i, are
none of them primary numbers. We have then the theorem, a number N is in one
way only =im(l +i)nA*& .... where m = 0, 1, 2, or 3, n is =0 or a positive integer,
A, B,... are primary primes, a, /3, ... positive integers. Here i is a unit of the theory,
1 + i is a special prime having reference to the number 2, but which might, by an
extension of the definition, be called a primary prime, and so reckoned as one of the
numbers A, B,...; the theorem stated broadly still is that the number N is, and
that in one way only, a product of prime factors, but the foregoing complete state
ment shows the precise sense in which this theorem must be understood. A like
explanation is required in other complex theories; we have to select out of each set
of primes differing only by unit factors some one number as a primary prime, and
the general theorem then is that every number N is, and that in one way only,
= P.A*BI*CY..., where P is a product of unities, and A, B, C,... are primary primes.
34. We have in the simplex theory (ante, No. 10) the theorem that, p being an
odd prime, there exists a system of p - 1 residues, that is, that any number not
divisible by p is, to the modulus p, congruent to one, and only one, of the p — l
numbers 1, 2, 3,..., p — l. The analogous theorem in the complex theory is that, for
any prime number p other than ±l±i, there exists a system of N(p)-l residues,
that is, that every number not divisible by p is, to the modulus p, congruent to one
of these N(p) — l numbers.
But p may be a real prime such as 3, or a complex prime such as 3 + 2i ; and
the system of residues presents itself naturally under very different forms in the two
cases respectively. Thus in the case p = 3, N(3) = 9, the residues may be taken to be
1,2,
i, l+i , 2 + i ,
being in number ^(3) -1=8. And for p = 3 + 2i, JV(3 + 2i) = 13, they may be taken
to be the system of residues of 13 in the simplex theory, viz. the real numbers
1, 2, 3,..., 12. We have in fact 5 + i = (2 + 3i)(l -i), that is, 5 + is 0 (mod 2 + 3i), and
consequently a+bi = a-5b, a real number which, when a + bi is not divisible by 3 + 2i,
may have any one of the foregoing values 1, 2, 3,..., 12.
795] NUMBERS. 611
Taking then any number x not divisible by p, the N (p) — 1 residues each
multiplied by x are, to the modulus p, congruent to the series of residues in a
different order; and we thus have, — say this is Format's theorem for the complex
theory — xN(p]~l — 1 = 0 (mod. p), with all its consequences, in particular, the theory of
prime roots.
In the case of a complex modulus such as 3 + 2t, the theory is hardly to be
distinguished from its analogue in the ordinary theorem ; a prime root is = 2, and the
series of powers is 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, for the modulus 3 + 2i as for
the modulus 13. But for a real prime such as 3 the prime root is a complex
number; taking it to be =2+i, we have (2 + i)8 — 1 = 0 (mod. 3), and the series of
powers in fact is 2 + i, i, 2 + 2i, 2, 1 + 2i, 2i, 1 + i, 1, viz. we thus have the system
of residues (mod. 3).
We have in like manner a theory of quadratic residues ; a Legendrian symbol
(which, if p, q, are uneven primes not necessarily primary but subject to the
condition that their imaginary parts are even, denotes +1 or — 1 according as ptWQ-u
r«"i
is =1 or = — 1 (mod. q), so that = + 1 or — 1 according as p is or is not a
residue of q), a law of reciprocity expressed by the very simple form of equation
- = " , and generally a system of properties such as that which exists in the
simplex theory.
The theory of quadratic forms (a, b, c) has been studied in this complex theory;
the results correspond to those of the simplex theory.
35. The complex theory with the imaginary cube root of unity has also been
studied ; the imaginary element is here y, = \ (— 1 + v7 — 3), a root of the equation
T + y + 1 = 0 ; the form of the complex number is thus a + by, where a and b are
any positive or negative integers, including zero. The conjugate number is a + by2,
= 0-6-67, and the product (a + by) (a + by2), = a2-ab + b2, is the norm of each of the
factors a + by, a+ by2. The whole theory corresponds very closely to, but is somewhat
more simple than, that of the complex numbers a + bi.
36. The last-mentioned theory is a particular case of the complex theory for the
imaginary Xth roots of unity, X being an odd prime. Here a is determined by the
a* — 1
equation -——=0, that is, ox~1 + «A~a+ ... + a + 1 =0, and the form of the complex
number is /(a), = a + 6a + ca2 + ... + ka.*~2, where a, b, c, ..., k, are any positive or negative
integers, including zero. We have X, - 1 conjugate forms, viz. /(a), /(a2), ... , /(a*"1), and
the product of these is the norm of each of the factors Nf(a), =Nf(a?), =..., = Nf(<&-1}.
Taking g any prime root of X, g^~l -1=0 (mod. X), the roots a, a2, . . . , a*-1, may be
arranged in the order a, a?, aP", ..., a^~'2 ; and we have thence a grouping of the roots
in periods, viz. if X — 1 be in any manner whatever expressed as a product of two
factors, \—l=ef, we may with the X — 1 roots form e periods rjQ, 171,..., ye-\, each of
77—2
612 NUMBERS. [795
/ roots. For instance, when X = 13, a prime root is #=2, and X-l = e/"=3.4: then
the three periods each of four roots are
?70 = a + a8 + aia + a5,
Vl = oi2 + a3 + a11 + a10,
rj.-, = a4 + a" + a9 + a7 .
So also, if e/= 2 . 6, then the 2 periods each of 6 roots are
rj0 = a + a4 4- a3 + a12 + a9 + a10,
77! = a2 + a8 + a6 + a11 + a5 + a7 ;
and so in other cases. In particular, if /= 1 and consequently e = X — ] , the e periods
each of/ roots are, in fact, the single roots' a, a?,..., a^~2. We may, in place of the
original form of the complex number
f («) = a + ba + ca2 + . . . + A;ax~2,
consider the new form f(n) = arj + b^ +...+ ltje-i, which when /= 1 is equivalent to
the original form, but in any other case denotes a special form of complex number;
instead of X— 1 we have only e conjugate numbers, and the product of these e numbers
may be regarded as the norm of /(T/).
37. The theory for the roots a includes as part of itself the theory for the periods
corresponding to every decomposition whatever \ — l = ef of X — 1 into two factors, but
each of these may be treated apart from the others as a theory complete in itself.
In particular, a simple case is that of the half- periods e = 2, /=£(X-1); and,
inasmuch as the characteristic phenomenon of ideal numbers presents itself in this
theory of the half-periods (first for the value X = 23), it will be sufficient, by way of
illustration of the general theory, to consider only this more special and far easier
theory; we may even assume X=23.*
For the case in question, X- 1 =ef= 2 . |(X - 1), we have the two periods rj0, r)lt
each of |(X — 1) roots; from the expressions for rj0, TJI, in terms of the roots we obtain
at once 770 + 77! = — 1, and with a little more difficulty r)0r}1= — |(X — 1) or \ (X + 1),
according as X is =1 or 3 (mod. 4), that is, in the two cases respectively 770, 77^ are
the roots of the equation 772+ 77 - £(X - 1) = 0, and rj- + 77 + J (X + 1) = 0. And this
equation once obtained, there is no longer any occasion to consider the original equation
of the order X — 1, but the theory is that of the complex numbers ar]0 4- 6771, or
* In the theory of the roots a, ideal numbers do not present themselves for the values X = 3, 5, or 7 ;
they do for the value X=23. It is stated in Smith's "Beport on the Theory of Numbers," Brit. Assoc.
Report for 1860, p. 136, [Collected Works, vol. i. p. 114], that "for X = ll, X = 13, X = 17, and X = 19, it is
not possible to say whether this is or is not the case for these values also." The writer is not aware
whether this question has been settled; but in Reuschle's Tafeln, 1875, no ideal factors present themselves
for these values of X; and it is easy to see that, in the theory of the half-periods, the ideal factors first
present themselves for the value X = 23. It may be remarked that the solution of the question depends on
the determination of a system of fundamental units for the values in question X = ll, 13, 17, and 19; the
theory of the units in the several complex theories is an important and difficult part of the theory, not
presenting itself in the theory of the half-periods, which is alone attended to in the text.
795] NUMBERS. 613
if we please a + by, composed with the roots of this quadric equation, — say the com
plex numbers a + brj, where a and b are any positive or negative integer numbers,
including zero. In the case X=23, the quadric equation is 17* + 17 +6=0. We have
N (a + bij) = (a + br}0)(a + br]l) = a2-ab+$(\+l)b2; and for X = 23, this is N(a+bi))
= a? - ab + 662. It may be remarked that there is a connexion with the theory of the
quadratic forms of the determinant - 23, viz. there are here the three improperly
primitive forms (2, 1, 12), (4, 1, 6), (4, -1, 6), 23 being the smallest prime number
for which there exists more than one improperly primitive form.
38. Considering then the case X = 23, we have 770, vjlt the roots of the equation
772 + 77 + 6 = 0; and a real number P is composite when it is =(a + br)0)(a + bij1)t
= a2-ab + 6fr, viz. if 4P = (2a - 6)3 + 2363. Hence no number, and in particular no
positive real prime P, can be composite unless it is a (quadratic) residue of 23; the
residues of 23 are 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18; and we have thus, for instance,
5, 7, 11, as numbers which are not composite, while 2, 3, 13, are numbers which are
not by the condition precluded from being composite: they are not, according to the
foregoing signification of the word, composite (for 8, 12, 52, are none of them of the
form #2 + 23?/2), but some such numbers, residues that is of 23, are composite, for
instance 59, = (5 - 2%) (5 - 2^). And we have an indication, so to speak, of the com
posite nature of all such numbers; take for instance 13, we have (77 - 4) (77 + 5)=- 2. 13,
where 13 does not divide either 77-4 or 77 + 5, and we are led to conceive it as the
product of two ideal factors, one of them dividing 77-4, the other dividing 77 + 5. It
appears, moreover, that a power 13s is in fact composite, viz. we have
133 = (31 - 12770) (31 - 12770, (2197 = 961 + 372 + 864) ;
and writing 13 = v/31 - 12^. v/ST^l^ we have 13 as the product of two ideal
numbers each represented as a cube root; it is to be observed that, 13 being in the
simplex theory a prime number, these are regarded as prime ideal numbers. We have
in like manner
2 = v/T^To . ^r^i, 3 = A^T^o . A^l - 2^, &c. ;
every positive real prime which is a residue of 23 is thus a product of two factors
ideal or actual. And, reverting to the equation (77 - 4) (77 + 5)= - 2 . 13, or as this may
be written
(77, - 4) (77! + 5) = - v/1 -TTO^I -
we have ^ - 4)3 and (1 -170)(31 -12i70) each =14 + 55^, or say
T?I - 4 = vT^o v/ 31^12770 ,
and similarly
7?! + 5 = - \Xf -^ v/31-1277,,
so that we verify that 77, - 4, 77! + 5, do thus in fact each of them contain an ideal
factor of 13.
39. We have 2 =\/\ -^\/\ -r),, viz. the ideal multiplier \/l-r)0 renders actual
one of the ideal factors \/l-^ of 2, and it is found that this same ideal multiplier
614 NUMBERS. [795
\/ ]_ — rjQ renders actual one of the two ideal factors of any other decomposable number
3, 13, &c.,
_
-To = 1 + rjo, v/31 - 12^0 vT'^o = - 5 - 1/0, &c.
Similarly the conjugate multiplier Vl—r)i renders actual the other ideal factor of any
number 2, 3, 13, &c. We have thus two classes, or, reckoning also actual numbers,
three classes of prime numbers, viz. (i) ideal primes rendered actual by the multiplier
v/1 -tjo, (ii) ideal primes rendered actual by the multiplier v/l-^, (iii) actual primes.
This is a general property in the several complex theories; there is always a finite
number of classes of ideal numbers, distinguished according to the multipliers by which
they are rendered actual ; the actual numbers form a " principal " class.
40. General theory of congruences — irreducible functions. In the complex theory
relating to the roots of the equation rf + 77 + 6 = 0, there has just been occasion to con
sider the equation (77 - 4)(?/ + 5) = - 2.13, or say the congruence (?7-4)(t7 + 5)=0(mod. 13);
in this form the relation rf + r) + 6 = 0 is presupposed, but if, dropping this equation,
77 be regarded as arbitrary, then there is the congruence rf + 77+ 6 = (77 — 4)(T; + 5)(mod. 13).
For a different modulus, for instance 11, there is not any such congruence exhibiting
a decomposition of rf + 77 + 6 into factors. The function r)'2 + 77 + 6 is irreducible, that
is, it is not a product of factors with integer coefficients ; in respect of the modulus
13 it becomes reducible, that is, it breaks up into factors having integer coefficients,
while for the modulus 11 it continues irreducible. And there is a like general theory
in regard to any rational and integral function F (x) with integer coefficients; such
function, assumed to be irreducible, may for a given prime modulus p continue irre
ducible, that is, it may not admit of any decomposition into factors with integer coefficients ;
or it may become reducible, that is, admit of a decomposition F(x) = <f}(x)-^r(x)^(x)...(mod.p}.
And, when this is so, it is thus a product, in one way only, of factors <f> (x}, ty(x), %(#), •••,
which are each of them irreducible in regard to the same modulus p ; any such factor
may be a linear function of x, and as such irreducible ; or it may be an irreducible
function of the second or any higher degree. It is hardly necessary to remark that,
in this theory, functions which are congruent to the modulus p are regarded as identical,
and that in the expression of F (x} an irreducible function <£ (x) may present itself
either as a simple i'actor, or as a multiple factor, with any exponent. The decom
position is analogous to that of a number into its prime factors ; and the whole theory
of the rational and integral function F(x) in regard to the modulus p is in many
respects analogous to that of a prime number regarded as a modulus. The theory has
also been studied where the modulus is a power pv.
41. The congruence-imaginaries of Galois. If F (x) be an irreducible function to
a given prime modulus p, this implies that there is no integer value of x satisfying
the congruence F (x) = 0 (mod. p) ; we assume such a value and call it i, that is, we
assume F(i) = 0(mod. p) ; the step is exactly analogous to that by which, starting from
the notion of a real root, we introduce into algebra the ordinary imaginary z = V-l.
For instance, y? - x + 3 is an irreducible function to the modulus 7 : there is no integer
solution of the congruence x2 - x + 3 = 0 (mod. 7). Assuming a solution i such that
NUMBERS. 615
i* — i + 3 = 0 (mod. 7), we have, always to this modulus, i2 = i — 3, and thence i3, i*, &c.,
each of them equal to a linear function of i. We consider the numbers of the form
a + bi, where a and b are ordinary integers which may be regarded as having each of
them the values 0, 1, 2, 3, 4, 5, or 6 ; there are thus 72, = 49, such numbers, or,
excluding zero, 48 numbers ; and it is easy to verify that these are, in fact, the
numbers i, i2,..., i47, i4S, = 1, that is, we have i a prime root of the congruence
a?48— 1 = 0 (mod. 7). The irreducible function may be of the third or any higher degree;
thus for the same modulus 7 there is the cubic function a? — x + 2, giving rise to a
theory of the numbers of the form a + bi + ci2, where i is a congruence-imaginary such
that i3 — i + 2 = 0 (mod. 7) ; and instead of 7 the modulus may be any other odd
prime p.
Ordinary Theory, Third Part.
42. In what precedes, no mention has been made of the so-called Pellian equation
x2- Dy- = 1 (where D is a given positive number)., and of the allied equations x2- Dy2 = - 1,
or = ± 4. The equations with the sign + have always a series of solutions, those with
the sign - only for certain values of D ; in every case where the solutions exist, a
least solution is obtainable by a process depending on the expression of VI) as a
continued fraction, and from this least solution the whole series of solutions can be
obtained without difficulty. The equations are very interesting, as well for their own
sake as in connexion with the theory of the binary quadratic forms of a positive
non-square determinant.
43. The theory of the expression of a number as a sum of squares or polygonal
numbers has been developed apart from the general theory of the binary, ternary, and
other quadratic forms to which it might be considered as belonging. The theorem for
two squares, that every prime number of the form 4?i + 1 is, and that in one way
only, a sum of two squares, is a fundamental theorem in relation to the complex
numbers a + bi. A sum of two squares multiplied by a sum of two squares is always
a sum of two squares, and hence it appears that every number of the form 2a(4n + l)
is (in general, in a variety of ways) a sum of two squares.
Every number of the form 4<n + 2 or Sn + 3 is a sum of three squares ; even in
the case of a prime number 8w+3 there is in general more than one decomposition,
thus 59 = 25 + 25 + 9 and =49 + 9+1. Since a sum of three squares multiplied by a
sum of three squares is not a sum of three squares, it is not enough to prove the
theorem in regard to the primes of the form 8n + 3.
Every prime number is (in general, in more than one way) a sum of four squares;
and therefore every number is (in general, in more than one way) a sum of four squares,
for a sum of four squares multiplied by a sum of four squares is always a sum of four
squares.
Every number is (in general, in several ways) a sum of m + 2 (ra + 2)gonal numbers,
that is, of numbers of the form \m (#2 -#) + #; and of these ra-2 may be at
pleasure equal to 0 or 1 ; in particular, every number is a sum of three triangular
numbers (a theorem of Fermat's).
616 NUMBERS. [795
The theorems in regard to three triangular numbers and to four square numbers
are exhibited by certain remarkable identities in the Theory of Elliptic Functions; and
generally there is in this subject a great mass of formulae connected with the theory
of the representation of numbers by quadratic forms. The various theorems in regard
to the number of representations of a number as the sum of a definite number of
squares cannot be here referred to.
44. The equation xh + y^ = zK, where X. is any positive integer greater than 2, is
not resoluble in whole numbers (a theorem of Fermat's). The general proof depends
on the theory of the complex numbers composed of the Xth roots of unity, and pre
sents very great difficulty ; in particular, distinctions arise according as the number X
does or does not divide certain of Bernoulli's numbers.
45. Lejeune-Dirichlet employs, for the determination of the number of quadratic
forms of a given positive or negative determinant, a remarkable method depending on
the summation of a series 2/~s, where the index s is greater than but indefinitely near
to unity.
46. Very remarkable formulae have been given by Legendre, Tchebycheff, and
Riemann for the approximate determination of the number of prime numbers less than
a given large number x. Factor tables have been formed for the first nine million
numbers, and the number of primes counted for successive intervals of 50,000; and
these are found to agree very closely with the numbers calculated from the approximate
sc
formulae. Legendre's expression is of the form , — - j , where A is a constant not
very different from unity; Tchebycheff 's depends on the logarithm-integral li(a?); and
Riemann's, which is the most accurate, but is of a much more complicated form, con
tains a series of terms depending on the same integral.
The classical works on the Theory of Numbers are Legendre, The'orie des Nombres,
1st ed. 1798, 3rd ed. 1830 ; Gauss, Disquisitiones Arithmetical, Brunswick, 1801 (reprinted
in the collected works, vol. I., Gottingen, 1863 ; French translation, under the title Recherches
Arithmetiques, by Poullet-Delisle, Paris, 1807) ; and Lejeune-Dirichlet, Vorlesungen uber
Zahlentheorie, 3rd ed., with extensive and valuable additions by Dedekind, Brunswick,
1879 — 81. We have by the late Prof. H. J. S. Smith the extremely valuable series
of " Reports on the Theory of Numbers," Parts I. to VI., British Association Reports,
1859 — 62, 1864 — 65, which, with his own original researches, [are] printed in the [first
volume of the] collected works [published in 1894] by the Clarendon Press. See also Cayley,
" Report of the Mathematical Tables Committee," Brit. Assoc. Report, 1875, pp. 305 — 336,
[611], for a list of tables relating to the Theory of Numbers, and Mr J. W. L. Glaisher's
introduction to the Factor Table for the Sixth Million, London, 1883, in regard to the
approximate formula? for the number of prime numbers.
796] 617
796.
SERIES.
[From the Encyclopedia Britannica, Ninth Edition, vol. XXI. (1886), pp. 677 — 682.]
A SERIES is a set of terms considered as arranged in order. Usually the terms
are or represent numerical magnitudes, and we are concerned with the sum of the
series. The number of terms may be limited or without limit; and we have thus the
two theories, finite series and infinite series. The notions of convergency and divergency
present themselves only in the latter theory.
Finite Series.
1. Taking the terms to be numerical magnitudes, or say numbers, if there be a
definite number of terms, then the sum of the series is nothing else than the number
obtained by the addition of the terms; e.g. 4 + 9 + 10 = 23, 1 + 2 + 4 + 8=15. In the
first example there is no apparent law for the successive terms; in the second example
there is an apparent law. But it is important to notice that in neither case is there
a determinate law : we can in an infinity of ways form series beginning with the
apparently irregular succession of terms 4, 9, 10, or with the apparently regular
succession of terms 1, 2, 4, 8. For instance, in the latter case we may have a series with
the general term 2n, when for n = 0, 1, 2, 3, 4, 5, ... the series will be 1, 2, 4, 8, 16, 32,...;
or a series with the general term £ (n3 + 5n + 6). where for the same values of n the
series will be 1, 2, 4, 8, 15, 26,... The series may contain negative terms, and in
forming the sum each term is of course to be taken with the proper sign.
2. But we may have a given law, such as either of those just mentioned, and
the question then arises, to find the sum of an indefinite number of terms, or say of
n terms (n standing for any positive integer number at pleasure) of the series. The
expression for the sum cannot in this case be obtained by actual addition ; the
formation by addition of the sum of two terms, of three terms, &c., will, it may be,
suggest (but it cannot do more than suggest) the expression for the sum of n terms
c. xi. 78
618 SERIES. [796
of the series. For instance, for the series of odd numbers 1 + 3 + 5+7+..., we have 1 = 1,
1 + 3 = 4, 1 + 3 + 5 = 9, &c. These results at once suggest the law, l+3 + 5 + ...+(2w-l)=n2,
which is in fact the true expression for the sum of n terms of the series ; and this
general expression, once obtained, can afterwards be verified.
3. We have here the theory of finite series : the general problem is, un being
a given function of the positive integer n, to determine as a function of n the sum
MO + MJ + M2 + ... + un, or, in order to have n instead of n + 1 terms, say the sum
M0 + Wi + MS + • • • + Mn_lt
Simple cases are the three which follow.
(i) The arithmetic series,
a + (a + 6) + (a + 26) + . . . + (a + n- 1) b ;
writing here the terms in the reverse order, it at once appears that twice the sum
is =2a + n— 16 taken n times: that is, the sum =na + ^n(n— 1)6. In particular, we
have an expression for the sum of the natural numbers
1 + 2 + 3 + . .. + n = \n (n + 1),
and an expression for the sum of the odd numbers
1 + 3 + 5 + .. .+('2n-l) = n2.
(ii) The geometric series,
a + ar + ar2 + . . . + arn~l •
here the difference between the sum and r times the sum is at once seen to be
I —rn
= a — arn, and the sum is thus = a -^ - ; in particular, the sum of the series
I _ rn
1 + r + r- + . . . + r'1-
- — .
i — r
(iii) But the harmonic series,
111 1
r ; j + . — ?ri r • • • +
a ' a + b a +26 a+(n-l)b'
or say | + £ + £ . . . + - , does not admit of summation ; there is no algebraical function of
IV
n which is equal to the sum of the series.
4. If the general term be a given function un, and we can find vn a function of n
such that vn+i — vn = un, then we have MO = i^ — v0, i(>i = v2 — vl, u.2 = v3 — v2,. . . , un = vn+1 — vn ;
and hence MO + Ui+ u.2 + ... +un = vn+1 — v0, — an expression for the required sum. This
is in fact an application of the Calculus of Finite Differences. In the notation of
this calculus vn+1 — vn is written Avn ; and the general inverse problem, or problem of
integration, is from the equation of differences Avn = un (where un is a given function
of ri) to find vn. The general solution contains an arbitrary constant, vn=Vn + C, but
this disappears in the difference vn+1 — v0. As an example consider the series
UO+H. + ... + w-n = 0 + l + 3+ ... +|w (w + 1);
796] SERIES. 619
here, observing that
n (n + 1) (w + 2) - (n - 1) n (n+ 1) = n (n + 1) (n + 2 - n - 1), = 3n (n + 1 ),
we have
and hence
1 + 3 + 6 + ... + i» (« + 1)- $n (*+ !)(»+ 2),
as may be at once verified for any particular value of n.
Similarly, when the general term is a factorial of the order r, we have
1 4. r + l _i_ + n(n+ 1) --(n + r- 1) _ n(n + l)... (n + r)
I 1.2 ... ~T~ ~T72 ...(r + 1)'
5. If the general term wrt be any rational and integral function of n, we have
n A ?z (ft — 1) . n (n — 1) . . . (n — n + 1 ) A
wn=wo + :j-AM0 + _Y-^AX + ... + - ^ 2 v — ^ -A%.O,
where the series is continued only up to the term depending on p, the degree of the
function un, for all the subsequent terms vanish. The series is thus decomposed into
a set of series which have each a factorial for the general term, and which can be
summed by the last formula; thus we obtain
(n + I ) n . (n + 1 ) n (n — 1 ) . . . (n — v + 1 )
u0 + Ul ... + un = (n + 1) tib + v— 3-y— Au0 + ... + ^ -- ' \ > ] ^
» • * 1 • * • w ••• \p ~r JL)
which is a function of the degree p+ 1.
Thus for the before-mentioned series 1 + 2 + 4 + 8 + ..., if it be assumed that the
general term un is a cubic function of n, and writing down the given terms and
forming the differences, 1, 2, 4, 8 ; 1, 2, 4; 1, 2 ; 1, we have
n n(n — ~L) n(n — l}(n — 2) ,
j + ^ 2 + -- 193 -- ^ * ^ + 5n + 6^' as abovel '
and the sum
34n + 24).
As particular cases we have expressions for the sums of the powers of the natural
numbers —
l); P + 23+ ... + n3= n*n + 1":
observe that the latter =(1 + 2 ... +?i)2; and so on.
6. We may, from the expression for the sum of the geometric series, obtain by
differentiation other results : thus
1 + r + r- + . . . + rn-1 =
1 — r
78—2
620 SERIES. [796
gives
q 2 / i\ n-2_^ 1— ?'n _l—nrn~1+(n — l)rn
~ dr I -r ' (1 - r)2
and we might in this way find the sum u0 + u^r + . . . -f unrn, where un is any rational
and integral function of n.
7. The expression for the sum u0+ ^ + ... -f un of an indefinite number of terms
will in many cases lead to the sum of the infinite series u-0 + u-^ + . . . ; but the theory
of infinite series requires to be considered separately. Often in dealing apparently
with an infinite series u0 + u^ + . . . we consider rather an indefinite than an infinite
series, and are not in any wise really concerned with the sum of the series or the
question of its convergency : thus the equation
m(m — 1) \f^ n(n—I)
a? + ...
n + n)(m + n - 1)
I . _
really means the series of identities
(ra + n) = m + n,
(m + n) (m + n - 1) _ im(m--l) m n n(n-l)
1 .2 ~T7sT~ + T T 4 ~T72~ ' '
obtained by multiplying together the two series of the left-hand side. Again, in the method
of generating functions we are concerned with an equation <f>(t) = A0 + AJ+ ...+Antn+...,
where the function <f>(t) is used only to express the law of formation of the successive
coefficients.
It is an obvious remark that, although according to the original definition of a
series the terms are considered as arranged in a determinate order, yet in a finite
series (whether the number of terms be definite or indefinite) the sum is independent
of the order of arrangement.
Infinite Series.
8. We consider an infinite series u0 + 1^ + u2 + ... of terms proceeding according
to a given law, that is, the general term un is given as a function of n. To fix
the ideas the terms may be taken to be positive numerical magnitudes, or say
numbers continually diminishing to zero; that is, M»>MB+I, and un is, moreover, such
a function of n that, by taking n sufficiently large, un can be made as small as we
please.
Forming the successive sums S0 = uQ, S^u. + u,, Sa= u0 -t- u, + «,, ..., these sums
S0, 81, S2,... will be a series of continually increasing terms, and if they increase up
to a determinate finite limit 8 (that is, if there exists a determinate numerical
magnitude 8 such that, by taking n sufficiently large, we can make S-8n as small
as we please), 8 is said to be the sum of the infinite series. To show that we can
7.96] SERIES. 621
actually have an infinite series with a given sum S, take u0 any number less than S,
then S—un is positive, and taking HI any numerical magnitude less than S — u0, then
S — u0 — U-L is positive. And going on continually in this manner we obtain a series
u0 + ztj + u2 + ..., such that for any value of n however large S — u0 — u^ ... — un is
positive; and if as n increases this difference diminishes to zero, we have w0 + M1+w2+... ,
an infinite series having S for its sum. Thus, if 8=2, and we take u0< 2, say
?/0=l; z^<2-l, say MI = £; w2<2-l-|, say w2 = i ; and so on, we have
or, more generally, if r be any positive number less than 1, then 1 + r + r2 + . . . = — — ,
1 — r
that is, the infinite geometric series with the first term =1, and with a ratio r< 1,
\ ~^ _ ytn.
has the finite sum - - . This, in fact, follows from the expression 1 + r +r2... + rn~l = -
1 — r 1 — r
for the sum of the finite series ; taking r < 1, then as n increases rn decreases to
zero, and the sum becomes more and more nearly = - - .
1 — r
9. An infinite series of positive numbers can, it is clear, have a sum only if
the terms continually diminish to zero ; but it is not conversely true that, if this
condition be satisfied, there will be a sum. For instance, in the case of the harmonic
series !+£ + £+..., it can be shown that by taking a sufficient number of terms the
sum of the finite series may be made as large as we please. For, writing the series
in the form 1 + \ + (! + £) + (£ + £ + | + |)+ ..., the number of terms in the brackets
being doubled at each successive step, it is clear that the sum of the terms in any
bracket is always > \ ; hence by sufficiently increasing the number of brackets the
sum may be made as large as we please. In the foregoing series, by grouping the
terms in a different manner 1 + (\ + £) + (£ + £ + £ + f ) + . . . , the sum of the terms in
any bracket is always < 1 ; we thus arrive at the result that (n = 3 at least) the
sum of 2n terms of the series is > 1 + ^n and < n.
10. An infinite series may contain negative terms; suppose in the first instance
that the terms are alternately positive and negative. Here the absolute magnitudes
of the terms must decrease down to zero, but this is a sufficient condition in order
that the series may have a sum. The case in question is that of a series v0 - vl + v.,-...,
where v0, vlt v2, ... are all positive and decrease down to zero. Here, forming the
successive sums S0 = v0, 8l = v0-v1, S.2=v0-v1 + v2, ... , S0, 8lt £,,... are all positive, and
we have S0>81, Sl<82, £,>£,,..., and 8n+l-Sn tends continually to zero. Hence the
sums S0, 8lt 8a,... tend continually to a positive limit 8 in such wise that S0, S2, S4,...
are each of them greater and S1} Sa, £5, ... are each of them less than S-, and we
thus have S as the sum of the series. The series 1— ^ + ^ — ^ + ... will serve as an
example. The case just considered includes the apparently more general one where
the series consists of alternate groups of positive and negative terms respectively; the
terms of the same group may be united into a single term + vn, and the original
series will have a sum only if the resulting series v0 — vl + v2 ... has a sum, that is,
if the positive partial sums v0, vlt v2,... decrease down to zero.
The terms at the beginning of a series may be irregular as regards their signs;
but, when this is so, all the terms in question (assumed to be finite in number) may
622 SERIES. [796
be united into a single term, which is of course finite, and instead of the original
series only the remaining terms of the series need be considered. Every infinite series
whatever is thus substantially included under the two forms, — terms all positive and
terms alternately positive and negative.
11. In brief, the sum (if any) of the infinite series u0 + u1 + u»+ ... is the finite
limit (if any) of the successive sums u0, WO + MI, u0 + ii! + u,2, ... ; if there is no such
limit, then there is no sum. Observe that the assumed order u0, u1} u?, ... of the
terms is part of and essential to the definition; the terms in any other order may
have a different sum, or may have no sum. A series having a sum is said to be
" convergent " ; a series which has no sum is " divergent."
If a series of positive terms be convergent, the terms cannot, it is clear, continually
increase, nor can they tend to a fixed limit : the series 1 4- 1 + 1 + ... is divergent.
For the convergency of the series it is necessary (but, as has been shown, not sufficient)
that the terms shall decrease to zero. So, if a series with alternately positive and
negative terms be convergent, the absolute magnitudes cannot, it is clear, continually
increase. In reference to such a series Abel remarks, " Peut-on imaginer rien de plus
horrible que de debiter 0= ln — 2n + 3" — 4>n + , &c., ou n est un nombre entier positif?"
Neither is it allowable that the absolute magnitudes shall tend to a fixed limit. The
so-called " neutral " series 1 — 1 + 1—1... is divergent : the successive sums do not tend
to a determinate limit, but are alternately + 1 and 0 ; it is necessary (and also sufficient)
that the absolute magnitudes shall decrease to zero.
In the so-called semi-convergent series, we have an equation of the form
S= U.-U^ U,-...,
where the positive values U0, £71( U.,,... decrease to a minimum value, suppose Upt
and afterwards increase ; the series is divergent and has no sum, and thus S is not
the sum of the series. 8 is only a number or function calculable approximately by
means of the series regarded as a finite series terminating with the term + Up. The
successive sums U0, U0— U1} U0 — U1+ U2, ... up to that containing + Up, give alternately
superior and inferior limits of the number or function S.
12. The condition of convergency may be presented under a different form : let
the series u0 + u^ + n.2 + ... be convergent, then, taking ra sufficiently large, the sum is
the limit not only of u0 + u^ + ...+ um but also of u0 + u^ + . . . + um+r, where r is any
number as large as we please. The difference of these two expressions must therefore
be indefinitely small; by taking ra sufficiently large the sum um+1 + um+2+ . . . -\- um+r (where
r is any number however large) can be made as small as we please ; or, as this may
also be stated, the sum of the infinite series um+l + um+z + . . . can be made as small
as we please. If the terms are all positive (but not otherwise), we may take, instead
of the entire series um+1 + um+2 + ..., any set of terms (not of necessity consecutive
terms) subsequent to um; that is, for a convergent series of positive terms the sum
of any set of terms subsequent to uin can, by taking m sufficiently large, be made as
small as we please.
796] SERIES. 623
13. It follows that, in a convergent series of positive terms, the terms may be grouped
together in any manner so as to form a finite number of partial series which will be
each of them convergent, and such that the sum of their sums will "be the sum of
the given series. For instance, if the given series be tt0-)-ul + u2+ ... , then the two
series u0 + u2 + u4 + ... and r*1 + w3 + ... will each be convergent and the sum of their
sums will be the sum of the original series.
14. Obviously the conclusion does not hold good in general for series of positive
and negative terms: for instance, the series 1— £+^— £ + ... is convergent, but the
two series 1+i + £+... and — ^ — i~ ... are each divergent, and thus without a sum.
In order that the conclusion may be applicable to a series of positive and negative
terms the series must be " absolutely convergent," that is, it must be convergent when
all the terms are made positive. This implies that the positive terms taken by them
selves are a convergent series, and also that the negative terms taken by themselves
are a convergent series. It is hardly necessary to remark that a convergent series of
positive terms is absolutely convergent. The question of the convergency or divergency
of a series of positive and negative terms is of less importance than the question
whether it is or is not absolutely convergent. But in this latter question we regard
the terms as all positive, and the question in effect relates to series containing positive
terms only.
15. Consider, then, a series of positive terms u0 + MJ + w2 + ... ; if they are increas
ing — that is, if in the limit un+ljun be greater than 1 — the series is divergent, but if
less than 1 the series is convergent. This may be called a first criterion ; but there
is the doubtful case where the limit =1. A second criterion was given by Cauchy
and Raabe ; but there is here again a doubtful case when the limit considered = 1.
A succession of criteria was established by De Morgan, which it seems proper to give
in the original form ; but the equivalent criteria established by Bertrand are somewhat
more convenient. In what follows Ix is for shortness written to denote the logarithm
of x, no matter to what base. De Morgan's form is as follows : — Writing un — —
*(»)'
/"/'//) />«
put p0 = ^— - ; if for x = oo the limit a0 of p0 be greater than 1 the series is conver
gent, but if less than 1 it is divergent. If the limit a0 = 1 , seek for the limit of
Pi, = (PO — !)&»; if this limit ctj be greater than 1 the series is convergent, but if
less than 1 it is divergent. If the limit 0^ = 1, seek for the limit p2, = (pi — 1) llx ;
if this limit a2 be greater than 1 the series is convergent, but if less than 1 it is
divergent. And so on indefinitely.
16. Bertrand's form is: — If, in the limit for n = oo , I — In be negative or less
Un'
than 1 the series is divergent, but if greater than 1 it is convergent. If it =1, then
if I -- - Un be negative or less than 1 the series is divergent, but if greater than 1
nUnl
it is convergent. If it =1, then if I —r- Ittn be negative or less than 1 the series
WUtfiLl'lt I
is divergent, but if greater than 1 it is convergent. And so on indefinitely.
624 SERIES. [796
The last-mentioned criteria follow at once from the theorem that the several series
having the general terms -- , , , , . . . respectively are each
?i° w (In)* nln (tin)* nlnlln (llln)*
of them convergent if a be greater than 1, but divergent if a be negative or less than
1 or = 1. In the simplest case, the series having the general term — , the theorem
may be proved nearly in the manner in which it is shown above (cf. § 9) that the
harmonic series is divergent.
17. Two or more absolutely convergent series may be added together,
{w0 + MI + MS +.-•} + K + V! + v.2 +...} = (>0 + v0) + O, + O + . . . ;
that is, the resulting series is absolutely convergent and has for its sum the sum of
the two sums. And similarly two or more absolutely convergent series may be multiplied
together
U0 + M! + W + . . . X V + fl + V +...= UV + WV + U + UV U
that is, the resulting series is absolutely convergent and has for its sum the product
of the two sums. But more properly the multiplication gives rise to a doubly infinite
series —
U0V0, U0Vlt M0V2...
— which is a kind of series which will be presently considered.
18. But it is, in the first instance, proper to consider a single series extend
ing backwards and forwards to infinity, or say a back-and-forwards infinite series
. . . + w_2 + it-i + u0 + i<j + u2 + . . . ; such a series may be absolutely convergent, and the sum
is then independent of the order of the terms, and in fact equal to the sum of the
sums of the two series u0 + ul + u2 + ... and u_± + u_.2 + w_3 + . . . respectively. But, if not
absolutely convergent, the expression has no definite meaning until it is explained in
what manner the terms are intended to be grouped together ; for instance, the expression
may be used to denote the foregoing sum of two series, or to denote the series
u0 + (u^ + u_j) + (u2 + u_2) + ... and the sum may have different values, or there may
be no sum, accordingly. Thus, if the series be ... — ^ — ^- + 0 + { + ^ + ..., in the former
meaning the two series 0 + 1 + £ + . . . and — | — £—... are each divergent, and there
is not any sum. But in the latter meaning the series is 0 + 0 + 0 + . . . , which has a
sum = 0. So, if the series be taken to denote the limit of
(U0 + U-L + U2 + . . . + Um) + (W_! -f- W_2 + . . . + U_rn>),
where m, m are each of them ultimately infinite, there may be a sum depending on
the ratio m : m', which sum consequently acquires a determinate value only when this
ratio is given.
19. In a singly infinite series we have a general term un, where n is an integer
positive in the case of an ordinary series, and positive or negative in the case of a
796] SERIES. 625
back-and-forwards series. Similarly for a doubly infinite series, we have a general term
um, n, where m, n are integers which may be each of them positive, and the form of
the series is then
0, 2
or m, n may be each of them positive or negative. The latter is the more general
supposition, and includes the former, since um.t n may =0 for m or n each or either
of them negative. To put a definite meaning on the notion of a sum, we may regard
in, n as the rectangular coordinates of a point in a plane ; that is, if m, n are each
of them positive, we attend only to the positive quadrant of the plane, but otherwise
to the whole plane ; and we have thus a doubly infinite system or lattice-work of
points. We may imagine a boundary depending on a parameter T which for T = oo is
at every point thereof at an infinite distance from the origin ; for instance, the boundary
may be the circle x- + y2=T, or the four sides of a rectangle, x — ± aT, y = ± /3T.
Suppose the form is given and the value of T, and let the sum 2wm> n be understood
to denote the sum of those terms um>n which correspond to points within the boundary,
then, if as T increases without limit the sum in question continually approaches a
determinate limit (dependent, it may be, on the form of the boundary), for such form
of boundary the series is said to be convergent, and the sum of the doubly infinite
series is« the aforesaid limit of the sum 2ttOT, n- The condition of convergency may be
otherwise stated : it must be possible to take T so large that the sum 2wm> n for all
terms um> n which correspond to points outside the boundary shall be as small as we
please.
It is easy to see that, if the terms umyn be all of them positive, and the series
be convergent for any particular form of boundary, it will be convergent for any other
form of boundary, and the sum will be the same in each case. Thus, let the boundary
be in the first instance the circle xz + yz = T ; by taking T sufficiently large the sum
2*V n for points outside the circle may be made as small as we please. Consider
any other form of boundary — for instance, an ellipse of given eccentricity, — and let such
an ellipse be drawn including within it the circle oc- + y-= T. Then the sum 2um> n
for terms um>n corresponding to points outside the ellipse will be smaller than the
sum for points outside the circle, and the difference of the two sums — that is, the
sum for points outside the circle and inside the ellipse — will also be less than that
for points outside the circle, and can thus be made as small as we please. Hence
finally the sum ^um,n, whether restricted to terms umjU corresponding to points inside
the circle or to terms corresponding to points inside the ellipse, will have the same
value, or the sum of the series is independent of the form of the boundary. Such a
series, viz. a doubly infinite convergent series of positive terms, is said to be absolutely
convergent; and similarly a doubly infinite series of positive and negative terms which
is convergent when the terms are all taken as positive is absolutely convergent.
20. We have in the preceding theory the foundation of the theorem (§ 17) as to
the product of two absolutely convergent series. The product is in the first instance
C. XI. 79
626 SERIES. [796
expressed as a doubly infinite series ; and, if we sum this for the boundary x + y = T,
this is in effect a summation of the series u0v0 + (u^ +u1v0) + ..., which is the product
of the two series. It may be further remarked that, starting with the doubly infinite
series and summing for the rectangular boundary x = aT, y = @T, we obtain the sum
as the product of the sums of the two single series. For series not absolutely con
vergent, the theorem is not true. A striking instance is given by Cauchy : the series
1 7o + ~7o 7Z + • • • *s convergent' and has a calculable sum, but it can be shown
\ & \ o Y T
2/2 \
without difficulty that its square, viz. the series 1 — -. - + ( , o + i )—•••> is divergent.
21. The case where the terms of a series are imaginary comes under that where
they are real. Suppose the general term is pn + qni, then the series will have a sum,
or will be convergent, if and only if the series having for its general term pn and
the series having for its general term qn be each convergent ; then the sum = sum of
first series -f i multiplied by sum of second series. The notion of absolute convergence
will of course apply to each of the series separately ; further, if the series having for its
general term the modulus *Jp\ + q\ be convergent (that is, absolutely convergent, since
the terms are all positive), each of the component series will be absolutely convergent;
but the condition is not necessary for the convergence, or the absolute convergence, of
the two component series respectively.
22. In the series thus far considered, the terms are actual numbers, or are at
least regarded as constant ; but we may have a series u0 + u^ + u.2 + . . . , where the
successive terms are functions of a parameter z ; in particular, we may have a series
a0 + a1z + a2zz+ ... arranged in powers of z. It is in view of a complete theory necessary
to consider z as having the imaginary value x + iy = r (cos </> + i sin <£). The two com
ponent series will then have the general terms anrncosn<J> and anrnsmn<f> respectively;
accordingly each of these series will be absolutely convergent for any value whatever
of <f>, provided the series with the general term anrn be absolutely convergent. More
over, the series, if thus absolutely convergent for any particular value R of r, will be
absolutely convergent for any smaller value of r, that is, for any value of x + iy
having a modulus not exceeding R ; or, representing as usual x + iy by the point
whose rectangular coordinates are x, y, the series will be absolutely convergent for any
point whatever inside or on the circumference of the circle having the origin for centre
and its radius = R. The origin is of course an arbitrary point : or, what is the same
thing, instead of a series in powers of z, we may consider a series in powers of z — c
(where c is a given imaginary value =« + /&). Starting from the series, we may within
the aforesaid limit of absolute convergency consider the series as the definition of a
function of the variable z ; in particular, the series may be absolutely convergent for
every finite value of the modulus, and we have then a function defined for every
finite value whatever x + iy of the variable. Conversely, starting from a given function
of the variable, we may inquire under what conditions it admits of expansion in a
series of powers of z (or z — c\ and seek to determine the expansion of the function
in a series of this form. But in all this, however, we are travelling out of the theory
of series into the general theory of functions.
796] SERIES. 627
23. Considering the modulus r as a given quantity and the several powers of r as
included in the coefficients, the component series are of the forms a0+a1 cos$+a2cos 2<£-f ...
and al sin (f> + a., sin 2<£ + . . . respectively. The theory of these trigonometrical or multiple
sine and cosine series, and of the development, under proper conditions, of an arbitrary
function in series of these forms, constitutes an important and interesting branch of
analysis.
24. In the case of a real variable z, we may have a series a0 + alz + a2z2 + ..., where
the series a0+a1+a2+... is a divergent series of decreasing positive terms (or as a
limiting case where this series is 1 + 1 + 1 + ...). For a value of z inferior but indefinitely
near to + 1, say z = ± (1 — e), where e is indefinitely small and positive, the series will
be convergent and have a determinate sum (f> (z), and we may write $ (+ 1) to denote
the limit of </>(+(!— e)) as e diminishes to zero; but unless the series be convergent
for the value z = ± 1, it cannot for this value have a sum, nor consequently a sum
2& «-3
= <£(±1). For instance, let the series be z + -5-+ ~-+ ... , which for values of z between
£ o
the limits +1 (both limits excluded) = — log(l— z). For z = + \ the series is divergent
and has no sum ; but for z = 1 — e, as e diminishes to zero, we have — log e and
(1 — e)+ ^(1 — e)2+ ..., each positive and increasing without limit; for z = — 1, the series
1— i + 3 — i + . . . is convergent, and we have at the limit log 2 = 1 — |- + 1- — £ + As
a second example, consider the series 1 + z + z2 + . . . , which for values of z between the
limits + 1 (both limits excluded) = . For 2= + !, the series is divergent and has no
± — z
sum ; but for z = 1 — e, as e diminishes to zero, we have - and 1 4- (1 — e) + (1 — e)2 +
each positive and increasing without limit ; for z = — 1 the series is divergent and has
no sum; the equation — — = 1 - (1 -e) +(1 - e)2- ... is true for any positive value of e
however small, but not for the value e = 0.
The following memoirs and works may be consulted: — Cauchy, Cours d' Analyse de
I'Scole Poll/technique — part I., Analyse Algebrique, 8vo. Paris, 1821; Abel, "Unter-
suchungen liber die Reihe 1 + ^ x + mC"l~ ) x*+ ..." in Crelle's Journ. de Math., vol. I.
-i- J. . Zi
(1826), pp. 211—239, and (Euvres (French trans.), vol. I. ; De Morgan, Treatise on the
Differential and Integral Calculus, 8vo. London, 1842; Id., "On Divergent Series and
various Points of Analysis connected with them " (1844), in Camb. Phil. Trans., vol. viu.
(1849), and other memoirs in Camb. Phil. Trans.; Bertrand, "Regies sur la Conver
gence des Series," in Liouv. Journ. de Math., vol. vn. (1842), pp. 35 — 54 ; Cayley, " On
the Inverse Elliptic Functions," Camb. Math. Journ., vol. iv. (1845), pp. 257 — 277, [24],
and " Me'moire sur les Fonctions doublement periodiques," in Liouv. Journ. de Math.,
vol. x. (1845), pp. 385 — 420, [25], (as to the boundary for a doubly infinite series):
Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe,"
in Gott. Abh.} vol. xin. (1854), and Werke, Leipsic, 1876, pp. 213 — 253 (contains an
account of preceding researches by Euler, D'Alembert, Fourier, Lejeune-Dirichlet, &c.);
Catalan, Traite Ele'mentaire des Series, 8vo. Paris, 1860 ; Boole, Treatise on the Calculus
of Finite Differences, 2nd ed. by Moulton, 8vo. London, 1872.
79—2
628
797.
SURFACE, CONGRUENCE, COMPLEX.
[From the Encyclopaedia Britannica, Ninth Edition, vol. XXIL (1887), pp. 668 — 672.]
IN the article Curve [785], the subject was treated from an historical point of view
for the purpose of showing how the leading ideas of the theory were successively
arrived at. These leading ideas apply to surfaces, but the ideas peculiar to surfaces
are scarcely of the like fundamental nature, being rather developments of the former
set in their application to a more advanced portion of geometry; there is consequently
less occasion for the historical mode of treatment. Curves in space were briefly con
sidered in the same article, and they will not be discussed here ; but it is proper to
refer to them in connexion with the other notions of solid geometry. In plane
geometry the elementary figures are the point and the line; and we then have the
curve, which may be regarded as a singly infinite system of points, and also as a
singly infinite system of lines. In solid geometry the elementary figures are the point,
the line, and the plane; we have, moreover, first, that which under one aspect is the
curve and under another aspect the developable (or torse), and which may be regarded
as a singly infinite system of points, of lines, or of planes; and secondly, the surface,
which may be regarded as a doubly infinite system of points or of planes, and also
as a special triply infinite system of lines. (The tangent lines of a surface are a
special complex.) As distinct particular cases of the first figure, we have the plane
curve and the cone: and as a particular case of the second figure, the ruled surface,
regulus, or singly infinite system of lines ; we have, besides, the congruence or doubly
infinite system of lines, and the complex or triply infinite system of lines. And thus
crowds of theories arise which have hardly any analogues in plane geometry: the
relation of a curve to the various surfaces which can be drawn through it, and that
of a surface to the various curves which can be drawn upon it, are different in kind
from those which in plane geometry most nearly correspond to them, — the relation of
a system of points to the different curves through them and that of a curve to the
systems of points upon it. In particular, there is nothing in plane geometry to corre
spond to the theory of the curves of curvature of a surface. Again, to the single
797] SURFACE. 629
theorem of plane geometry, that a line is the shortest distance between two points,
there correspond in solid geometry two extensive and difficult theories, — that of the
geodesic lines on a surface and that of the minimal surface, or surface of minimum
area, for a given boundary. And it would be easy to say more in illustration of the
great extent and complexity of the subject.
Surfaces in General; Torses, &c.
1. A surface may be regarded as the locus of a doubly infinite system of
points, — that is, the locus of the system of points determined by a single equation
£/"=(*]£#, y, z, l)w, =0, between the Cartesian coordinates (to fix the ideas, say rect
angular coordinates) x, y, z\ or, if we please, by a single homogeneous relation
~{J=(*§x, y, z, w)n, = 0, between the quadriplanar coordinates x, y, z, w. The degree n
of the equation is the order of the surface; and this definition of the order agrees
with the geometrical one, that the order of the surface is equal to the number of
the intersections of the surface by an arbitrary line. Starting from the foregoing point
definition of the surface, we might develop the notions of the tangent line and the
tangent plane; but it will be more convenient to consider the surface ab initio from
the more general point of view in its relation to the point, the line, and the plane.
2. Mention has been made of the plane curve and the cone; it is proper to
recall that the order of a plane curve is equal to the number of its intersections
by an arbitrary line (in the plane of the curve), and that its class is equal to the
number of tangents to the curve which pass through an arbitrary point (in the plane
of the curve). The cone is a figure correlative to the plane curve: corresponding to
the plane of the curve we have the vertex of the cone, to its tangents the generating
lines of the cone, and to its points the tangent planes of the cone. But from a
different point of view, we may consider the generating lines of the cone as corre
sponding to the points of the curve and its tangent planes as corresponding to the
tangents of the curve. From this point of view, we define the order of the cone as
equal to the number of its intersections (generating lines) by an arbitrary plane through
the vertex, and its class as equal to the number of the tangent planes which pass
through an arbitrary line through the vertex. And in the same way that a plane
curve has singularities (singular points and singular tangents), so a cone has singularities
(singular generating lines and singular tangent planes).
3. Consider now a surface in connexion with an arbitrary line. The line meets
the surface in a certain number of points, and, as already mentioned, the order of the
surface is equal to the number of these intersections. We have through the line a
certain number of tangent planes of the surface, and the class of the surface is equal
to the number of these tangent planes.
But, further, through the line imagine a plane ; this meets the surface in a curve
the order of which is equal (as is at once seen) to the order of the surface. Again,
on the line imagine a point; this is the vertex of a cone circumscribing the surface,
and the class of this cone is equal (as is at once seen) to the class of the surface.
630 SURFACE. [797
The tangent lines of the surface, which lie in the plane, are nothing else than the
tangents of the plane section, and thus form a singly infinite series of lines ; similarly,
the tangent lines of the surface, which pass through the point, are nothing else than
the generating lines of the circumscribed cone, and thus form a singly infinite series
of lines. But, if we consider those tangent lines of the surface which are at once
in the plane and through the point, we see that they are finite in number; and we
define the rank of a surface as equal to the number of tangent lines which lie in
a given plane and pass through a given point in that plane. It at once follows that
the class of the plane section and the order of the circumscribed cone are each equal
to the rank of the surface, and are thus equal to each other. It may be noticed that
for a general surface (* $#, y, z, w)n, = 0, of order n without point singularities the
rank is a, =n(n—l), and the class is ri, =n(n— I)2; this implies (what is, in fact,
the case) that the circumscribed cone has line singularities, for otherwise its class,
that is, the class of the surface, would be a (a— 1), which is not =n(n — I)2.
4. In the last preceding number, the notions of the tangent line and the tangent
plane have been assumed as known, but they require to be further explained in
reference to the original point definition of the surface. Speaking generally, we may
say that the points of the surface consecutive to a given point on it lie in a plane
which is the tangent plane at the given point, and conversely the given point is the
point of contact of this tangent plane, and that any line through the point of contact
and in the tangent plane is a tangent line touching the surface at the point of
contact. Hence we see at once that the tangent line is any line meeting the surface
in two consecutive points, or — what is the same thing — a line meeting the surface in
the point of contact, counting as two intersections, and in n — 2 other points. But,
from the foregoing notion of the tangent plane as a plane containing the point of
contact and the consecutive points of the surface, the passage to the true definition
of the tangent plane is not equally obvious. A plane in general meets the surface
of the order n in a curve of that order without double points ; but the plane may
be such that the curve has a double point, and when this is so the plane is a
tangent plane having the double point for its point of contact. The double point is
either an acnode (isolated point), then the surface at the point in question is convex
towards (that is, concave away from) the tangent plane ; or else it is a crunode, and
the surface at the point in question is then concavo-convex, that is, it has its two
curvatures in opposite senses (see infra, No. 16). Observe that, in either case, any line
whatever in the plane and through the point meets the surface in the points in
which it meets the plane curve, namely, in the point of contact, which qua double
point counts as two intersections, and in n — 2 other points ; that is, we have the
preceding definition of the tangent line.
5. The complete enumeration and discussion of the singularities of a surface is
a question of extreme difficulty which has not yet been solved*. A plane curve has
* In a plane curve, the only singularities which need to be considered are those that present themselves
in Pliicker's equations: for every higher singularity whatever is equivalent to a certain number of nodes,
cusps, inflexions, and double tangents. As regards a surface, no such reduction of the higher singularities
has as yet been made.
797] SURFACE. 631
point singularities and line singularities ; corresponding to these, we have for the surface
isolated point singularities and isolated plane singularities, but there are besides con
tinuous singularities applying to curves on or torses circumscribed to the surface, and
it is among these that we have the non-special singularities which play the most
important part in the theory. Thus the plane curve represented by the general equation
(*$#, y, z)n = 0, of any given order n, has the non-special line singularities of inflexions
and double tangents; corresponding to this, the surface represented by the general
equation ( *]£#, y, z, w}n = 0, of any given order n, has, not the isolated plane singul
arities, but the continuous singularities of the spinode curve or torse and the node-
couple curve or torse. A plane may meet the surface in a curve having (1) a cusp
(spinode) or (2) a pair of double points ; in each case, there is a singly infinite
system of such singular tangent planes, and the locus of the points of contact is the
curve, the envelope of the tangent planes the torse. The reciprocal singularities to
these are the nodal curve and the cuspidal curve : the surface may intersect or touch
itself along a curve in such wise that, cutting the surface by an arbitrary plane, the
curve of intersection has, at each intersection of the plane with the curve on the
surface. (1) a double point (node) or (2) a cusp. Observe that these are singularities
not occurring in the surface represented by the general equation (*][#, y, z, w)n = 0
of any order; observe further that, in the case of both or either of these singularities,
the definition of the tangent plane must be modified. A tangent plane is a plane
such that there is in the plane section a double point in addition to the nodes or
cusps at the intersections with the singular lines on the surface.
6. As regards isolated singularities, it will be sufficient to mention the point
singularity of the conical point (or cnicnode) and the corresponding plane singularity
of the conic of contact (or cnictrope). In the former case, we have a point such that
the consecutive points, instead of lying in a tangent plane, lie on a quadric cone,
having the point for its vertex ; in the latter case, wre have a plane touching the
surface along a conic, that is, the complete intersection of the surface by the plane
is made up of the conic taken twice and of a residual curve of the order n — 4.
7. We may, in the general theory of surfaces, consider either a surface and its
reciprocal surface, the reciprocal surface being taken to be the surface enveloped by
the polar planes (in regard to a given quadric surface) of the points of the original
surface ; or — what is better — we may consider a given surface in reference to the
reciprocal relations of its order, rank, class, and singularities. In either case, we have
a series of unaccented letters and a corresponding series of accented letters, and the
relations between them are such that we may in any equation interchange the accented
and the unaccented letters ; in some cases, an unaccented letter may be equal to the
corresponding accented letter. Thus, let n, n' be as before the order and the class of the
surface, but, instead of immediately defining the rank, let a be used to denote the
class of the plane section and a the order of the circumscribed cone ; also let S, S'
be numbers referring to the singularities. The form of the relations is a = a' (= rank
of surface); a' = n (n - 1) -S; n' = n (n- I)2 - 8', a = n (n' - 1) -S'; n = n' (ri - I)2 -S'.
In these last equations S, S' are merely written down to denote proper corresponding
combinations of the several numbers referring to the singularities collectively denoted
632 SURFACE. [797
by 8, S' respectively. The theory, as already mentioned, is a complex and difficult one,
and it is not the intention to further develop it here.
8. A developable or torse corresponds to a curve in space, in the same manner
as a cone corresponds to a plane curve : although capable of representation by an
equation [/"=(*$#, y, z, w)n, =0, and so of coming under the foregoing point definition
of a surface, it is an entirely distinct geometrical conception. We may indeed, qua
surface, regard it as a surface characterized by the property that each of its tangent
planes touches it, not at a single point, but along a line; this is equivalent to saying
that it is the envelope, not of a doubly infinite series of planes, as is a proper surface,
but of a singly infinite system of planes. But it is perhaps easier to regard it as
the locus of a singly infinite system of lines, each line meeting the consecutive line,
or, what is the same thing, the lines being tangent lines of a curve in space. The
tangent plane is then the plane through two consecutive lines, or, what is the same
thing, an osculating plane of the curve, whence also the tangent plane intersects the
surface in the generating line counting twice, and in a residual curve of the order
n — 2. The curve is said to be the edge of regression of the developable, and it is
a cuspidal curve thereof; that is to say, any plane section of the developable has at
each point of intersection with the edge of regression a cusp. A sheet of paper bent
in any manner without crumpling gives a developable ; but we cannot with a single
sheet of paper properly exhibit the form in the neighbourhood of the edge of
regression: we need two sheets connected along a plane curve, which, when the paper
is bent, becomes the edge of regression and appears as a cuspidal curve on the surface.
It may be mentioned that the condition which must be satisfied in order that
the equation U = 0 shall represent a developable is H ( U) = 0 ; that is, the Hessian
or functional determinant formed with the second differential coefficients of U must
vanish in virtue of the equation £7 = 0, or — what is the same thing — H (U) must
contain U as a factor. If in Cartesian coordinates the equation is taken in the form
z—f(x, 2/)=0, then the condition is rt-s- = 0 identically, where r, s, t denote as usual
the second differential coefficients of z in regard to x, y respectively.
9. A ruled surface or regulus is the locus of a singly infinite system of lines,
where the consecutive lines do not intersect ; this is a true surface, for there is a
doubly infinite series of tangent planes, — in fact, any plane through any one of the
lines is a tangent plane of the surface, touching it at a point on the line, and in
such wise that, as the tangent plane turns about the line, the point of contact moves
along the line. The complete intersection of the surface by the tangent plane is made
up of the line counting once and of a residual curve of the order n — 1. A quadric
surface is a regulus in a twofold manner, for there are on the surface two systems
of lines each of which is a regulus. A cubic surface may be a regulus (see No.
11 infra).
Surfaces of the Orders 2, 3, and 4.
10. A surface of the second order or a quadric surface is a surface such that
every line meets it in two points, or — what comes to the same thing — such that every
plane section thereof is a conic or quadric curve. Such surfaces have been studied
797] SURFACE. 633
from every point of view. The only singular forms are when there is (i) a conical
point (cnicnode), when the surface is a cone of the second order or quadricone ; (ii) a
conic of contact (cnictrope), when the surface is this conic ; from a different point of
view it is a surface aplatie or flattened surface. Excluding these degenerate forms,
the surface is of the order, rank, and class each = 2, and it has no singularities.
Distinguishing the forms according to reality, we have the ellipsoid, the hyperboloid of
two sheets, the hyperboloid of one sheet, the elliptic paraboloid, and the hyperbolic
paraboloid (see Geometry, Analytical, [790]). A particular case of the ellipsoid is the
sphere; in abstract geometry, this is a quadric surface passing through a given quadric
curve, the circle at infinity. The tangent plane of a quadric surface meets it in a
quadric curve having a node, that is, in a pair of lines; hence there are on the surface
two singly infinite sets of lines. Two lines of the same set do not meet, but each line
of the one set meets each line of the other set ; the surface is thus a regulus in a
twofold manner. The lines are real for the hyperboloid of one sheet and for the
hyperbolic paraboloid ; for the other forms of surface they are imaginary.
11. We have next the surface of the third order or cubic surface, which has also
been very completely studied. Such a surface may have isolated point singularities
(cnicnodes or points of higher singularity), or it may have a nodal line; we have thus
21 + 2, = 23 cases. In the general case of a surface without any singularities, the
order, rank, and class are =3, 6, 12 respectively. The surface has upon it 27 lines,
lying by threes in 45 planes, which are triple tangent planes. Observe that the
tangent plane is a plane meeting the surface in a curve having a node. For a
surface of any given order n there will be a certain number of planes each meeting
the surface in a curve with 3 nodes, that is, triple tangent planes; and, in the
particular case where n = 3, the cubic curve with 3 nodes is of course a set of 3 lines ;
it is found that the number of triple tangent planes is, as just mentioned, = 45. This
would give 135 lines, but through each line we have 5 such planes, and the number
of lines is thus = 27. The theory of the 27 lines is an extensive and interesting one ;
in particular, it may be noticed that we can, in thirty-six ways, select a system of
6x6 lines, or " double sixer," such that no 2 lines of the same set intersect each
other, but that each line of the one set intersects each line of the other set.
A cubic surface having a nodal line is a ruled surface or regulus; in fact, any
plane through the nodal line meets the surface in this line counting twice and in a
residual line, and there is thus on the surface a singly infinite set of lines. There
are two forms; but the distinction between them need not be referred to here.
12. As regards quartic surfaces, only particular forms have been much studied.
A quartic surface can have at most 16 conical points (cnicnodes); an instance of such
a surface is Fresnel's wave surface, which has 4 real cnicnodes in one of the principal
planes, 4x2 imaginary ones in the other two principal planes, and 4 imaginary ones at
infinity, — in all 16 cnicnodes; the same surface has also 4 real + 12 imaginary planes
each touching the surface along a circle (cnictropes), — in all 16 cnictropes. It was
easy by a mere homographic transformation to pass to the more general surface called
the tetrahedroid ; but this was itself only a particular form of the general surface
c. XI. 80
634 SURFACE. [797
with 16 cnicnodes and 16 cnictropes first studied by Kummer. Quartic surfaces with
a smaller number of cnicnodes have also been considered.
Another very important form is the quartic surface having a nodal conic ; the
nodal conic may be the circle at infinity, and we have then the so-called anallagmatic
surface, otherwise the cyclide (which includes the particular form called Dupin's cyclide).
These correspond to the bicircular quartic curve of plane geometry. Other forms of
quartic surface might be referred to.
Congruences and Complexes.
13. A congruence is a doubly infinite system of lines. A line depends on four
parameters and can therefore be determined so as to satisfy four conditions ; if only
two conditions are imposed on the line, we have a doubly infinite system of lines or
a congruence. For instance, the lines meeting each of two given lines form a con
gruence. It is hardly necessary to remark that, imposing on the line one more
condition, we have a ruled surface or regulus ; thus we can in an infinity of ways
separate the congruence into a singly infinite system of reguli or of torses (see infra,
No. 16).
Considering in connexion with the congruence two arbitrary lines, there will be in
the congruence a determinate number of lines which meet each of these two lines ;
and the number of lines thus meeting the two lines is said to be the order-class
of the congruence. If the two arbitrary lines are taken to intersect each other, the
congruence lines which meet each of the two lines separate themselves into two
sets, — those which lie in the plane of the two lines and those which pass through
their intersection. There will be in the former set a determinate number of congruence
lines which is the order of the congruence, and in the latter set a determinate number
of congruence lines which is the class of the congruence. In other words, the order
of the congruence is equal to the number of congruence lines lying in an arbitrary
plane, and its class to the number of congruence lines passing through an arbitrary
point.
The following systems of lines form each of them a congruence : — (A) lines meeting
each of two given curves ; (B) lines meeting a given curve twice ; (C) lines meeting
a given curve and touching a given surface ; (D) lines touching each of two given
surfaces ; (E) lines touching a given surface twice, or, say, the bitangents of a given
surface.
The last case is the most general one ; and conversely, for a given congruence,
there will be in general a surface having the congruence lines for bitangents. This
surface is said to be the focal surface of the congruence ; the general surface with
16 cnicnodes first presented itself in this manner as the focal surface of a congruence.
But the focal surface may degenerate into the forms belonging to the other cases
A, B, C, D.
14. A complex is a triply infinite system of lines, — for instance, the tangent lines
of a surface. Considering an arbitrary point in connexion with the complex, the com-
797] SURFACE. 635
plex lines which pass through the point form a cone ; considering a plane in connexion
with it, the complex lines which lie in the plane envelope a curve. It is easy to
see that the class of the curve is equal to the order of the cone ; in fact, each of
these numbers is equal to the number of complex lines which lie in an arbitrary
plane and pass through an arbitrary point of that plane; and we then say order of
complex = order of curve ; rank of complex = class of curve = order of cone ; class of
complex = class of cone. It is to be observed that, while for a congruence there is
in general a surface having the congruence lines for bitangents, for a complex there
is not in general any surface having the complex lines for tangents ; the tangent
lines of a surface are thus only a special form of complex. The theory of complexes
first presented itself in the researches of Malus on systems of rays of light in connexion
with double refraction.
15. The analytical theory as well of congruences as of complexes is most easily
carried out by means of the six coordinates of a line; viz. there are coordinates
(a, b, c, f, g, h) connected by the equation af+bg + ch = 0, and therefore such that the
ratios a : b : c : f : g : h constitute a system of four arbitrary parameters. We have
thus a congruence of the order n represented by a single homogeneous equation of
that order (*$a, b, c, f, g, h)n = 0 between the six coordinates ; two such relations
determine a congruence. But we have in regard to congruences the same difficulty
as that which presents itself in regard to curves in space : it is not every congruence
which can be represented completely and precisely by two such equations.
The linear equation (*$a, b, c, f, g, h) = 0 represents a congruence of the first
order or linear congruence; such congruences are interesting both in geometry and in
connexion with the theory of forces acting on a rigid body.
Curves of Curvature; Asymptotic Lines.
16. The normals of a surface form a congruence. In any congruence, the lines
consecutive to a given congruence line do not in general meet this line ; but there is
a determinate number of consecutive lines which do meet it ; or, attending for the
moment to only one of these, say the congruence line is met by a consecutive con
gruence line. In particular, each normal is met by a consecutive normal; this again
is met by a consecutive normal, and so on. That is, we have a singly infinite system
of normals each meeting the consecutive normal, and so forming a torse ; starting from
different normals successively, we obtain a singly infinite system of such torses. But
each normal is in fact met by two consecutive normals, and, using in the construction
first the one and then the other of these, we obtain two singly infinite systems of
torses each intersecting the given surface at right angles. In other words, if in place
of the normal we consider the point on the surface, we obtain on the surface two
singly infinite systems of curves such that for any curve of either system the normals
at consecutive points intersect each other ; moreover, for each normal the torses of the
two systems intersect each other at right angles ; and therefore for each point of the
surface the curves of the two systems intersect each other at right angles. The two
systems of curves are said to be the curves of curvature of the surface.
80—2
636 SURFACE. [797
The normal is met by the two consecutive normals in two points which are the
centres of curvature for the point on the surface ; these lie either on the same side
of the point or on opposite sides, and the surface has at the point in question like
curvatures or opposite curvatures in the two cases respectively (see supra, No. 4).
17. In immediate connexion with the curves of curvature, we have the so-called
asymptotic curves (Haupt-tangenten-linien). The tangent plane at a point of the surface
cuts the surface in a curve having at that point a node. Thus we have at the point
of the surface two directions of passage to a consecutive point, or, say, two elements
of arc ; and, passing along one of these to the consecutive point, and thence to a
consecutive point, and so on, we obtain on the surface a curve. Starting successively
from different points of the surface we thus obtain a singly infinite system of curves ;
or, using first one and then the other of the two directions, we obtain two singly
infinite systems of curves, which are the curves above referred to. The two curves
at any point are equally inclined to the two curves of curvature at that point, or —
what is the same thing — the supplementary angles formed by the two asymptotic lines
are bisected by the two curves of curvature. In the case of a quadric surface, the
asymptotic curves are the two systems of lines on the surface.
Geodesic Lines.
18. A geodesic line (or curve) is a shortest curve on a surface ; more accurately,
the element of arc between two consecutive points of a geodesic line is a shortest arc
on the surface. We are thus led to the fundamental property that, at each point of
the curve, the osculating plane of the curve passes through the normal of the surface ;
in other words, any two consecutive arcs PP', P'P" are in piano with the normal at
P'. Starting from a given point P on the surface, we have a singly infinite system
of geodesies proceeding along the surface in the direction of the several tangent lines
at the point P ; and, if the direction PP' is given, the property gives a construction
by successive elements of arc for the required geodesic line.
Considering the geodesic lines which proceed from a given point P of the surface,
any particular geodesic line is or is not again intersected by the consecutive gene
rating line ; if it is thus intersected, the generating line is a shortest line on the
surface up to, but not beyond, the point at which it is first intersected by the con
secutive generating line ; if it is not intersected, it continues a shortest line for the
whole course.
In the analytical theory both of geodesic lines and of the curves of curvature,
and in other parts of the theory of surfaces, it is very convenient to consider the
rectangular coordinates x, y, z of a point of the surface as given functions of two
independent parameters p, q ; the form of these functions of course determines the
surface, since by the elimination of p, q from the three equations we obtain the equation
in the coordinates x, y, z. We have for the geodesic lines a differential equation of the
second order between p and q ; the general solution contains two arbitrary constants,
797] , SURFACE. 637
and is thus capable of representing the geodesic line which can be drawn from a
given point in a given direction on the surface. In the case of a quadric surface, the
solution involves hyperelliptic integrals of the first kind, depending on the square root
of a sextic function.
Curvilinear Coordinates.
19. The expressions of the coordinates x, y, z in terms of p, q may contain a
parameter r, and, if this is regarded as a given constant, these expressions will as
before refer to a point on a given surface. But, if p, q, r are regarded as three
independent parameters, x, y, z will be the coordinates of a point in space, determined
by means of the three parameters p, q, r ; these parameters are said to be the
curvilinear coordinates, or (in a generalized sense of the term) simply the coordinates
of the point. We arrive otherwise at the notion by taking p, q, r each as a given
function of x, y, z; say we have p =/i (x, y, z), q =/2 (x, y, z), r=f3 (x, y, z), which
equations of course lead to expressions for p, q, r each as a function of x, y, z. The
first equation determines a singly infinite set of surfaces : for any given value of p
we have a surface ; and similarly the second and third equations determine each a
singly infinite set of surfaces. If, to fix the ideas, f1} f2, f3 are taken to denote each
a rational and integral function of x, y, z, then two surfaces of the same set will not
intersect each other, and through a given point of space there will pass one surface
of each set; that is, the point will be determined as a point of intersection of three
surfaces belonging to the three sets respectively ; moreover, the whole of space will be
divided by the three sets of surfaces into a triply infinite system of elements, each
of them being a parallelepiped.
Orthotomic Surfaces; Parallel Surfaces.
20. The three sets of surfaces may be such that the three surfaces through any
point of space whatever intersect each other at right angles ; and they are in this
case said to be orthotomic. The term curvilinear coordinates was almost appropriated
by Lame, to whom this theory is chiefly due, to the case in question : assuming that
the equations p =f^ (x, y, z\ q =/2 (x, y, z), r =/3 (x, y, z) refer to a system of ortho
tomic surfaces, we have in the restricted sense p, q, r as the curvilinear coordinates
of the point.
An interesting special case is that of confocal quadric surfaces. The general
equation of a surface confocal with the ellipsoid
2 ^ Z*2 ' 2 2 i /3 7 ** i /]
and, if in this equation we consider x, y, z as given, we have for 6 a cubic equation
with three real roots p, q, r, and thus we have through the point three real surfaces,
one an ellipsoid, one a hyperboloid of one sheet, and one a hyperboloid of two sheets.
638 SUBFACE. [797
21. The theory is connected with that of curves of curvature by Dupin's theorem.
Thus in any system of orthotomic surfaces, each surface of any one of the three sets
is intersected by the surfaces of the other two sets in its curves of curvature.
22. No one of the three sets of surfaces is altogether arbitrary : in the equation
p=fl(x, y, z), p is not an arbitrary function of x, y, z, but it must satisfy a certain
partial differential equation of the third order. Assuming that p has this value, we
have q =/2 (x, y, z} and r =fs (x, y, z} determinate functions of x, y, z, such that the
three sets of surfaces form an orthotomic system.
23. Starting from a given surface, it has been seen (No. 16) that the normals
along the curves of curvature form two systems of torses intersecting each other, and
also the given surface, at right angles. But there are, intersecting the two systems
of torses at right angles, not only the given surface, but a singly infinite system of
surfaces. If at each point of the given surface we measure off along the normal one
and the same distance at pleasure, then the locus of the points thus obtained is a
surface cutting all the normals of the given surface at right angles, or, in other words,
having the same normals as the given surface ; and it is therefore a parallel surface
to the given surface. Hence the singly infinite system of parallel surfaces and the two
singly infinite systems of torses form together a set of orthotomic surfaces.
The Minimal Surface.
24. This is the surface of minimum area — more accurately, a surface such that,
for any indefinitely small closed curve which can be drawn on it round any point, the
area of the surface is less than it is for any other surface whatever through the closed
curve. It at once follows that the surface at every point is concavo-convex ; for, if
at any point this was not the case, we could, by cutting the surface by a plane,
describe round the point an indefinitely small closed plane curve, and the plane area
within the closed curve would then be less than the area of the element of surface
within the same curve. The condition leads to a partial differential equation of the
second order for the determination of the minimal surface : considering z as a function
of x, y, and writing as usual p, q, r, s, t for the first and second differential coefficients
of z in regard to x, y respectively, the equation (as first shown by Lagrange) is
(1 -f q2)r — 2pqs + (l +p'2)t= 0, or, as this may also be written,
. .
dy Vl +p* + q2 dx Vl + p2 + q2
The general integral contains of course arbitrary functions, and, if we imagine these
so determined that the surface may pass through a given closed curve, and if, more
over, there is but one minimal surface passing through that curve, we have the
solution of the problem of finding the surface of minimum area within the same
curve. The surface continued beyond the closed curve is a minimal surface, but it is
not of necessity or in general a surface of minimum area for an arbitrary bounding
curve not wholly included within the given closed curve. It is hardly necessary to
797] SURFACE. 639
remark that the plane is a minimal surface, and that, if the given closed curve is a
plane curve, the plane is the proper solution ; that is, the plane area within the given
closed curve is less than the area for any other surface through the same curve. The
given closed curve is not of necessity a single curve : it may be, for instance, a skew
polygon of four or more sides.
The partial differential equation was dealt with in a very remarkable manner by
Riemann. From the second form given above it appears that we have , ^ JL = a
Vi +p* + <?
complete differential, or, putting this =d£, we introduce into the solution a variable f,
which combines with z in the forms z ± i% (i = V — 1 as usual). The boundary conditions
have to be satisfied by the determination of the conjugate variables 77, rf as functions
of z + i£, z—i%, or, say, of Z, Z' respectively. By writing S, S' to denote x + iy, x—iy
respectively, Riemann obtains finally two ordinary differential equations of the first
order in S, S', 77, 77', Z, Z', and the results are completely worked out in some very
interesting special cases.
The memoirs on various parts of the general subject are very numerous ; references
to many of them will be found in Salmon's Treatise on the Analytic Geometry of Three
Dimensions, 4th ed., Dublin, 1882 (the most comprehensive work on solid geometry);
for the minimal surface (which is not considered there) see Memoirs xvn. and xxvi.
in Riemann's Gesammelte mathematische Werke, Leipsic, 1876 ; the former — " Ueber die
Flache vom kleinsten Inhalt bei gegebener Begrenzung," as published in Gott. Abhandl.,
vol. XTII. (1866 — 67) — contains an introduction by Hattendorff giving the history of the
question.
€40 [798
798.
WALLIS (JOHN).
[From the Encyclopedia Britannica, Ninth Edition, vol. xxiv. (1888), pp. 331, 332.]
WALLIS, JOHN (1616 — 1703), an eminent English mathematician, logician, and
grammarian, was born on the 23rd November 1616 at Ashford, in Kent, of which
parish his father was then incumbent. Having been previously instructed in Latin,
Greek, and Hebrew, he was in 1632 sent to Emmanuel College, Cambridge, and after
wards was chosen fellow of Queens' College. Having been admitted to holy orders, he
left the university in 1641 to act as chaplain to Sir William Barley, and in the
following year accepted a similar appointment from the widow of Sir Horatio Vere.
It was about this period that he displayed surprising talents in deciphering the inter
cepted letters and papers of the Royalists. His adherence to the Parliamentary party
was in 1643 rewarded by the living of St Gabriel, Fenchurch Street, London. In
1644 he was appointed one of the scribes or secretaries of the Assembly of Divines
at Westminster. During the same year he married Susanna Glyde, and thus vacated
his fellowship; but the death of his mother had left him in possession of a handsome
fortune. In 1645 he attended those scientific meetings which led to the establishment
of the Royal Society. When the Independents obtained the superiority, Wallis adhered
to the Solemn League and Covenant. The living of St Gabriel he exchanged for that
of St Martin, Ironmonger Lane; and, as rector of that parish, he in 1648 subscribed
the Remonstrance against putting Charles I. to death. Notwithstanding this act of
opposition, he was in June 1649 appointed Savilian professor of geometry at Oxford.
In 1654 he there took the degree of D.D., and four years later succeeded Dr Langbaine
as keeper of the archives. After the Restoration, he was named one of the king's
chaplains in ordinary. While complying with the terms of the Act of Uniformity,
Wallis seems always to have retained moderate and rational notions of ecclesiastical
polity. He died at Oxford on the 28th of October 1703, in the eighty-seventh year of
his age.
798] WALLIS. 641
The works of Wallis are numerous, and relate to a multiplicity of subjects. His
Institutio Logicw, published in 1687, was very popular, and in his Grammatica Lingua
Anglicance we find indications of an acute and philosophic intellect. The mathematical
works are published some of them in a small 4to volume, Oxford, 1657, and a com
plete collection in three thick folio volumes, Oxford, 1695-93-99. The third volume
includes, however, some theological treatises, and the first part of it is occupied with
editions of treatises on harmonics and other works of Greek geometers, some of them
first editions from the MSS., and in general with Latin versions and notes (Ptolemy,
Porphyrius, Briennius, Archimedes, Eutocius, Aristarchus, and Pappus). The second and
third volumes include also two collections of letters to and from Brouncker, Frenicle,
Leibnitz, Newton, Oldenburg, Schooten, and others; and there is a tract on trigonometry
by Caswell. Excluding all these, the mathematical works contained in the first and
second volumes occupy about 1800 pages. The titles in the order adopted, but with
date of publication, are as follows: — " Oratio Inauguralis," on his appointment (1649) as
Savilian professor, 1657 ; " Mathesis Universalis, seu Opus Arithmeticum Philologice et
Mathematice Traditum, Arithmeticam Numerosam et Speciosam Aliaque Continens,"
1657 ; "Adversus Meibomium, de Proportionibus Dialogus," 1657 ; "De Sectionibus Conicis
Nova Methodo Expositis," 1655; " Arithmetica lufinitorum, sive Nova Methodus Inqui-
rendi in Curvilineorum Quadraturam Aliaque Difficiliora Matheseos Problemata," 1655 ;
"Eclipsis Solaris Observatio Oxonii Habita 2nd Aug. 1654," 1655; "Tractatus Duo,
prior de Cycloide, posterior de Cissoide et de Curvarum turn Linearum Ei50iWei turn
Superficierum HXarvcrfjuw" 1659 ; " Mechanica, sive de Motu Tractatus Geometricus,"
three parts, 1669-70-71; " De Algebra Tractatus Historicus et Practicus, ejusdem orig-
inem et progressus varios ostendens," English, 1685 ; "De Combinationibus Alternationibus
et Partibus Aliquotis Tractatus," English, 1685 ; " De Sectionibus Angularibus Tractatus,"
English, 1685 ; " De Angulo Contactus et Semicirculi Tractatus," 1656 ; " Ejusdem
Tractatus Defensio," 1685; "De Postulate Quinto, et Quinta Definitione, Lib. VI.
Euclidis, Disceptatio Geometrica," ?1663; " Cuno-Cuneus, seu Corpus partim Conum
partim Cuneum Representans Geometrice Consideratum," English, 1685; " De Gravitate
et Gravitatione Disquisitio Geometrica," 1662 (English, 1674); "De JSstu Maris Hypo
thesis Nova," 1666—69.
The Arithmetica Infinitorum relates chiefly to the quadrature of curves by the
so-called method of indivisibles established by Cavalieri, 1629, and cultivated in the
interval by him, Fermat, Descartes, and Roberval. The method is substantially that of
the integral calculus ; thus, e.g., for the curve y = #2 to find the area from an = 0 to
#=1, the base is divided into n equal parts, and the area is obtained as
= i, (1= + 22 + ... + O, = ^ n (n + 1) (2n + 1),
which, taking n indefinitely large, is = £. The case of the general parabola y = xm
(in a positive integer or fraction), where the area is ^ri > na(^ keen Previ°usly solved.
Wallis made the important remark that the reciprocal of such a power of x could be
regarded as a power with a negative exponent (••— = x~m\ , and he was thus enabled
\3C I
C. XT. 81
642 WALLIS. [798
to extend the theorem to certain hyperbolic curves, but the case m a negative value
larger than 1 presented a difficulty which he did not succeed in overcoming. It should
be noticed that Wallis. although not using the notation xm in the case of a positive
or negative fractional value, nor indeed in the case of a negative integer value of m,
deals continually with such powers, and speaks of the positive or negative integer or
fractional value of m as the index of the power. The area of a curve, y = sum of
a finite number of terms Ax™, was at once obtained from that for the case of a
single term ; and Wallis, after thus establishing the several results which would now
f1 f1 f1 f1
be written I (x — #2)° dx = 1, I (x — x2)1 dx = ^, I (x — x2)2 dx = -fa, I (x — x2)* dx = Ti^, &c.,
Jo Jo •' o .' o
proposed to himself to interpolate from these the value of I (x — x2)^ dx, which is the
.'o
expression for the area (= ^TT) of a semicircle, diameter = 1 ; making a slight trans-
(4\
= - ] , the term halfway
between 1 and 2, in the series of terms 1, 2, 6, 20, 70,...; and he thus obtained the
2.4.4.6.6.8.8... .
remarkable expression TT = — =— = — =— ^ — , together with a succession or superior
and inferior limits for the number TT.
fWj
In the same work, Wallis obtained the expression which would now be written
ds = dx A/1 + ( j^J for the length of the element of a curve, thus reducing the
problem of rectification to that of quadrature. An application of this formula to an
algebraical curve was first made a few years later by W. Neil ; the investigation is
reproduced in the " Tractatus de Cissoide, &c." (1659, as above), and Wallis adds the
remark that the curve thus rectified is in fact the semicubical parabola.
The Mathesis Universalis is a more elementary work intended for learners. It
contains copious dissertations on fundamental points of algebra, arithmetic, and geometry,
and critical remarks.
The De Algebra Tractatus contains (chapters 66 — 69) the idea of the interpretation
of imaginary quantities in geometry. This is given somewhat as follows: the distance
represented by the square root of a negative quantity cannot be measured in the line
backwards or forwards, but can be measured in the same plane above the line, or (as
appears elsewhere) at right angles to the line either in the plane, or in the plane at
right angles thereto. Considered as a history of algebra, this work is strongly objected
to by Montucla, on the ground of its unfairness as against the early Italian algebraists
and also ^7'ieta and Descartes, and in favour of Harriot ; but De Morgan, while
admitting this, attributes to it considerable merit.
The two treatises on the cycloid and on the cissoid, &c., and the Mechanica
contain many results which were then new and valuable. The latter work contains
elaborate investigations in regard to the centre of gravity, and it is remarkable also
for the employment of the principle of virtual velocities. The cuno-cuneus is a highly
798] WALLIS. 643
interesting surface ; it is a ruled quartic surface, the equation of which may be written
cy = (c- zf (a2 - so2).
Among the letters in volume HI., there is one to the editor of the Leipsic Acts,
giving the decipherment of two letters in secret characters. The ciphers are different,
but on the same principle : the characters in each are either single digits or com
binations of two or three digits, standing some of them for letters, others for syllables
or words, — the number of distinct characters which had to be deciphered being thus
very considerable.
For the prolonged conflict between Hobbes and Wallis, see the article Hobbes,
[Encyclopcedia Britannica, ninth edition,] vol. XH. pp. 36 — 38.
END OF VOL. XI.
CAMBRIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
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