MATHEMATICAL PAPEKS.
JLonHon: 0. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE,
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MATH KM ATI
\ I'l
THE COLLECTED
MATHEMATICAL PAPERS
OF
AETHUE CAYLEY, Sc.D., F.E.S.,
LATE 8ADLERIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE.
VOL. XI.
CAMBRIDGE :
AT THE UNIVERSITY PRESS.
1896
[All Riff/Us reserved.]
CAMBKIDGE :
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.
3
ADVEETISEMENT.
rMHE 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 Encyclopaedia
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.
vi 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,
M VIII.
486
555,
ix.
556
629,
,, X. ,,
630
705,
XL
706
798.
A. R. FORSYTH.
*
21 November, 1896.
Vll
CONTENTS.
[An Asterisk means that the paper is not printed in full.]
PAOK
706. On the distribution of electricity on two spherical surfaces . 1
Phil. Mag., Ser. 5, t. v. (1878), pp. 5460
707. On the colouring of maps ........ 7
Geogr. Soc. Proc., t. i. (1879), pp. 259261
708. Note sur la theorie des courbes de I'espace .... 9
Assoc. Fran<;., Compt. Rend., t. ix. (1880), pp. 135139
709. On the number of constants in the equation of the surface
PS-QR = 14
Tidsskrift for Mathematik, Ser. 4, t. IV. (1880), pp. 145148
710. On a differential equation . .... . . . 17
Collectanea Mathematica, in memoriam Dominici Ohelini, (Milan,
Hoepli, 1881), pp. 1726
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. Rowe's "Memoir on Abel's theorem" . . 29
Phil. Trans., t. CLXXII. (1881), pp. 751758
Vlll CONTENTS.
PAOI;
714. Various notes 37
Messenger of Mathematics, t. vu. (1878), pp. 69: 115: 124: 125
715. Note on a system of algebraical equations . . . . 39
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 ....... 47
Messenger of Mathematics, t. vn. (1878), pp. 48 50
718. Addition to Mr. Genese's 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. Formulae involving tlie 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: lL'7
724. On the deformation of the model of a hyperboloid ... 66
Messenger of Mathematics, t. vin. (1879), pp. 51, 52
dx du
725. New formulce for the integration of -r^+ -7^ = 68
JA. V i
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. vin. (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. Lend. Math. Soc., t. x. (1879), pp. 4348
729. On a theorem relating to conformable figures .... 78
Proc. Lond. Math. Soc., t. x. (1879), pp. 143146
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. 194196
731. On the binomial equation x p 1 = 0; trisection and quarti-
section ........... 84
Proc. Lond. Math. Soc., t. xi. (1880), pp. 417
732. A theorem in spherical trigonometry ..... 97
Proc. Lond. Math. Soc., t. xi. (1880), pp. 4850
733. On a formula of elimination ....... 100
Proc. Lond. Math. Soc., t. xi. (1880), pp. 139141
734. On the kinematics of a plane . . . . . . . 103
Quart. Math. Journ., t. xvi. (1879), pp. 18
735. Note on the theory of apsidal surfaces . . . . . Ill
Quart. Math. Journ., t. xvi. (1879), pp. 109112
736. Application of the Newton- Fourier method to an imaginary
root of an equation . . . . . . . . 114
Quart. Math. Journ., t. xvi. (1879), pp. 179185
737. On a covariant formula ........ 122
Quart. Math. Journ., t. xvi. (1879), pp. 224226
738. Note on a hypergeometric series . . . . . . 125
Quart. Math. Journ., t. xvi. (1879), pp. 268270
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 Abets relating to a quintic equation . 132
Oamb. Phil. Soc. Proc., t. in. (1880), pp. 155159
742. On the transformation of coordinates 136
Camb. Phil: Soc. Proc., t. in. (1880), pp. 178184
743. On the Newton- Fourier problem ... . 143
Camb. Phil. Soc. Proc., t. in. (1880), pp. 231, 232
744. Table of A m O n -i-n (m) up to m = n = 20 144
Camb. Phil. Trans., t. xm. (1883), pp. 14
745. On the Schwarzian derivative, and the polyhedral functions . 148
Camb. Phil. Trans., t. xm. (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. 383385
748. On the bitangents of a quartic . . . . . . . 221
Salmon's Higher Plane Curves, (3rd ed., 1879), pp. 387389
*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. 539550
751. Note on Riemann's paper " Versuch 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 vanable ;
with a correction . . . . . . . . . 237
Mathematische Annalen, t. xvi. (1880), pp. 260263; 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 formulae 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
\c,d\
ax + b 052
7* * . . Lt\) *j
cx + d
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. xi. (1882), pp. 15 18
758. Solution of a Senate-House problem ..... 265
Messenger of Mathematics, t. xi. (1882), pp. 23 25
759. Illustration of a theorem in the theory of equations . , 268
Messenger of Mathematics, t. xi. (1882), pp. Ill 113
760. Reduction
( dx
ion of \-T- -^i to elliptic integrals . . . . 270
J ( 1 or)
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. 137147
762. On Schubert's method for the contacts of a line with a surface 281
Quart. Math. Journ., t. xvn. (1881), pp. 244258
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 x p 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. 103108
766. On the geodesic curvature of a curve on a surface . . . 323
Proc. Lond. Math. Soc., t. xn. (1881), pp. 110117
62
XJi CONTENTS.
PAOE
767. On the Gaussian theory of surf aces . . 331
Proc. Lond. Math. Soc., t xn. (1881), pp. 187192
768. Note on Landen's theorem ... . 337
Proc. Lond. Math. Soc., t xm. (1882), pp. 47, 48
769. On a formula relating to elliptic integrals of the third kind . 340
Proc. Lond. Math. Soc., t. xui. (1882), pp. 175, 176
770. On the 34 concomitants of the ternary cubic .... 342
American Journal of Mathematics, t. IV. (1881), pp. 1 15
771. Specimen of a literal table for binary quantics, 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. 266268
773. On the 8-square imaginaries . . . . . . . 368
American Journal of Mathematics, t iv. (1881), pp. 293296
774. Tables for the binary sextic 372
American Journal of Mathematics, t. iv. (1881), pp. 379 384
775. Tables of covariants of the binary sextic. .... 377
Written in 1894 : now first published.
776. On the Jacobian sextic equation . . . . . . 389
Quart. Math. Journ., t. xvm. (1882), pp. 5265
777. A solvable case of the quintic equation ..... 402
Quart. Math. Journ., t. xvm. (1882), pp. 154157
778. [Addition to Mr. Hudson's paper "On equal roots of equations"] 405
Quart. Math. Journ., t. xvm. (1882), pp. 226229
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"] 408
Proc. Lond. Math. Soc., t. xiv. (1883), p. 85
CONTENTS. Xlll
PAGE
780. [Addition to Mr. Hammond's paper "Note on an exceptional
case in which the fundamental postulate of Professor
Sylvesters theory of tamisage fails "] . . . . . 409
Proc. Lond. Math. Soc., t. xiv. (1883), pp. 8891
781. On the automorphic transformation of the binary cubic
function .....'..... 411
Proc. Lond. Math. Soc., t. xiv. (1883), pp. 103108
782. On Monge's " Memoire sur la theorie des deblais et des
remblais" . . . . . . . . . . 417
Proc. Lond. Math. Soc., t. xiv. (1883), pp. 139142
783. On Mr. Wilkinson's rectangular transformation . . . 421
Proc. Lond. Math. Soc., t. xiv. (1883), pp. 222229
784. Presidential Address to the British Association, Southport,
September 1883 429
British Association Report, 1883, pp. 3 37
785. Curve 460
Encyclopaedia Britannica, 9th ed., t. vi. (1878), pp. 716728
786. Equation 490
Encyclopaedia Britannica, 9th ed., t. vm. (1878), pp. 497 509
787. Function 522
Encyclopaedia Britannica, 9th ed., t. IX. (1879), pp. 818824
788. Galois 543
Encyclopaedia Britannica, 9th ed., t. x. (1879), p. 48
789. Gauss 544
Encyclopaedia Britannica, 9th ed., t. x. (1879), p. 116
790. Geometry (analytical) ........ 546
Encyclopaedia Britannica, 9th ed., t. x. (1879), pp. 408420
791. Landen 583
Encyclopaedia Britannica, 9th ed., t. xiv. (1882), p. 271
792. Locus 585
Encyclopaedia Britannica, 9th ed., t. xiv. (1882), pp. 764, 765
XIV CONTENTS.
PAOB
793. Monge 586
Encyclopaedia Britannica, 9th ed., t. xvi. (1883), pp. 738, 739
794. Numbers (partition of) . . . . . . . . 589
Encyclopaedia Britannica, 9th ed., t. XVH. (1884), p. 614
795. Numbers (theory of) 592
Encyclopaedia Britannica, 9th ed., t. XVH. (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. 668672
798. Wallis (John) . 640
Encyclopaedia Britannica, 9th ed., t. xxiv. (1888), pp. 331, 332
Portrait To face Title.
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 A m O" H- II (TO) ; 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.
DC-Main et remMais, 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 1'e'lectricite' a la surface des corps
conducteurs," M6m. 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 $>(x) the question depends
on the solution of the functional equations
6*
*)
c - x c-x
C X ' \C
where of course the x of either equation may be replaced by a different variable.
It is proper to consider the meaning of these equations : for a point on the axis,
at the distance x from the centre of the first sphere, or say from the point A, the
potential of the electricity on this spherical surface is a<f>x or < ( ) , according as
x \ x J
the point is interior or exterior ; and, similarly, if x now denote the distance from
the centre of the second sphere (or, say, from the point B), then the potential of
the electricity on this spherical surface is b<$>x or 4>( ), according as the point is
oc \sc /
interior or exterior ; <f> (x) is thus the same function of (x, a, b) that <I> (x) 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 x 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
c
and, secondly, for a point interior to the sphere B, if x denote the distance from B
and therefore cx the distance of the same point from A, the potential of the
point is
c -
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
a 2 . /a a \ b" . f b 3 \
= d) ( -} + - * ;
x ^ \x) cx \c-xl
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 <i>(x) gives thus the complete solution of the question.
I obtain Poisson's solution by a different process as follows: Consider the two
functions
o a (c x) a# + b
- , = - j , suppose,
c 3 -b" cx cx + d
and
b 1 (c - x) cuK +
-
and let the nth functions be
, , suppose;
c 2 a" ex yx+ S
T^ and
d n 7,, x -f S n
respectively.
Observing that the values of the coefficients are
(a, b ) = ( -a 2 , a 2 c ), and (a, /3 ) = ( -6 2 , 6 2 c
c, d -c, c*-b- 7, 8 -c, <?-a-
so that we have
a + d = a + 8, = c- a 2 6 2 , ad be = aS /3-y, = a-b-,
and consequently that the two equations
(a + d) a (X + I) 2 = (q + 8y
ad-bc' X a8-/3y'
706] ON TWO SPHERICAL SURFACES.
are in fact one and the same equation
for the determination of X, then (by a theorem which [686, 687] I have recently
obtained) we have the following equations for the coefficients
, b n ),
c, d n
of the nth functions ; viz. these are :
7,,,
*' + b n = -j y {(V+ 1 - 1) (a* + b) + (V - X) (- d# + b)J,
c n x + d n = {(\+i_i)(ca; + d) + (\-\)( ex -a)};
and similar!}-
"' 1 - X) (- &* + )},
7* -a)}.
Observe that these equations give, as they ought to do,
and similarly
>2
Substituting in the first two equations - - in place of x, and in the second two
C 3s
b'
equations - - in place of x, we obtain the following results which will be useful :
C ~~ 00
ana 2 + b n (c - x) = a 2 (y n x + 8 n ),
c n a 2 + d n (c - x) = ^ (a n+1 * + /8 B+1 ),
n (C - X) = b 1 (C n x + d n ),
S n (c - a;) = - (a n+1 a; + b n+1 ),
a
the last 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
--
b n (c - *) = {(X+> - 1) [a 2 + b (c - *)] + (X - X) [- da 2 + b (c - *)]},
12
4 ON THE DISTRIBUTION OF ELECTRICITY [706
where the term in { } is
= (X"- 1 - 1 - 1) [- a* + a'c (c - *)] + (X" - X) [a j (b* - c 2 ) + a'c (c - x)] ;
viz. this is
= a' {(X+' - 1) (c 3 - a 2 - ex) + (X - X) (b> - ex)} ;
or it is
= a? {(\ n+l - 1) (yx +8) + (X - X) (yx - a)},
whence the relation in question.
The proof of the second equation is a little more complicated. We have
1 /a -i- H \ n ~ 1
Cna' + d (c - x) = ^~ gJ) {(X - 1) [ca + d (c - *)] + (X - X) [ca 2 - a (c - *)]},
where the term in { } is
= (X+' - 1) [- ca? + (c 2 - V) (c - x)] + (X - X) [- ca 2 + a 2 (c - )].
Comparing this with
" > - X) (- S* + ft)},
where the term in { } is
= (X+> - 1) [6 s (c - *)] + (X"-*- 1 - X) [- c (c 2 - a 2 - 6 2 ) + (c 2 - a 2 ) (c - x)],
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
(X n+1 - 1) (- ca 2 ) + (X - X) (-co 2 ) = a !i (X+ 1 -l+X >> -X) o(X + l)
(X n + 1 -X){-c(c 2 -a a -& ! )} ~(X+ 1 -X)(c 3 -a 1! -6 2 )' "(c 2 - a 2 -6 a )X '
and when x = Q is
(X"+'-l)c(c 2 -a 2 -6 2 ) (X n+1
(X" +a - 1) 6 2 c + (X" +1 - X) b-c ' ~(X n+2 -l+X n+1 -X)6 2 '
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
c 2 a? 6 2
always the same value ; say it is = , , . Hence, observing that a + d = a + 8,
^X T 1)
= c 2 a 2 6 J , the quotient, c n a 2 + d n (c x) divided by ttn+iX + ft n+1 , is
X + l c 3 -o 2 -6 2
or we have the required equation
d (c - x) = - a (ctn^x
706] OX TWO SPHERICAL SURFACES. 5
Considering now the functional equations, suppose for the moment that g is = ;
the two equations may be satisfied by assuming
We in fact, from the foregoing relations, at once obtain
a 2 . a 2 , [ to to 2 I a 2 6 2
^ i _ . a ~" .. .. t Q '" \ ~f\ '
C
...}*
c x c x \c l x + d l C 2 aj+d 2
To satisfy the first equation we must have M=aL; viz. this being so, the equation
becomes
6" \ aLh
6 2 ,f 6 2 \ aLh
a<i>x + --<&- = j- ;
c a; \c-xj c x + d
or, since c + d =l, the equation will be satisfied if only aL = l, whence also M=l.
And the second equation will be satisfied if only - = bM ; viz. substituting for L, M
their value, we find (o = 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
c - x \c - x
viz. the solution is
te)= M_L_ ab ) ab
|
'" ^
t g _ 1_ ab }
A + '- + "
We have a general solution containing an arbitrary constant P by adding to the
foregoing values for if>x a term
Pb(a-b)
Va a (c - x) - x (c 2 - 6 3 - ex) '
and for <f>x a term
Pa (6 -a) ______
6 THE DISTRIBUTION OF ELECTRICITY ON TWO SPHERICAL SURFACES. [706
as may be easily verified if we observe that the function
a 1 (c x) - x (c 1 6 2 ex),
writing therein - for x, becomes
and similarly that
6 J (c x) x (c 2 o 2 ex),
writing therein - - for x, becomes
c x
= ^ {a?(c-x)-x(c?--cx)}.
More generally, the terms to be added are for fac a term as above, where P denotes
a? (c x)
a function of x which remains unaltered when x is changed into - , and for
C 2 > CX
<&x a term as above with P' instead of P, where P 1 denotes what P becomes when
x is changed into - . But these additional terms vanish for the electrical problem,
C ^ X
and the correct values of <f>x, <# are the particular values given above.
It is to be remarked that^the function
a 8 (c - x) . a"
~~~
c
c x
viz. considering x as the distance of a point X from A, then taking the image of A'
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
&'- (c - x)
c 2 a a ex
c
c x
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. 259261.]
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 ia 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.
8 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 (n+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 + 1 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 COURBES DE L'ESPACE.
[From the Compte Rendu de I' Association Franfaise pour I'Avancement des Sciences (1880),
pp. 135139.]
EN consideYant dans 1'espace une courbe d'espece donnee, de'terminee 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 coustantes et sera ainsi
determine'e par 4p points (le cas p = 1 est une exception) : on ne connait pas, je
pense, le nombre des courbes par les 4/> 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 : theorerne ddmontre" par moi depuis
longtemps par des considerations geometriques. (Voir Salmon, Geometry of three
dimensions, 3" e"d. 1874, p. 319.) Ce n'est que dernierement que j'ai considers la
question analytique, de trouver les Equations d'une courbe excubo-quartique qui passe
par 8 points donnes ; et meme j'ai pris pour les 8 points une disposition qui n'est
pas tout a fait generate : 1'investigation elle-meme, et la forme du resultat, m'ont
paru assez interessantes pour que je les soumette a 1'Association.
En conside'rant sur une courbe excubo-quartique 4 points donnas, le plan passant
par 3 quelconques de ces points rencontre la courbe dans un seul point ; et Ton
obtient ainsi encore 4 points sur la courbe : voila mon systeme de 8 points donne"s,
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 * = 0, y = 0, z = 0, w = 0, pour les equations des plans du tetraedre
forme* par les 4 premiers points, les coordonne'es de ces points seront (1, 0, 0, 0),
(0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) : et pour les coordonne'es des 4 autres points,
je prends (0, y lt z lt w,), (.r 2 , 0, z.,, w s ), (x 3 , y 3 , 0, w t ), (x t , y t , z,, 0).
C. XI. 2
10
NOTE SUR LA THEORIE DBS COURSES DE 1/ESPACE. [708
Les equations de la courbe sont x : y : z : w = P : Q : R : S, ou P, Q, R, S
sont des fonctions (*)(0, I) 4 d'un parametre variable 0; il s'agit de faire passer une
telle courbe par les 8 points.
Je prends a, P, y, S, a, b, c, d pour les valeurs du parametre 6 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
.B-a V 0~b n 6-c n 6-d
* V * w = A 8^a ' B 0-p : C 0^~y = D J=$'
les conditions pour les autres 4 points seront alors
=
B a-b
c a ~ c
D a-d
a- 7
a-8'
b-a
c b ~ c
b-d
~~~ ** I.
~~ ot
6-7 '
D b-*'
. c a
.B c ~ b
:
c-d
c-a
o-p
V c-S'
. d a
= A . r -
da
B d~P
~d c
d 7
x, : y s . :
x 4 : y t : z t
^videmment il y a deux Equations qui donnent la valeur de B : C, et qui servent
ainsi pour e"liminer cette quantite. De cette maniere on obtient six equations que
j'ecris comme voici :
a 6 . d c QI 7 . d P
_
a c.d b' a P .d 7 '
_w,y 3 _a d.c b a fi.cS
y,w 3 a b.cd'a 6.c@'
_ ^w., _a c .b d a S .b y
z#Ui ~ a d.b c' a 7.6 8 '
Zf>s t _b c. d a b a. d y
z.Xi b a.d c' b y .d a.'
x,iv a _b a.e d b S . c - a
b d.c a' b a. c B'
_
_ #/4 _ c a.d b c @.d a.
x $3 o b.d a'c a.d /S '
savoir X, p, v, w, K, p d^notent ici les quantite's donnees \ = - l - , etc. Le nombre
yA
des Equations ind^pendantes est 5, car Ton a identiquement \nv&icp = 1. Je remar-
que que Ton peut faire sur le parametre Q une transformation lineaire quelconque
(h0 + i) : (jd + k), et introduire ainsi 3 constantes arbitraires ; on peut done prendre a
708] NOTE SUR LA THEORIE DBS COURSES DE L/ESPACE. 11
volonte" 3 valenrs du parametre 6, c'est-a-dire les valeurs de 3 quelconques des quantite*s
a, ft, 7, B, a, b, c, d; et cela e'tant les 5 equations donneront les valeurs des autres
o quantitds. Si au moyen des equations on elimine a, ft, 7, S, on obtient entre
a, b, c, d une equation qui sera, comme on va voir, de 1'ordre 4 par rapport a
chacune de ces quantites : en prenant comme donnees a, b, c il y aura done 4 valeurs
de d; et pour 1'une quelconque de ces valeurs, celles de a, ft, 7, B seront de'termine'es
uniquement : il y aura ainsi 4 courbes qui passent chacune par les 8 points ; ce qui
est le the'oreme dont il s'agit.
J'introduis, pour abreger, la notation
a d, b d, c d, b c, c a, a b,
= f, g, h, a, b, c:
on a done identiquement
a, b, c = g-h, h-f, f-g,
a + b + c = 0,
fa -f gb + he = 0.
Les Equations prennent ainsi la forme
he a 7 . d ft
\ = -- r a ,-- , etc. ;
gb a-ft.d-y'
on, en introduisant pour plus de commodity, les symboles
L, M, N, P, Q, R,
pour designer respectivement
les equations seront
gb he fa he gb fa
F \, -*-!*>, -- c v > ~ ~r~ w > f If, -- ;- p.
he fa ^ gb fa he gb r
M _a-fl.c-&
a-S.c-ft'
P =
c-
b a . d 7
6-7. d-o'
b- S.c-a
b-a.c-8'
c ft .d a.
avec la relation identique LMNPQR 1 ; il s'agit entre ces 5 Equations d'e"liminer
a, ft, 7- 8 -
22
12 XOTE 8UR LA THEORIE DBS COURSES DE L' ESP ACE. [708
J'$cris a = a tj>, les facteurs b tt, c a, d a de P, Q, R deviennent ainsi
respectivement -c + <f>, g + <f>, -f+</>; cela e"tant, les valeurs de P, Q, R servent a
exprimer $, y, 8 en fonction de <f> : substituant ces valeurs de y9, y, B dans celles
de L, M, N, on obtient sans peine
h f(
j/ = _JL b
jy__ JL ^\
valeurs qui donnent, comme cela doit tre, LMNPQR = 1 : il faut entre ces equations
i : liiiiiiii.T <f>.
En retablissant X, p, v, w, K, p au lieu de L, M, N, P, Q, R, ces equations
deviennent
X, + Y
,
6 g X+Y<f>'
(^videmment ^f=l), ou j'ecris f, 7;, f pour ddnoter les expressions -Xor, etc., et ou
c
les valeurs des coefficients X, Y, etc., sout
X = fc (fa + crhc), 7 = - Pa - 13-hc 2 ,
X I = fb (gb + pfa), F, = gb a + pf 2 a,
Z s = be (he + gb), 7 2 = he 2 - /cgb 2 .
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
Z, F, - Z,F, = fb (gb + pfa) (he 2 - gb 2 ) - be (he + /cgb) (- gb + pf'a)
= fghbV - fgb< + fhabc'p - fgab'/cp + ghb'c" + g 2 b*c - f'habc'p - f 2 gab 2 c/j
= ghb'c 5 (f + b) + g'b' (- f + c) p - f 2 gab 2 (b + c) Kp
= ghb'c'h + g>b 4 (-
708] NOTE SUE LA THEORIE DES COURBES DE I/ESPACE. 13
et de meme
- X. 2 Y= he 2 (Pa 2 -
j - X,Y= Pa (g 2 b 2 - Ftfp + tfcfrsrp).
Done
gb 2 (h 2 c 2 - g 2 b 2 *
+ - fip he 2 (f 2 a 2 -
+ f 2 a (g 2 b 2 - Pa-'p + h 2 c 2 oT/j) = 0,
ou enfin en multipliant par av, et dans un terme g^cr^c-^vptsK, au lieu de fj.vp-&K
e'crivant - , liquation devient
A,
(fa)< vp + (gb)^ + (hey 1 - (gb) 2 (he) 2 (i + 1
- (he) 2 (fa) 2 vp ( + /*) - (fa) 2 (gb) 2 ( + p) = 0,
ou, comme on peut 1'ecrire,
, , ,
2 , (gb) 2 ,
C'est la deuxieme d'un systeme de trois equations e'quivalentes ; savoir, en multipliant
par - - et en reduisant par \/*fsr<p = 1, on obtient la premiere forme : et, en multipliant
par \K et reduisant de meme, on obtient la troisieme forme : le systeme est
, ~, f^, -/* w (X + ), - (|t + w ), - (^ + J))((fa) 2 , (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) : (6 d) (c a) : en consideYant a, b, c comme donnees, il y a done 4
valeurs de d : et Ton a deja vu que les valeurs a, /3. 7, 8 sont donnees rationnelle-
ment en fonctions de a, b, c, d: le theoreme est done ddmontre".
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. 4, t. iv. (1880), pp. 145 148.]
THE very important results contained in Mr H. Valentiner's paper " Nogle
Ssetninger om fuldstsendige Skjseringskurver 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,np 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
r,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
P = HP+ 1)0>+ 2)0' + 3) - 1, = ^(^ + 6^ + 11),
the number of constants contained in a function of the order p is = a p + 1 ; or if
we take one of the coefficients (for instance that of at* 1 ) to be unity, then the number
* Idet vi med stor Gliede optage Prof. Cayley'a simple Forklariug sf den Reduktion af Konstanttallet i
Ligningen PS-QR=0, som Hr. Valentiiier havde paavist (Tidtikr. f. Math. 1879, S. 22), skulle vi dog
bemserke, at Grunden til, at dennes Bevis er bleven Baa vanskeligt, er den, at ban tillige bar villet bevise,
at der ikke finder nogen yderligere Reduktion Sted.
709] ON THE NUMBER OF CONSTANTS IN THE EQUATION OF A SURFACE. 15
of the remaining constants is = a p viz. a p is the number of constants in the equation
of a surface of the order p. As regards the surface in question
P, Q
R, S
we may it is clear take P, Q, R each with a coefficient unity as above, but in the
remaining function S, the coefficient must remain arbitrary : the apparent number of
constants is thus = Op + a g + a,n- p + a }l _ 9 + 1 ; but there is a deduction from this number.
The equation may in fact be written in the form
+ Q, Q
= 0,
where a represents an arbitrary function of the order p q, and /8 an arbitrary function
of the degree np q: we thus introduce (aj,_ 9 + l) + (a,i_p_ 9 + 1), = a p _ q + a_p_ 5 + 2,
constants, and by means of these we can impose the like number of arbitrary relations
upon the constants originally contained in the functions P, Q, R, S respectively (say
we can reduce to zero this number a p _ ? + dn^ p ^ q + 2 of the original constants) : hence
the real number of constants is
dp + d q + d n - p + dn-q + 1 (dp-q + ffn-p-, + 2),
= ft> suppose ;
viz. this is the required number in the case n > p + q, p>q.
If however n=p + q, or p = q, or if these relations are both satisfied, then there is a
P Q'
further deduction of 1, 1, or 2: in fact, calling the last-mentioned determinant | ' , ,
then the four cases are
n > p + q, p> q,
n = p + q, p > q,
n > p -t- q, p = q,
n=p + q, p = q,
where k, I denote arbitrary constants: these, like the constants of a and {$, may be
used to impose arbitrary relations upon the original constants of P, Q, R, & ', aid
hence the number of constants is = &>, <a - 1, <u 1, a> - 2 in the four cases respectively;
where as above
F, Q'
R, S'
=
P, Q'
R', S'
ry /y
* , v
T>' O'
It , o
=
R', S'
P', <?
R, S'
=
F, Q' + kF
R', 8' + kR'
P', Of
R', S'
=
R, S'+ IR'
n -p
16 ON THE NUMBER OF CONSTANTS IN THE EQUATION OF A SURFACE. [709
If M = 4, there is in each of the four cases one system of values of p, q ; viz. the
cases are
P. <1 =
21 No. = a s + a 1 + a 2 + a 3 -a 1 -a,-l= 9 + 3+ 9 + 19 -3 - 3 - 1, = 33,
31 a, + a, + a, + a, - a, - a. - 2 = 19 + 3 + 3 + 19 - 9 - - 2, = 33,
1 1 a, + a, + a, + a, - a,, - a, - 2 = 3 + 3 + 19 + 19 - - 9 - 2, = 33,
22 a,+ a, + a J + a s -a -a -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 = 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 = 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 DIFFERENTIAL EQUATION.
[From Collectanea Mathematica : in memoriam Dominid Chelini, (Milan, Hoepli, 1881),
pp. 1726.]
IN the Memoir on hypergeometric series, Crelle, t. xv. (1836), Kummer in effect
considers a differential equation
(a'z- + 2b'z + c) dz* _ (oaf + 2bx + c) dm?
z*(z-iy> a?(ic-iy ~'
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,
or 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 determinable in terms of a, b, c ; and z is to
be found as a function of x independent of a, b, c: so that when these coefficients
are 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
the same value of z ; 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
general case where the coefficients a, b, c are regarded as having any given values
whatever.
Writing for shortness X = an? + 2bx + c, in general the integral
Ndx
f
'
where D is the product of any number n of distinct linear factors x p, and N is
a rational and integral function of x of the order n at most, and therefore also the
integral
NXdx
t
J
D
c. xi. 3
18 ON A DIFFERENTIAL EQUATION. [710
where A T is now of the order n 2 at most, is expressible as the logarithm of a
i|ua-i-algebrnical fum-timi, that is, a function containing powers the exponents of which
are incommensurable (for instance, .X* is a quasi-algebraical function): in fact, the integral
is of the form
+ ...*
p x-q
where each term is separately integrable,
*+ -+JL + ...)*?
x- x-q J-JX
C dx
J (x-p)*
where P is written to denote ap' + 2bp + c : the integral is thus = log SI, where SI
is a product of factors
(IX +
xp
raised to powers -7= , = , etc. : hence, if we have a differential equation
va V7 J
N'dz Ndx N'^Zdz _ N^Xdx
' ( ~~ '
where Z (= a'z* + 2b'z + c'), and N', D/ are functions of z such as X, N, D are of
x; then, taking log C for the constant of integration, the general integral is
\og SI' = log C+ \ogfl:
viz. we have the quasi-algebraical integral SI' CSl = 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
I4x+Idx
the general integral is in the first instance obtained in the form
which, observing that (2x+ 2)*- A" = 8 (x - I) 3 , may also be written
(z + 1) (z 1 - 3-te+l) -f =
"J~Z (Z - iy >Jx (IK -
710] ON A DIFFERENTIAL EQUATION.
I had previously obtained the solution
19
Z=[ ' Yr-1
and I wish to show that this is, in fact, the particular integral belonging to the value
C'=l of the constant of integration: for this purpose I proceed to rationalise the general
integral as regards z.
Writing for a moment
where
the integral is
we have
(x + 1) (a? -
(of +
Va; (#-!)
= 0; or rationalising, it is
= ;
and thence
P- = (l, -66, 1023, 2180, 1023, -66, l\z, I) 6 ,
Q- = (l, 42, 591, 2828, 591, 42, l\z, I) 6 ,
=_^- = (0, -108, 432, -648, 432, -108, 0$*, I) 6 ,
= - 108*(* -I) 1 ;
= 2(1, -12, 807, 2504, 807, - 12, \\z, If.
Writing the equation in the form
it thus becomes
(1, -12, 807, 2504, 807, -12, 1^, l)-z(z-\)' jjf^ + = 0,
where M has its above-mentioned value; and if we now assume (7=1, then
108 (x+
Va; (#-
)-^ +
+ 14^ + 1
and thence
^ 1 ' ~ 12< 807> 2504> 807> ~ 12> 1 ^ flr> 1)6;
32
20 ON A DIFFERENTIAL EQUATION. [710
and the rationalised equation is
(1, - 12, 807, 2504, 807, - 12, 1$*, I) 8
> ~ 12 ' 8 7 ' 25 4 ' 8 7 ' ~
This is a sextic equation in z, of the form
where
X, /*, v = - 12 - fl, 807 + 4fl, 2504 - 6fl,
if fl denote the function of x which enters into the equation ; and writing z + - = 0, this
z
becomes
0s _ 30 + x. (#2 _ 2) + fj.0 + z> = 0.
N
But the equation in z is satisfied by the value z = x, and therefore the equation in by
the value 6 = a; + - o suppose, we have therefore
sc
of - 3a + X (a 2 - 2) + /xa + v = 0,
and thence subtracting, and throwing out the factor a,
viz. writing for X, /t, a their values, this is
x
0* + ( x + - - 12 - fl) + a? - I + i - f x + -} ( 1 2 + fl) + 807 + 4fi = 0,
\ X J 3, \ X/
or, what is the same thing,
0* + 6 ( x - 1 2 + - - fl} + a? - 1 2* + 806 - + ^ - f x - 4 - -} fl = 0,
V x / x a? \ x)
where
fl= . 1 ...(1. -12, 807, 2504, 807, -12, I'Sx, l) s .
x(x I) 4
Hence in the quadric equation, the coefficients, each multiplied by (x I) 4 , are
and
- i (l, -12, 807, 2504, 807, -12, 1$, 1) B ,
SCl 3)
(* - 1 ) 4 f a? - IZx + 806 - + 1 )
\ 57 *Z /
- - fa; - 4 + -^i (1, -12, 807, 2504, 807, -12, l\x, I) 6 ,
x \ x/
which are respectively rational and integral quartic functions of x ; and, writing for its
value, the equation finally is
, l\(l, 188. 646, 188, l\x, I) 4 , (1. -644, 3334, -644. \\x, 1) 4 _
__
710] ON A DIFFERENTIAL EQUATION. 21
Writing
f-tf* ^=J-| *-l|, <?--* D = \-3, (i = V- las usual)/
this is
(z - A') (z - ') (z - C") (z - &) = 0,
or, what is the same thing,
that is,
for we have
And substituting these values, the coefficients will be rational functions of f 4 , that is, of
j;, and it is easy to verify that they have in fact their foregoing values.
It thus appears that for (7=1, besides the values x and -, we have for z only the
36
values
viz. that the only solution is
The example shows that although the differential equation
+ 2b'z + c'dz \/cwr' + 2bas + c dx
z(z-\) x(x-\)
can be integi-ated 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 dx, dz, the relation between
x and z is of the form
p^_p
Q'~Q'
22 ON A DIFFERENTIAL EQUATION. [710
where P, Q are quadric functions of x; P 1 , Q 1 quadric functions of z. But P and
Q may contain a common factor, and the integral is then expressible in the form
p>
x = -^ , the quotient of two quadric functions of z ; or P 1 and Q' may have a common
p
factor, and the integral is then expressible in the form z= j- , 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 1 and Q', and the integral is then of the form z = ~, the quotient of two
linear functions of x.
In the general case the differential equation is
X (aP ' + bQ') dz 3 (aP + bQ) da?
where a, b are arbitrary constants, X is a constant the value of which can in each
particular c
equation is
p
particular case be at once determined ; so when the integral is z -^ , the differential
X (az + b) dz* _ (aP + bQ) dx*
z- (z-Tf~ a?(x-\y '
where , b are arbitrary constants, but X is now a linear function of z the value
of which can in each particular case be at once determined. When the integral
is 2=T>. the differential equation is
+ c) dz* (aL* + ZbLM + cM*) dx-
z* (z - iy a?(x-\f
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 = Tjr for which the differential equation
p /
contains three arbitrary constants: 18 integrals of the form z = j-. (and of course the
P'x P P'
same number of integrals of the form # = 7y), and 9 integrals of the form Q=7y, fo r all
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
1 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 a; 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'z* + 2b'z + c' and aa? + 2bx + c
which enter into the differential equation are as follows:
as 2 + 26' + c' =
1.
2.
3.
z
X
l-x
1
X
1
as 2 + 26s + c
>
)>
5J
J
acc 2 -f 2bx + c
a(x-\f-'2b(x-\) + c
a + 2bx + car
9A //* 1 \ _l_ y / 1 \2
1-0!
X
^1
(B-l
a 2 +26a;(x-l) + c(a;-l) 2
(a;-l) !! +26a;(a;-l) + car i
X
/*+iy
az 2 + bz
v
j
>
)
a(a;+ l) 2 + 6(-l) 2
a(2x-l) 2 + 6
a (x - 2) 2 + bo?
a (x + 1 ) 2 + 46o:
a(2*-l) 2 + 46a;(a:- 1)
/->. 9\2 4.7. /o- T\
U- l)
(2x-iy
/-2\.
V x J
(*+l) 3
4a;
(2* -I) 2
4a>(-l)
(a-'-2) 3
4(x-l)
/*-IY
62;+ c
)j
;>
H
))
6(a;-l) 2 + c(a;+l) 2
t> j- / C9/>- 1 V
U+i/
/ i y
V2X-1J
( x V
Av 2 A- r (r *>\i
U-2J
tr
4bx + c(x+ I) 2
4ia;(x'-l) + c(2a;-l) 2
1A / 1 \ . x. / r O'i 2
(X+\Y
4(0-l)
(2x-l) 2
4(^-1)
( a ,--2) 2
24
ON A DIFFERENTIAL EQUATION.
a'z" -i- 26'z + c' = aa? -t- 26* 4- c =
(*-!)=
4*
4a^
-1
a?
as? (a -f c) z + c
5. }
6.
V.
- same as 2, 3, 4 interchanging a; and s.
6.
9.
10.
a (x I) 2 + 4e
4aa;(a;- 1) + c
- 4 (a; - 1 ) + car 2
4o. + c (a; - 1 ) 2
a + 4ca; (x 1)
ax* 4c (a; 1 )
ace 2 +
[710
(* I) 2 4a;
4o + 6 (a; I) 2
Irt f-j- _ 1 ^ fcr 2
z* 4 (x 1 )
4 ( 1) ar 8
4~ /. i \
0^ + 4o(* 1)
a + 46aj(a;- 1)
___l (1-y^-r- 1\
4aa: (x- 1) + 6
4oa; (a; 1 ) + b
4a (a; 1) + bo?
4 ' '
4~ i~, -\\
_._5 A/./* i\
4z a; 3
/(./. l\ V 21 "^)
4az(z l) + 6
4a^(2-l)+6
a(x- 1 ) 2 + 46a;
oar 2 -46 (a; -1)
l-(- 1)- ^
4 (z-1) (a;-!) 2
z 9 4a;
10] ON A DIFFERENTIAL EQUATION. 25
The six functions of the set (1), that is,
1 _
t ~~ t
x' 1 a;' x 1 ' x
form a group : and by operating with the substitutions of this group, and of the like
group
1 1 z z-\
&t A 2, , i _ " > -i ' '
(X + 1\ 2
Y) , we form all
x I/
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,
and y=( --, ) , whence | -r) = ( \ ,
or
/a + ly (*+ 1) 2 AB + iy
w = and v = - - , whence =
\x-lj 4>z \x-lj
4,z
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-\f _ 4,x
~te~ "(aT^i)-'
we may obtain
(z+iy = fx+_ iv
4,2 \Z-~l) '
If, to get rid of this variety of form, we multiply out the denominators, the 9
equations are
0= x-z-- Zc-z- 2xz*+ x*
0= a?z- -I6xz
Q = lGa*'z---16a?z-Wa;z* +I6xz - I,
0= a?z-- %a?z + x"+16xz - 16z ,
0= I6a?z -I6xz- z- + 2z- I,
0= IQafz -16* 8 -16^+ z*+16te
0= a?z- 2xz- +ltjxz+ z--Wx ,
= 16a.'2 2 - x 2 - ISxz + 2x +1,
0= 16*2 2 + ^
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. XI. 4
[711
711.
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 x = i, and x = \ ; 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.
The analytical theorem is that the whole series of transformations, o> into v- ,
where a, $, 7, 6 are integers such that aS $7=!, can be obtained by combination
of the transformations to into w + 1 and o> into .
712]
27
712.
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 X
pp. 534, 535.]
CONSIDER a given cubic curve cut by a line in the points (*,, yj, (i, y,),
(**, y); taking the first and second points at pleasure, these determine uniquely the
third point. Analytically, the equation of the curve determines y, as a function of
x l . and y, as a function of ,: writing in the equation
*i-X*,+(l-X)*,. y,= Xy,+(l-X)y t ,
we have X by a simple equation, and thence a;', viz. jr 3 is found as a function of
*i, ,, and of the nine constants of the equation. Hence forming the derived equal iou.-
(in regard to a:,, xj of the first, second, and third orders, we have (1 + 2 + 8 + 4=) 10
equations from which to eliminate the 9 constants; x a , considered as a function (
x t and ,, 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,
dx t dx* dx,
x + r. + r,- '
where A', is a given function of a:, and y u that is, of *, ; X t and X t are the like
functions of a, and x t respectively. Hence, considering .r. as a function of .c, and ./...
we have
42
28 A PARTIAL DIFFERENTIAL EQUATION. [712
and consequently
dx tj _dz ; _.\.
rfa:, ' Ac, ^ Z, '
where X t , Xt are functions of #,, x, respectively : hence taking the logarithm and
differentiating successively with regard to a;, and x.,, we have
_<^ d^ . /da ___ dx,\ _
dx l dx 3 " \dx t ' dxj
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 : qud 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. 751758. Received May 27, Read June 10, 1880.]
IN Abel's general theorem y is an irrational function of x determined by an
equation x(?/) = 0, or say x(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 ^ (#> y) = 0, and given in his equation (62), [CEuvres 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*
3i
n-iHi series each of the form y = ;#*' + ...,
tAt .. # = #**+...,
m,
* The several powers of x have coefficients: the form really is y = A l x ltt + ..., which is regarded as
1
representing the /*, different values of y obtained by giving to the radical .r^ 1 each of its /xj 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 n, such 'series each representing MI values of y. It is assumed that the
I
series contains onli/ the radical xf', that is, the indices after the leading index 1 are , , ... ; a
Hi Mi Mi
series such as y = A 1 x^ + B 1 x^ + ... , depending on the two radicals x 7 ', a;i represents 15 different values, and
would be written y = A l xt + ..., or the values of ?, and /i t would be 20 and 15 respectively: in a case like
this where is not in its least terms, the number of values of the leading coefficient /I, is equal, not to
/tj. but to a submultiple of AI,. But the case is excluded by Abel's assumption that , '"-',..., are fractions
Mi Ma
each of them in its least terms.
30 ADDITION TO MR ROWE's [713
(so that )( = H,^, +,/*,+ ... + HMt), tnen 7 i 8 a determinate function of n,, ?,, MI?
Mr Rowe has expressed Abel's 7 in the following form, viz. assuming
Mi Ms Mi '
then this expression is
or, what is the same thing, for n writing its value
7 =
where in the first sum r, s have each of them the values 1, 2, ...,k, subject to the
condition s > r ; in each of the other sums n, m, and ft 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 x( x > y) : an ^ ' l
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. 212223, [374].
M
It is necessary to distinguish between the values of : which are >, =, and < 1 ;
and to fix the ideas I assume k = 7, and
m, m* m,
j t ' - t (it 11 ^ J.,
Ml Ma Ma
(H
= 1 ; say m t = /*, = X, and 4 = # ;
MI
^, ^, T,each<l,
Me Ms Mr
but it will be easily seen that the reasoning is quite general. I use ' 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 = S'?IM + \0 + 2'V-
Introducing a third coordinate z for homogeneity, the equation ^ (x, y) = of
the curve will be
where it is to be observed that ( y*' is written to denote the product of
j_j "'
different series each of the form yz^ A^' ... ; these divide themselves into ,
713] MEMOIR ON ABEL'S THEOREM. 31
groups, each a product of /*, series; and in each such product the fj,, coefficients A l
i
are in general the /u, 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 ,^, 1 coefficients A^ are different from each other* : the like
remarks apply to the other factors. It applies in particular to the term
viz. it is assumed that the coefficients A in the \0 series y = Ax* + ... are all of
them different from each other. These assumptions as to the leading coefficients
really imply Abel's assumption that --?, . . . , - k are all of them fractions in their least
terms, and in particular that - is a fraction in its least terms, viz. that X = 1 : I
A.
retain however 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(x, y, z) of the coordinates, which on putting therein z=l becomes =%(#, y).
The equation of the curve thus is F(x, y, z)=0; and the order is
4- + X# + H.U..J + . . . , = m, , + ... +\d + 5 M 5 + . . . ;
viz. if K is the order of the curve ^ (x, y) = 0, then K = S'it -f \0 + ^"np.
The curve has singularities (singular points) at infinity, that is, on the line z = :
vz.
First, a singularity at (z = 0, x = 0), where the tangent is x = 0, and which,
wiiting for convenience y = 1, is denoted by the function
where observe that the expressed factor indicates n, branches ( z *>-'" 1 , or
.,
say H! (m, /*,) partial branches z x m >~*> , that is, ,(?! /,) partial branches
2 = A l x m > -*> + ..., with in all M,(m, /*,) distinct values of A l : and the like as regards
the unexpressed factors with the suffixes 2 and 3.
Secondly, a singularity at (z=0, y = 0), where the tangent is y = 0, and which,
writing for convenience x = l, is denoted by the function
* This assumption is virtually made by Abel, (/. c.) p. 162, in the expression "alors on aura en general,
excepte quelques cas particuliers que je me dispense de considerer : h(ij' -y") = lii/', &c." : viz. the meaning is
that the degree of ?/' 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 KOWE's [713
(M: \Mj-J
z y**-" 1 *} , or
Ms
say 9 (/t5 ni 5 ) partial branches z y"*" 1 " 1 , that is, n 5 (ft, ) partial branches
"
2 = A t y*>- n >+ ..., with in all ?i 8 (/*, 7 5 ) distinct values of .4 5 : and the like as
regards the unexpressed factors with the suffixes 6 and 7.
Thirdly, singularities at the 6 points (z = 0, y Ax = 0), A having here 6 distinct
values, at any one of which the tangent is y Ax Q, and which are denoted by
the function
x\x
but in the case ultimately considered X is = 1 ; and these are then the 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 8' + K of nodes and cusps ;
viz. we have
hence
S' + K '
Assuming that there are no other singularities, the deficiency
This should be equal to the before-mentioned value of 7 ; viz. we ought to have
(K - 1) (K - 2) - M + 2 (a - 1) = 22X.m r M (l /i, + In-vip - Inm - Inn - 2 + 2,
t>r
or, as it will be convenient to write it,
Af = K- SK + 2 (a - 1 )
which is the equation which ought to be satisfied by the values of M and 2 (a 1)
calculated, according to the method of my paper, for the foregoing singularities of
the curve.
We have as before
The term ^.n r m r n t ft,,, written at length, is
>r
= , m,
713] MEMOIR ON ABEL'S THEOREM. 33
which is
0\ (S'nm + 2'V) + 2'?i7/t . 2'V + 2"
s>r
We have moreover
2nm = 2'nm + 0X + 2"?im,
2n/i = 2'n/x, + 0X + 2"n/ti,
2n =2'm +0 +2"n.
We next calculate 2 (a 1).
For the singularity
( - s^y 1 '" 1 "* 1 '
\z x )
each branch (z x m >~^j gives a = m l fa, and the value of 2 (a 1) for this
singularity is
n t (TO! fa 1) + ru (m^ ^l) + n 3 (m 3 p 3 1),
which is
^' ^/ ^/
For the singularity
fa \"s ((*s-
2 -V s "
/ _ftl_\C5-"5
each branch ( z y^- m '} gives a = /u 5 - TO,, and the value of 2 (a 1) for this
singularity is
?! 5 (> 5 - TO S - 1) -I- n 6 (ft, -m 6 -l) + n 7 ( f ^-m 7 - 1),
which is
= 2'V-2"nm-2".
For each of the ff singularities
we have a = \ and the value of 2 (a 1) is =0(\ 1): this is = for the value
X = 1, which is ultimately attributed to X.
The complete value of 2 (a 1) is thus
= 2'nm - 2"nm - 2V + 2"n/t - 2'n - 2"w + 0X - 0.
Substituting all these values, we have
M= (2'nm + 2'V) 2 + 2#X (2'nm + 2"n/x) + (^) 2
- 3 (2'nm + 2'V) - 30X
+ 2'nm 2"nTO 2V + 2'V ~ ^' n 'Z"n+6\6
- 2^,'n r m r n g fj, s - 20X (2'wm + 2'V) 22'nm . 2'V - 22" r TO r n,/i,
+ 2'wm + ^X + 2"nTO
+ 2V + ^x + 2'V
+ 2'n + 6 + 2"n,
c. xi.
34 ADDITION TO MR ROWE*S [713
or, reducing,
M= (Z'nmy - "2'nm S'/i'm/i 22,'n r m r n f (i. t
t>r
+ (2' V) 1 - S"n/t - 2"re a m/i - 22"n r vnM;
r
and it is to be shown that the two lines of this expression are in fact the values
of M belonging to the singularities
m, xn,(m,-Mi) / __M \(f
..., and ( z y >~ m ' j
respectively. We assume \ = 1, and there is thus no singularity (y x-
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 2, = -4,3r pi , z^= A^P* (where p lt p 3 are each equal to or
greater than 1), then if p t = pi, and z l z,= (A l A^x^ where .4, A 3 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 p t and p t are unequal,
then taking p? to be the greater of them, the leading term of z l z t ia = A^x^, and
the number of intersections" is taken to be =p r ; viz. in the case of unequal ex-
ponents, it is equal to the smaller exponent.
Consider now the singularity \ss-x m >-^J ...; and first the intersections of
m,
a partial branch z x m *~*> by each of the remaining HI (m, ^t,) 1 partial branches
of the same set : the number of intersections with any one of these is =
m,-/*,'
and consequently the number with all of them is = [, (m^ ^ 1]. But we
Wlj /A!
obtain this same number from each of the n l (m, /^) partial branches, and thus the
whole number is
- Mi) - 1]. =n 1 m l [>, (TO, - /*,) - 1].
TO, /
Taking account of the other sets, each with itself, the whole number of such
intersections is
,TO, [H, (m, - /*,) - 1] + njTO, [n a (TO, - /^) - 1] + n 8 m 3 [jj, (m, - /& 3 ) - 1],
which is
713] MEMOIR ON ABEL'S THEOREM. 35
Observe now that > , that is, ^i<^, and that, these being each < 1, we
fii fr m 1 m. 2
thence have 1 _ > i _ , that is, 1 ^L^ > ^^* : and we thus have
Considering now the intersections of partial branches of the two sets
[z a;"* 1 "' 1 ') and \z a;' 2 "** 2 )
respectively, a partial branch z x^~^ gives with each partial branch of the other
set a number = - ; and in this way taking each partial branch of each set,
**l~A*i
the number is
1YL
and thus for all the sets the number is
= niWh2 (m, -fr) + n^rij (m 3 - /ts) + n 2 m 2 rz 3 (m 3 - /A 3 ),
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
S'ftrrn 2 2'n 2 m^ S'nm,
the whole number for the singularity in question is
s>r
(_es\n,(Mj-m,)
z yps-m,} >g< j taking each set with itself, the
number of intersections is
n>fJ.i [n, (ft, - m s ) - 1] + n,p, [n. (/i, - rw.) - 1] + n^ [ 7 (/*, -m,)- 1],
which is
= S"V - 2"n 2 m/i - 2'V-
52
36 ADDITION TO MR ROWfi's MEMOIR ON ABEL'S THEOREM. [713
We have here > ; each of these being less than 1, we have 1 - - < 1 -- ,
M. Me M. M.
that is, * = *<*-*!, or -- >---; and so
Ms M. M*-"*. M.-TW,
M? - *
Hence considering the two sets
and U y-"l ,
a partial branch of the first set gives with a partial branch of the second set -
Me -
intersections: and the number thus obtained is
5 (M. - wt) e (M. - m)
M ~" TO
For all the sets the number is
n,Ti,M (MS ~ m,) + n t n,fr (MS - m,) + W.^MT <Me -
or taking this twice, the number is
where in the first sum the 2" refers to each pair of suffixes. Adding the foregoing
value
the whole number for the singularity in question is
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 consideVer," 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,
= 1+1
then
On two related quadric functions.
Assume
(ftx = a 2 (c x) x (c 2 6 2 ex),
yfrtc = b- (c x) x (c 2 a 2 c) :
In the first of these for x write - ; then
c x
a'(c-x)\_ a^c-xy fr 3 _j?b ,
38 VARIOUS NOTES. [714
A Trigonometrical Identity.
cos (b - c) cos (b + c + d) + cos a cos (a + d)
= cos (c a) cos (c + a + d) + cos b cos (b + d)
cos (a - 6) cos (a + b + d) + cos c cos (c + d)
= cos a cos (a + d) + cos b cos (6 + d) + cos c cos (c + d) cos d.
Extract from a Letter.
" I wish to construct a correspondence such as
(x + iyY + (x + iy) = X + iY,
or, say, 1 for greater convenience
3(x + iy) = X + iY;
viz. if
x + iy = cos u,
then
Suppose 3 is a value of 3w corresponding to a given value of X + iY, then the
() \
Mo-r); but I am afraid that the cal-
o /
culation of , even with cosh and sinh tables, would be very laborious. Writing
X + iY = R (cos @> + i sin ),
the intervals for 6 might be 5, 10 or even 15, those of R, say 01 from 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 3."
October 31, 1877.
715] 39
715.
NOTE ON A SYSTEM OF ALGEBRAICAL EQUATIONS.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 17, 18.]
ASSUME
x +y + 2 = P,
yz + zx + xy = Q ,
xyz = R,
A=x (nyz + Q) w> (mx + P),
B =y (nzx + Q) - w 2 (my + P),
C = z (nxy + Q) - w- (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-x),
(my+P)A-(mx + P)B=(x- y) @.
Hence any two of the equations ^=0, B = 0, (7=0 imply the third equation.
40 NOTE ON A SYSTEM OF ALGEBRAICAL EQUATIONS. [715
We have
A = x \(n + 1) yz + zx + xy} up {(m + 1) x + (y + z)}
= (a? - w 1 ) (y + z) - x [(m + 1) up - (n + 1) yz],
and similarly for B and C. The three equations therefore are
a? 11? (m
V =
z*-ur> (m + l)w"- (n + l)xy'
and any two of these equations imply the third equation.
716] 41
716.
AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 27 32.]
IF X be a given quartic function of x, and if u, or for convenience a constant
r (fa
multiple , be the value of the integral I -r^-p--. taken from a given inferior limit to
J v(-X-)
the superior limit x; then, conversely, x is expressible as a function of w, viz. it is
expressible in terms of ^-functions of u, where ^, or say ^(M, g) (g a parameter
upon which the function depends), is given by definition as the sum of a series of
f dx
exponentials of u ; and it is possible from the assumed equation au = I ~,T^- , and the
definition of S-M, to obtain 'by general theory the actual formulae 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 ^ru, (as given by
/dx
by a
V(-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 S-M or, when the parameter is expressed, ^ (u, g) is
where s has all positive or negative integer values, zero included, from oo to + oo
(that is, from S to +8, & = oo ) ; the parameter g, or (if imaginary) its real part,
must be ositive.
must be positive.
c. XI.
N
42 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. [716
Evidently *u is an even function: ^(-u) = ^w. Moreover, it is at once seen that
we have
^ (u + TT) = ^u, ^ (u + i8) = e*~* u < &u,
whence also
^ (u + rmr + nig),
where m and n are any positive or negative integers, is the product of ^tt by an
exponential factor, or say simply that it is a multiple of <bu.
Writing w = -ig, we have ^ (- Jig) = * (ig), that is,
and therefore also
The above properties are general, but if g be real, then &, A", K', q being as in
Jacobi (consequently k being real, positive, and less than 1, and K and #' real and
ir
positive), and assuming g=-^-, or, what is the same thing,
the function ^ is given in terms of Jacobi's by the equation ^w = ^- J; or,
what is the same thing, w =
We hence at once obtain expressions of the elliptic functions sn u, en u, dn u in
terms of ^, viz. these are
+ *(m\
* (ZK) '
- V ( " ^
r|. T J.
Consider now the integral
dx dx
8
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 <J(X) is taken so that y is equal to a positive multiple of i,
and this being so the integral is taken from the inferior limit a to the superior
limit as, which is real.
Take x a linear function of y, such that for
x = a, b, c, d,
y = Q, 1, 5 , oc , respectively,
so that, x increasing continuously from a to d, y will increase continuously from to oo .
We have
b a.d c
, 2
K"
y
' d b . c a'
b d x a
.. _d a x b
b a x d'
c a x d'
and, thence,
d - a /fd-l)\ <J(X)
v (V . 1 v . 1 Km = A / I T^ i >
c a V \c-al (x-df
where */( ) is taken to be positive, and the sign of >J(X) is fixed as above. Then
^ \c &/
for y between and 1 or > , , y . 1 y .1 k*y will be positive, and vXy 1 y 1 ~ ^y)
will also be positive ; but y being between 1 and -=-, y.ly.lfc*y will be negative,
and the sign of the radical is such that ~r. = = = -r is a positive multiple of i.
^(y.l-y.l- %)
We have moreover
7 d a . . dx
dy , , dx
= J(d - b . c - a)
and therefore
where >J(d b.c a) is positive ; or, say,
',-b.c-a]
62
44 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS.
Hence, writing y = z* = sn' u, we have
[716
and it is to be further noticed that to
x = a, b, c, d,
correspond
or we may say
Writing for shortness
we have
and moreover
sn=0, 1, p oo,
tt-0, K, K+iK',
2
au =
dx
v
or if for a moment we write
then these equations are
a (2K + i.
r* dx
dx
s
J II
A, &c.,
Hence B + C-2A = D-A, that is, A-B-C + D = 0, or B-A = D-C, that is,
dx d dx
where observe as before that x = a to x = b, or x = c to x = d, X is positive, and the
radical \J(X) is taken to be positive.
We have also
-i:
dx
vW
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
where, from b to c, X is positive, and >J(X) is also taken to be positive.
Collecting the results, we have
dx 2 , b a .d c
= ait, a =
J(d-b.c-a)' d-b.c-a'
and also
d b . c a'
and then conversely
_ a (d b) + d (b a) sn 2 u
(d-b) + (b-a)stfu '
or, what is the same thing,
b d . x a
sn 2 u = ,
o a . x a
da.x b
cn 2 M =
dn 2 u =
b a . x d'
d a .x c
c a . x d '
where, in place of the elliptic functions we are to substitute their ^--values ; it will
be recollected that g, the parameter of the ^-functions, has the value
_ dx ( b dx
fi r -K)-"] t J(-
and, as before,
1 f dx
1 f
Ja
Hence, finally, a, k, k', K, g denoting given functions of a, b, c, d, if as above
dx
we have conversely
b-d.x-a 1 -+ l-rru \ TTU
da.x
b a.x d k
-a.x-c k ,
c a.x d
which are the formulas in question.
46 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS.
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
and from the foregoing definition of the function *.
It may be recalled that the function *u is a doubly infinite product
u
m and n positive or negative integers from -co to +00; I purposely omit all further
explanations as to limits; or, what is the same thing,
JL^^J;
and consequently that, disregarding constant and exponential factors, the foregoing
expressions of
b d.x a d a.x b d a.x c
b-a.x-d' b-a.x-d' c-a.x-d'
are the squares of the expressions * , , , where X, Y, Z, W are respectively of
the form
i u 1 (_ u
uim
' (m, n)\ '
-,
(m, n))
where (m, n) = ZmK + ZniK', 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 +
this is a transformation which has apparently no application to the
more than one variable.
717]
47
717.
ON THE TKIPLE THETA-FUNCTIONS.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 4850.]
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) = 2exp.@,
where
= (A, B, C, F, 0, ff)(l, m, n)* + 2(U, V, W)(l, m , ),
2 denoting the sum in regard to all positive and negative integer values from
- oo to + oo (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. *t(U, V, W) = 0, for
, H, (?)(, ft, y)},
, B, F)(a, ft, 7 )j,
W=\{zTri + (G, F, C) (a, ft, 7 )J,
x, y z, a, 0, y being any positive or negative integer numbers, such that aat + ftv + v*
= odd number.
Demonstration. It is only necessary to show that to each term of & there corre-
a second term, such that the indices of the two exponentials differ by an odd
multiple of ?n.
4g ON THE TRIPLE THETA-FUNCTIONS.
Taking I, m, n as the integers which belong to the one term, those belonging to
the other term are
-(l + a), -(m+A\ ~(+7)>
f
, = (A, B, C, F, G, H)(l, m, n)'+2(E7. V, W)(l, m, n)
and
, B, C, F, G, H)(l + , m + A n + 7)<-2(^, 7,
viz. the value of 8' is
= (A, B, C, F, G, H)(l, m, nJ>+(A, B, C, F, G, #)(, A 7)'
+ 2(A, B, C, F, G, H)(l, m, n)(a, /3, 7)
-2(U, V,
and we then have
8'- 8 = 2(4, B, C, F, G, H)(l, m, n)(a, A 7)
+ (4, B, C, F, G, H)(a, ft, 7)'
7, F,
Substituting herein for IT, F, F their values, the last term is
-2 (A, B, C, F, G, H)(l, m, n)(a, A 7)
- (A, B, C, F, G, H)(*. A 7) 2 .
and thence , ,
@' _ 6 = - {(M + a)x + (Zm + A^ V +(2 + V)*\
which proves the theorem.
i off r (A B G F G H) has been once written
As to the notation, remark that, after (A, a, ^, *
out in full, we may instead of
(A, B, C, F, G, H)(l, m, n) 2 , &c., write (A, ...)(*, m, n)\ &c.,
and that we may use the like abbreviations
(A,...) (I, m, n), to denote (A, H, G)(l, m, n) respectively,
(H,...)(l,m, n), (H,B,F)(l,m,n)
(G ...)(*. m, n), (G,F, C)(l,m, n)
a-jr A iss-W-ar
which follows.
ON THE TRIPLE THETA-FUNCTIONS.
Second Property. If U ly V lt W, denote
(A, H, G)(a, 0, 7 ),
(H, B, F)(, /3, 7 ),
(G, F, C)( a , ff, 7 ),
49
or say
= exp.{- (4, ...)(, &
Writing *(Z7 ; Fl , ^ - 2 . erp. ,, then in the expression of 8,
ce of I, m , wnte ^- , i-/3 ( K - 7; we thus obtain
...)^-^ m-/3, n-
which is
..)(l, m , nf
..., m , n)(a, ft, 7 )
-2(A, ...)(/, m , n )( a> ^
which is
+ (A, ...)Ca
M '
= (A, ...)(l, m , ny-+2(lU+mV+nW)
-(A, ...)(, ft, 7) 2 -
Hence, rejecting the last line, which (as an even multiple of ) leaves the
unaltered, we see that *(tf lf F, TT l} is - W F IT) multiplied by the factor"
p.{-(4, ...)(, ft, 7 ) 2 }.exp. {-
which is the theorem in question.
In many cases a formula, which belongs to an indefinite number . of letters is
most easdy intelhpble when written out for three letters, but it is sometim
vement to speak of the . letters I, m , ..., , or even the , letters I, . ., nzTtl te
out the formulae accordingly.
C. XI.
50
718.
ADDITION TO MR GENESE'S NOTE ON THE THEORY
OF ENVELOPES.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 62, 63-1
THE example, although simple, is an instructive one. Introducing z, p for
homogeneity, the equation is
X s y (y - bz) + 2\fixy + fj?x (x - az) = 0,
giving the envelope
xy [(x - az) (y - bz) - xy] = ;
that is,
xy (bx + ay abz) z = 0;
viz. we have thus the four lines
' y ' ' a b
Writing these values successively in the equation of the curve, we find respectively
\*y (y - bz) = 0,
/j?x (x az) = 0,
(\y + fixf = ;
viz. in each case the equation in X, fj, has (as it should have) two equal roots; but
in the first three cases the values are constant ; viz. we find \ = 0, n = 0, b\ - ap, = 0,
respectively; and the curves a> = 0, y = 0, ?+|-*-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
\ = 0, fj. = 0, b\ a/j. = successively, we find respectively
x(x az) = 0,
y(y-bz) = 0,
(bx + ay) (b.i; + ay abz) = ;
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 \, /t is (\y + pa)- = 0, giving a variable
value \ + /* = x + y; hence = 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 qud curve of the order 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.
72
52 [719
<
719.
SUGGESTION OF A MECHANICAL INTEGRATOR FOR THE
CALCULATION OF ((Xdx+Ydy) ALONG AN ARBITRARY
PATH*.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 92 95 ; British Association
Report, 1877, pp. 1820.]
I CONSIDER an integral \(Xdx + Ydy), where X, Y are each of them a given
i
function of the variables (x, 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 = Q, y=l, x=0, x = \ respectively. In the
wall faces represented by these equations, we have the endless bands A, A', B, B 1
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 Z)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
* Read 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 , 17, 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
dx 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 \(Xdx+ Ydy).
It is obvious that we might, by means of a combination of two such mechanisms,
calculate the value of an integral \f(u) 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
j(Pdx-Qdy) and j(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 6' 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
ronnccted 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'
A.
2?r
through each angle - , there will correspond an entire revolution of A, and therefore,
A,
as before, a rotation of E through the determinate angle n . 2ir ; hence, \ being
sufficiently large to each increment of rotation of A', there corresponds in E an
increment of rotation which is nX 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 EC, 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 o 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 2tr, there will correspond a rotation of E through an angle n.2r 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
1 \
^ \ a 1.2 M '")
du dhi dhi ,
by the differentiation ot <pu, writing a, b, c, d, ... in place of u, j- , -y 2 , -j-, , &c. or
'/.'.' CLX Qj*Ki
say in place of u, u', u", u'", &c. respectively ; thus
<f>a, <f>'a . b, ^ {</>'a . c + <f>"a . 6 2 j, ^ f<f>'a . d + ff>"a . be
fa {<f>'a . d 4- <f>"a . be 1
'a . d + <f>"a . 3bc + <f>'"a . b 3 }, &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 </> (a + bx+ cx' + da? +...),
are obtained by the rule of the last and the last but one ; thus we have
<a, <j>'a .b, (f>'a.c + <f>"a . ^b'-, (f)'a.d + (j>"a . be + tf>'"a. & 3 , &c.,
where each numerical coefficient is found directly, without an addition in any case.
* See Messenger of Mathematics, vol. vii. (1878), pp. 142, 143.
[721
56
N
721.
FORMULA INVOLVING THE SEVENTH ROOTS OF UNITY.
[From the Messenger of Mathematics, vol. vn. (1878), pp. 177182..
LET o, be an imaginary cube root of unity, o>* + o,+ l =0, or say - i {- 1 + tV(3)} ;
a' = -7 (l + 3o,), ,3* = -7 (l + 3o,'), values giving a/3> = 343, and the cube
being such that /3 = 7 ; th^n a + A = + , is a three-valued function (since changing
the root o, we merely interchange a and ; and if r be an imaginary seventh root
of unity, then
3(r + ?") = + -1,
3 ( r s + ,-=)= wa + w'/S-l,
3 (r 4 + r 3 ) = ora + w/3-1.
Any one of these formulae gives the other two ; for observe that we have = -aft (1 + 3),
jS^a + Mk that is , % /8(l + 8.), /9-- d + 8^); hence, starUng for mstance
with the first formula, we deduce
14 - a (1 + 3<u') - 2a - 2^ + 1,
= - a (3 + 3a, s ) - ft (3 + 3w) 4- 15,
= 3wa + 3a> s /3 + 15,
that is,
3 ( r s + r 5 ) = wa + w 2 p - 1 ;
and in like manner by squaring each side of this we have the third formula
721]
FORMULAE INVOLVING THE SEVENTH BOOTS OF UNITY.
57
A 3 = G + 3 X + (1+ 3w 2 ) Y,
B* = 6 + 3 2 ;r + (1 + 3ew ) F,
The foregoing formulae apply to the combinations r + r*, r 2 + r 8 , r 4 + r 3 of the seventh
roots of unity, but we may investigate the theory for the roots themselves r, i a , r 3 , r*, r", r".
These depend on the new radical </( 7) or i V(7) ; introducing instead hereof X, Y,
where
then if
where
we have (Lagrange, Equations Numiriques, p. 294),
I found that, in order to bring this into connexion with the foregoing formula,
3(r+r) = a + /9-l, where as before a 3 = - 7 (1 + 3a>), /S 3 = - 7 (1 + 3w 2 ), a/9 = 7, it is
necessary that B, A should be linear multiples of a, /3 respectively, the coefficients
being rational functions of ta, X ; and that the actual relations are
B = {4 - a> + X (1 - 2ea)},
in verification of which, it may be remarked that these equations give
AB = ^ {(20 - - o>") + X (17 - 4a> - 4w') + JT (3 - 4 - 4*%
T*y
viz. in virtue of the equation w" + v> + 1 = 0, the term in { } is =21-f2lAT+ TX',
= 7(^'-t-3Z + 3), or since ^ + ^ + 2 = 0, this is =7(2Z+1), =7iV(7); the equation
thus is TAB = a/3.i V(7), which is true in virtue of AB = i-J(7) and a/9 = 7. The same
relations may also be written
- a = B ( + JST),
I found in the first instance
3r = JT + A + B,
'A + w B,
X),
<o X),
Zr 4 = X + <aA + o>'B,
3r>=-l-X+ A (I -uX)+B(l
c. xi.
58 FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. [721
which in fact gave the foregoing formula?
3 ( r + r) = - 1 + a + 0,
3 (r + r 5 ) = - 1 + a> + w'/3,
3 (H -(. r>) = - 1 + w *a + o>/3.
But there is a want of symmetry in these expressions for r, r 3 , &c., inasmuch as the
values of r, r*, r 4 are of a different form from those of r 6 , r 5 , r 8 ; to obtain the proper
forms, we must for A, B substitute their values in terms of a, ft, and we thus obtain
3r = X + j{ 4- < + X( l-2))+{ 5+ a) + JT( 3
{ 3+ + *(- l + 2a)))+y { 2- + AT (- 3 - 2w)},
- }- 3 - 2a> +
X + | {- 5 - 4 + X (- 3 - )} + y {- 1 + 4a> + X (- 2 + )},
3+ )} + ^{ l + 3 + A7( 2- )} ;
viz. each of the imaginary seventh roots is thus expressed as a linear function of the
cubic radicals a, /9 (involving w under the radical signs) with coefficients which are
functions of o>, X.
Recollecting the equations or =-/8(l +3o>), /3 3 = -o(l + 3a> 2 ), a/3 = 7; w a + a> -f 1 = 0,
.3f s + X + 2 = ; 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 9? 12
an expression of the like form ; viz. we thus in fact obtain the expression for Sr 2 ;
then from the expressions of 3r and 3r 2 , multiplying together and reducing, we should
obtain the expression for 3? J ; 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. 166182 ; [Ges. Werke, t. vi. pp. 254274]. The
gf _ J
starting-point is the following theorem : if x be a root of the equation - - = 0,
3C ~~ I
p a prime number, and if g is a prime root of p, and
F (a) = x +
a -_
where a is any root of i =0, we have
F(a m ) F(o. n ) = ^ (a)
721]
FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY.
59
where ty (a) is a rational and integral function of a with integral coefficients ; or, what
is the same thing, if a and /3 be any two roots of the above-mentioned equation, then
where ifr(y, /3) is a rational and integral function of a, /3 with integral coefficients.
As regards the proof of this, it may be remarked that, writing a? for x, F(OL), F(/3),
and F(a&) become respectively ar^a), 0-^(0), (a/3)- 1 F(a0); hence, F (a) F (0) + F (<*0)
remains unaltered, and it thus appears that the function in question is expressible
rationally in terms of the adjoint quantities a and /3. With this explanation the
following extract will be easily intelligible :
" The true form (never yet given) of the roots of the equation x p 1 = 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 1 and a* = 1, then it is further known that
{F(a)}*- is a mere function of a. But it is only necessary to know those values of
F(a) for which X is the power of a prime number. For suppose \\'\".,. is a factor
of p 1 ; 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
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, = p, n , suppose, the corresponding function F(a.) can be found as follows:
Assume
F(a) F(a i ) = ^ (a)
then
F(a) = $/{*, (a) *.()
and so on, up to
so that the formulae contain ultimately /tth roots only. It is remarked in a foot-
note that, when n = l, the /n 1 functions can always be reduced to one-sixth part in
number, and that by an induction continued as far as p = 31, Jacobi had found that
all the functions i/r could be expressed by means of the values of a single one of
these functions.
" The fj. 1 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 /a" 1 functions ^(a*) by means of the
powers of F(a); since all the fj. n 1 magnitudes [F (a)}* + F (a!) are each of them
82
60 FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. [721
equal to a product of several of the functions ifr (at). 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 $ gives simultaneously the quantities under the radical
signs, and the mutual dependency of the radicals. The formation of the functions i/r
is obtained by a very simple algorithm, which requires only that one should, from the
table for the residues of g m , form another table giving g m ' \+g m (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 of 1 =
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, a? + a? + a? + a; + 1 = 0.
According to Gauss, we have tc + tn* and a? + or", the roots of the equation u? + u 1 = ;
say x + sc t = b j-l + V(5)}, a?+ as 3 = \ {- 1 - V(5)}. The first of these, combined with
x.x t = l, gives a;- 4 = V[-i{5-l-\/(5)j]; and thence 4 = - 1 + V(5) + V[- 2 {5 -I- V(5)}] ;
if from the second of them, combined with a?.a? = \, we were in like manner to obtain
the values of of and a?, it would be necessary to investigate the signs to be given
to the radicals, in order that the values so obtained for a? 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=H, and writing for shortness F lt F 3 , F 3 , F t to denote F(a),
F(cf), F(a 3 ), F(a') respectively, these functions are
Ft = x x 1 + i (a? of),
Ft = x + a?- (of + o?),
F 3 = x & - i (a? of),
viz. we have F t = -l, F*=5, or say J t 2 = V(5), ^ = -(1 + 20^, =- (1 + 2i) V(5); and
similarly F t *=-(l-2i)F 3 , = - (1 - 2i) V(5) ; but also F l F 3 = -9, so that the values
.F 1 = V{-(l + 2i)V(5)}, .F 3 = V{- (1 - 2i) V(5)}, must be taken consistently with this last
equation F^F 3 = V(5). The values of F lt F 3 , F,, F 4 being thus known, the four equations
then give simultaneously x, a?, a?, a?, 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)} + V{- (1 - 20 V(5)J = V[~ 2 (5 + V(5))],
as is at once verified by squaring the two sides.
722]
61
722.
A PKOBLEM 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/S = 15, or 2a + /8=5.
The solutions of this last equation are (a = 0, y9=5), (a = l, /9 = 3), (a = 2, /9 = 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
* = 0, /3 = 5
a = l, /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 at partitions
into 5's and /8 partitions into 3's, then, if these contain all the duads, 4a + 2/9 = 14,
or what is the same 2a + /3 = 7 ; if a = 0, /3 = 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, ft, 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 o, /8, which are such that /3a = tfp* cannot give rise to a group made up of
symbols of the form a p /3i. In fact, the assumed relation gives /3a" = a-^a 2 /^ ; and
hence, if a 2 be of the form in question, = cfftv suppose, we have
= a- .
that is, l=a 2 /8 2 , and thence /?a=l, that is, @=a.~ 1 , viz. the symbols are commutative,
and the only group is that made up of the powers of a.
723]
63
723.
VARIOUS NOTES.
then
[From the Messenger of Mathematics, vol. vm. (1879), pp. 4546, 126, 127.]
An Algebraical Identity: p. 45.
Let a, b, c, f, g, h be the differences of four quantities a, /3, 7, , say
a, b, c,f, g, h = /8 - 7, 7-0, a - /3, a-S, /3 - B, 7 - 8 ;
-h .+f+b = 0,
n f . + c = 0,
j j
a b c . = 0.
Now Cauchy's identity
(a + iy - ; _ b 7 = Tab (a + b) (a- + ab + b-)-,
putting therein a + 6 = c, becomes
a 1 + b 7 + c 7 = labc ( be + ca + ab) 1 ;
hence we have
h 7 - g 7 + a 7 = - Tagh (- ga + ah- kg)*,
-h 7 . +f 7 + b 7 = - nttf (- hb + bf -fhf,
g 7 -f 7 . +c 7 = - Icfcj (- fc + eg - gf)*,
-a 7 -b 7 -c 7 . =-7abc( bc + ca+ab)-;
whence, adding,
agh (- ga + ah- %)' + bhf(- hb + bf-fKf + cfg (-fc + cg- gff + abc (be + ca + ab? = 0,
64 VARIOUS NOTES. [723
or, as this may also be written,
agh (g* + A' + a')" + bhf(li> +/' + b ' 2 )' + c /9 (/' + ff" + C 2 )' + abc (a* + b*- + c') 1 = 0,
an identity if a, b, c, f, g, h denote their values in terms of a, y9, y, 8.
Note on a Definite Integral : p. 126.
The integral
, = f 1 k*a?dx
~Jo <S(l-aP.l-iW)'
used by Weierstrass, is at once seen to be =K E\ but the proof that the other integral
i
,, = r* k-x-dx
~h \?(x?-l.l-kW)
is = E' is not so immediate.
We have
_
% Vd ->/) (1 - y^ (1 -
and thence
f> (l-2y+Ay)dy
/ a-2/ 2 )Hi-% 2 ) }
viz. replacing the numerator by
-TF+pflrW.
this becomes
1 p
^=)J * *. J
that is,
r * . I E<
Jo (l-2/ 2 )i(l-^/)5 A-"-'
or, writing i' for ,
The integral /' writing therein x = X becomes
~
viz. its value is thus =".
723] VARIOUS NOTES. 65
OK a Formula in Elliptic Functions : p. 127.
cn if
Writing enu = ^ , then the formulae p. 63 of my Elliptic Functions give
m __ mt D i jy
snu + v) = ,, en ( + ) = -,;
and, substituting for T, T', B, B', and C, C' their values, we obtain
sn u en v + sn v en u
sn
en (M + v) =
+ k- sn u en umvenv'
en en sn u sn D
1 &* sn M en M sn v en t> '
formulae which, as regards their numerators, correspond precisely with the formulae,
sin (u + v) = sin u cos v + sin v cos u
and
cos (u + v) = cos u cos w sin u sin t,
of the circular functions, and which in fact reduce themselves to these on putting k = 0.
The foregoing formulas, putting therein & 2 = 1, are the formulas given by Gauss,
Werke, t. in., p. 404, for the lemniscate functions sin lemn (a + b) and cos lemn (a + b) ;
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 = \/(l sin lemn 2 a) -=- \/(l + sin lemn 2 a).
C. XI.
66
[724
724.
ON THE DEFORMATION OF A MODEL OF A HYPERBOLOID.
[From the Messenger of Mathematics, vol. vm. (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 mpdel 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 (a;,, y lt 2,), (xj, y 3 , 2 2 ) be points on the generating line of
a- f f
+ > ~ 2 '
then
or, what is the same thing, if
then
P
#a
a'
724] ON THE DEFORMATION OF A MODEL OF A HYPERBOLOID. 67
Similarly, if (ft, %, ft), (ft, r}.,, ft) be points on generating line of
and if
then
fi _ - . 'k _ n n r .
' ' ~ P " q " " ' ' ~ P2> q2 ' "
Pl 2 + q, 2 - Fl 2 = 1,
p/ + q* 2 - rf = I,
p 1 p 2 + q 1 q.,-r 1 r 2 = l.
Hence if (a;,, y lt z t ), (ft, %, ft) be corresponding points on the two surfaces, that
is, if
and similarly, if (#, y 2 , ^.), (ft, ^2, ? 3 ) are corresponding points, that is, if
tf s y 3 ^_ft ^ ?s_
a' 6' " "y3' y-**' rs '
then we have, as before, the system of three equations
Then if the two surfaces are confocal, that is, if
a 2 , 0"; -7 2 =a 2 + A, b-+h, - c" + It,
(a, - a,? + (y, - y t y + (*, - * 2 ) 2 = (ft - ft) 2
For this equation is
we shall have
r = a
that is,
(J3i - Pi? + (?i - 90 2 - (n - r 8 ) 2 = 0,
an equation which is obviously true in virtue of the above system of three equations.
Hence, if on confocal surfaces
a? y- *_'_, f ?' C"
2 + 2 " ' 2 + a * '
we take two points P,, P 2 on the first, and Q t , Q 2 the corresponding points on the
second ; then P,, P., being on a generating line of the first surface, Qi, Q., will be
on a generating line of the second surface, and PiP 2 will be = QiQ 2 . The same
is evidently true for the quadrilaterals PjP^PaPi and QiQ 2 Q 3 Q 4 , where PiP 2 , P 2 P 3 ,
P S P 4 , P 4 P, are generating lines on the first surface : and therefore QiQ 2 , QtQ 3 , Q 3 Q t ,
Q t Q t are generating lines on the second surface, which proves the theorem.
92
68
[725
725.
NEW FORMULAE FOR THE INTEGRATION OF
[From the Messenger of Mathematics, vol. vin. (1879), pp. 60 62.]
I HAVE found in regard to the differential equation
~
J i
_ _ _ _ =
\J(a x.b a; . c x . d x) *J(a y.b y.c y.d y)
a system of formulas 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,
Oi, b,, c,, d, = a - y, b - y, c-y, d- y,
and (be, ad) to denote the determinant
1 , x + y, xy
1, b + c, be
1 , a + d, ad
and (cd, ab), (bd, ac) to denote the like determinants; then the formulae are
x/C
a- z\ _ V(a - b . a - c) { VfcdbA) + V(aid,bc))
d z) (be, ad)
_ \/(a b.a-c) (x y)
= V(adb lCl ) -
_ V(o - b . a - c) {V(abc t di) + V(aib,cd)|
(a - c) VCbdb.d,) - (b - d) V(aca,c,) '
_ V(o - b . a - c) { VCacb.d,) +
725]
NEW FORMULA FOR THE INTEGRATION OF
dii
69
A
/ /O ~ \ y \tt
V U^J =
(be, ad)
V(adb,c,)
<s/Ga) (cd ' ab)
(a - c) VCbdhA) - (6 -
(a ~ d) V(bcb ' Cl) + (6 ~ c
-
/ / C *~~ * \ y \0r- tt
V U - "z) =
(a - 6) V(cdc,d,) - (c -
cA) + (c - d) VC
- d) {(a ~
- < 6 - c)
(a - c) V(bdb,d,) - (b -
(a -
jdj) - (c -
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
a b.a c dj
d b.d c a, '
a b C]
d c &i '
a c
11
d
b
X =
- z
a,
a,
d,'
b,
c
b,
a
b,
x = c,
b a
Ci
z
d
c
-b
-b
a,
c,'
d,
d
b a. b
c
c
b,
d
c
z
z
d,'
d
c a .c
a
-b
d a.d
b
c
c
b,
d z dj ' d a .d b Oj
d a bj '
d - b a, '
viz. in the first case we have z = y, and in each of the other cases z equal to a
linear function \. of y.
Cambridge, July 3, 1878.
70 [726
viz. this is
726.
A FORMULA BY GAUSS FOR THE CALCULATION OF LOG 2
AND CERTAIN OTHER LOGARITHMS.
[From the Messenger of Mathematics, vol. VIII. (1879), pp. 125, 126.]
GAUSS has given, Werke, t. n., p. 501, a formula which is in effect as follows:
m = inai/'lPJ^Y ( /1048576 V /6560N 3 /I 5624 Y /980iy
U024/ U048575; \G56lJ U5625/ \9800J '
3 .41Y / 2 y /S^a^lV /2*.3*.7.3iy / 3MP y
2' ) U 2 .3.11.31.4lJ ^ 3" ) \ 5 J l2 3 .5'-.7V '
\
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 = 2 196 as it should be. The value of log 2 calculated from
2 U =10 51> is log 2 = -^r = -301020, viz. there is an error of a unit in fifth place of
decimals. The actual value of 2 196 has been given me by Mr Glaisher :
2 U = 10043 36277 66186 89222 13726 30771
32266 26576 37687 11142 45522 06336.*
Supposing log 2 calculated by the form, we then have
12 + 10 2 , giving log 41,
and
3 8 =10.&M*.2.41, 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 2 193 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. vin. (1879), pp. 190, 191.]
THE equation of the wave-surface
aa? j__ by- cz* _ A
a? + y* + z* - a a? + y 2 + ^- - b a? + y- + z*
when transformed to coordinates p, q, r, such that
a? y- z"
__ j_ _ y _ j __ = i
a + p -b+p c+p
_y a _
_ __ ___=
a + q b + q c + q
a? y- z*
I __ y. _ j
-a+r-b+r-c+r
(that is, to the elliptic coordinates belonging to the quadric surface '- + ^- + -- = 1),
^ a "~ o c
assumes the form
(q + / 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) = (u -a)(u b)(u c)
-a?(u- b) (u - c) - y 2 (u -c)(u- a) -z*(u- a) (u - b) ;
72 EQUATION OF THE WAVE-SURFACE IN ELLIPTIC COORDINATES. [727
whence, writing for shortness
A=a + b + c , P=p
B = be + ca + ab, Q = qr + rp +2>q,
C = abc , R = jMjr,
we have
y- + z' = P-A,
bca? + cay- + abz- = R C,
and thence also
by'-+ cz> = A(P-A)-(Q-B).
The equation of the wave-surface is
abc - [a (b + c) a? + b (c + a) f- + c (a + b) z*} + (a? + y- + z*) (oa? + by* + cz") = 0.
By the formulae just obtained, this is
that is,
that is,
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, 6, 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, x , 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. Arid 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, \/b),
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. 43 48.
Read January 8, 1879.]
THE theorem is as follows :
If u + v + r + s = 0, then
1 k' 3
k' 1 sn u sn v sn r sn s + en u en v en r en s j- dn dn v dn r dn s = =- .
K 2 A 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 sn 2 u sn 2 r, cn s u en 2 r, dn 2 u dn 2 r
is the relation in question
1 k'-
A/ 2 sn 2 u sn 2 r + en 2 u en 2 7 ^ dn 2 w dn- r = p .
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 + S) sn (7 - 8)
+ en (a + /3) en (a - /3) en (7 + S) en (7 - S)
C. XI.
k'- 2k'* (sn 2 a - sn" 7) (sn 2 ff - sn 2 8)
" & 1 - & 2 sn 2 a sn 2 /S . 1 - fc 2 sn 2 7 sn 2 2
10
74 A THEOREM IN ELLIPTIC FUNCTIONS. [728
In fact, writing herein + 7=0, that is, 7 = a, the right-hand side becomes = ;
and the arcs on the left-hand side are a. + ft, a ft, a + B, 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, ft, 7, 8
respectively, also
R = 1 - k'a?-
D = 1 -
the equation is
_
A? DD, A? DA
that is,
- AAPP, + QQ, - ~ RR, + ~ DA + 2i' 2 (a,- 2 - ) (^ - w s ) = 0.
It is easy to verify that the terms of the orders 0, 1, 2, 3 and 4 in y?, y 2 , z-,
separately destroy each other; for instance, for the terms of the order 2, we have
- fc' 2 (a? - y 2 ) (z* - w 2 ) + {(a? + f) ( 2 + w 2 ) + If (x?y- + s'wr')}
+ *L. {- jt> (^ + iW)} + 2k' 3 (a? - z 3 ) (y 2 - t^) = 0,
A*
that is,
- k' 1 (a? - y") (z* - w 3 ) + (1 - If) (a? + y s ) (z* + w 3 )
+ (k*-l- k'*) (a?y' + zW) + 2&' 2 (a? - z*) (y"- - w 2 ) = ;
or, omitting the factor A;' 2 , this is
_ (a? _ y!) (^ _ w s) + (a? + y 1 ) (z* + iu") - 2 (o?y* + zW) + 2 (a? - z*) (y 1 - w") = 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
a? w 2 z*
__ i j __ i __ __ i
a +p b+p c+p
a? u* z*
i y __ i
__ __ __
a + q b +q c + q
a? y"- z 1
_ + y + - = i
a+r 6 + r c +r
728] A THEOREM IN ELLIPTIC FUNCTIONS. 75
or, what is the same thing,
=a+p.a+q.a + r,
=b + p .b + q .b + r,
a/3z" = c +p . c + q . c + r,
where a, ft, y 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
a? if z*
-- hr 2 -- h =1,
a + r o +r c + r
and the differential equation of the right lines upon this surface is
_ dp _ _ dq
^a+p.b+p.c+p */a + q.b + q.c + q
Take #, y c , z the coordinates of a point on the surface, and p , q a the corresponding
values of p, q, so that
ySya-,, 3 = a + p . a + q a . a + r,
- 7yo 2 = b+p t .b+q,.b+r,
aftz,? = c + p . c + q, . c + r,
then the equation of the tangent plane at the point (x , y , z a ) is
<<> + yy<> + zz _ = i
a + r b + r c + r
or, substituting for a?, x, &c., their values, we have
and consequently the equation of the tangent plane is
, &c.,
+ q.b+p a .
+ 7 Vc +p .c + q.
the equation of a plane intersecting the ellipsoid in a pair of lines; hence this
equation (containing in appearance the two arbitrary constants p and (ft) is the integral
of the proposed differential equation.
Writing
sn 2 u = A (a + p), crfu = B(b+p), du-ii = C(c+p),
the values of A, B, G, k are determined ; and, assuming for q, p a , q a the like forms
with the arguments v , u , v , the differential equation becomes du = dv, having the
102
76 A THEOREM IN ELLIPTIC FUNCTIONS. [728
integral u = 1> ; while the foregoing integral equation, on reducing the constant
coefficients contained therein, takes the form
k' 2 sn M sn v sn u, sn v
+ en u en v en M O en v t
_ dn u dn v dn M O dn i>
viz. this equation holds good if u u l) = v v . 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, en u en v en r en s, dmidnvdnr 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.l - Ay, a = z Vl^wVl
A' = y Vl - a? . l-^a?, a = w Vl -
P =uf-y t , TS =z- - w 2 ,
D = 1 - te, S = 1 -
we have
sn (u + v)= - sn (r + s),
that is,
A+A/ P _a+a' _
S~~A~-A'~ ~S ~^~-
and consequently
DOT = - (a - a') (A + A'),
PS = - (a + a') (A - A') ;
whence
728]
that is,
A THEOREM IN ELLIPTIC FUNCTIONS.
(z 2 - w 2 ) (1 - #tey) - (a; 2 - y 2 ) (1 - teW)
Vl - *.l -%" . 1 - *". 1 - A*2 S - y* Vl - <eM -
77
- w" . 1 -
Rationalising, we obtain, as mentioned above, an equation containing only the squares
a?, y 2 , z*, w 2 ; it therefore is of a degree twice that of the equation containing
the product xyzw. I worked out in this way the equation in (x a , y\ z 3 , up}, 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,
A;' 2 xyzw
T 2 '
k'*
that is,
and then, writing
.i-kY.i-k'z'.i-,
M
k'* (1 - k*xyzw) = k- Vl - a? . i~-y* .l-z\l^w*
-\l\-tftf.\- k*y* .1-k-z 2 .!- k*w* ;
P = x- + f + z 1 - + w\
Q = a?y* + a?z- + 2 w 2
R =
Cf
and using >JS to denote the rational function xyzw, we have
or, if for a moment the radical is called \/A, then the factor & 2 divides out, and
the equation becomes
2 VA = 2 - (1 + Ji?)P+2k 2 Q -
whence
- {2 - (1 + fc 2 ) P + 2& 2 Q - (fc 2 + A*) R + 2& 4 S} 2 - 4,k'*S
= -2k t ^/S{2-(l+ k 1 ) P + 2A: 2 Q - (jfc + ^) -R -
The factor A;' 4 divides out; omitting it, we have
4Q _ p> _ 4 (i + If) R + iQk*S + 2k*PR - 4 (k* + ) PS - k*
-2</S{2-(l+k*)P + 2k*Q - (k*
or, as this may also be written,
{(_ P' + 4,Q - 4E) + A; 2 (- 4,R + 2PR + 168- 4>PS) + * (- R 2 + 4,QS - PS)}
2k<S},
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 1 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 a;, y, and we may consider the second figure as derived
from the first by altering the distance OP in the ratio Va;' a + y'* -f- *Jx* + y', and by
y> y . .
rotating it through the angle tan" 1 , tan" 1 - ; say by the Extension vx' 1 + y" 1 -e- va? 4- y',
SC 3C
y' 11
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+dx being arbitrary), then the coordinates of the
corresponding point Q' will be x' + dx, y' + dy', where
dx' . dx' . , d.
Writing -f-, and -~ instead of dy -5- dx' and dy -f- dx, the expressions
doc (tx
dy"' -5- */da?+dy*, and tan" 1 - , tan~' - ,
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
dtf = _dy' dj/^M.
dy da;' dy dx'
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,
' 2 + dy" 2 -f- Veto 2 + dy\ and tan" 1 -]-, - tan- 1 -/ ,
dx'
dx'
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"* -r- "Jda? + dy*, and rotating the element through the angle tan" 1 --, tan~' -~- ;
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' Ay' _ dx'
dy dx' dy dx'
the third figure is to have the Auxesis Va/ 2 + y' 2 ~ *Jy? + y*, and the Streblosis
tan- 1 ^7 - tan- 1 ^;
x x
viz. writing r for V^ 2 + y", we ought to have
dx" =
dx -
dy,
xii x'v
= - y
xx
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
_ _ =
dy dx ' dy ~ dx '
and the third figure is thus conformable with the first.
Writing, for shortness,
. _ xx' + yy 1 __ xy' - x'y
* ' ~ '
80 ON A THEOREM RELATING TO CONFORMABLE FIOURES. [729
the equations are
dx" = Adx-Bdy,
dy"=Bdx+Ady;
or the conditions for the existence of the functions x", y" are
dA dB = Q dA_dB = Q
dy dx dx dy
We, in fact, have
f dx'\ ,) 2 ,,
and similarly
dA_dB
dx dy
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
<M = dj = F dx L = _ d JL = _Q
dx dy dy dx
we have
dx = Fdx Gdy,
dy 1 = Gdx - Fdy ;
that is,
dx 1 + 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 <f> (x, y) + iifr (x, y) is
a function of x + 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
<j)(x, y) + ity(x, y);
and Cauchy, adopting this definition, introduced the expression " fonction monogene "
of x 4- 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 "The'orie des
fonctions elliptiques " (Paris, 1875), although not using Cauchy 's expression fonction
monog&ne, 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 admettent
une de'rive'e." 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 zf, = 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' r
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 "fonction 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
as + 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
fi
figure, in fact, the Auxesis and the Streblosis are = ^F- + G 1 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 MR 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. 194196.]
WRITE
U=(a, b, c, d, f, g, h, I, m, nfrx, y, z, ff,
#o = ( $> r,, ?, ),
w=( $*, y,
P = (a, ft, 7, 8$*, y, z, t\
P = (, ft, 7.
Then the equation of the cone, having for its vertex the arbitrary point (f, if, f, to), and
passing through the conic U = 0, P = 0, is
UP,? - 2 WPP,> + U P> = 0.
Or if, to put the coefficients f, ij, , o> in evidence, we write for a moment
A = (a, h, g, I $, y, z, t),
B=(h,b,f,mji ),
c = (9> / c, n $ ),
Z) = (I, m, n, d $ ),
and therefore
then the equation is
f -f ^ + 7?+ &) (-4f + ^ + C+ -Da)
P s (a, 6, c, d, /, 0, A, I, m, n%, ij, f, o>) 2 = 0.
730]
ADDITION TO MR SPOTTISWOODE S PAPER.
And if we expand first in f, 17, f, o>, and then in x, y, z, t, the final result is
of y*
yz
zx
xy
xt
zt
+ f
+ r
1 01
C
B
F
1A'
2L
2L'
c 1
A
G
IB'
1M'
2M
5
A
H
B
2C'
2N
2N'
I 1
G
H
IF'
2G'
1H'
2^1'
IF'
-2A
-1C 1
-2B'
2(Q-R)
-2M
-2N'
IB'
2G'
-2C"
-IB
-2A'
-2L'
1(R-P)
-2N
1C'
111'
-IB'
-2A'
-1C
- 2L
-2M'
2(P-Q)
2M'
2N
2(Q-Jt)
-2L'
-1L
-2F
-IE'
-29'
2L
IN'
-2M
-2(B-P)
-2M'
- Ill'
-2G
-IF'
2L'
2M
-2N'
-IN
2(l'-Q]
- 1G'
-IF'
-2H
= 0.
In particular, if ^ = 0, f=0, o> = 0, then we have the foregoing equation X=0', and the
like for the equations F=0, Z=0, and W=Q respectively.
Take a, b, c, f, g, h for the six coordinates of the line through the points
x, y, z, t
that is, write
a = 7/f zi), f = xa> t!;,
b = 2%-x!;, g = y<a - in,
c = xij y%, h = zu> t,
where, of course,
af+bg + ch = 0.
Then the foregoing equation of the cone is
Aaf + b 2 + Cc 2 + F? 1 + Gg 2 + -ffh 2
- 2A 'be - 25'ca - 2C'ab + ZF'gh + 2(?'hf + 2H'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].
112
84
[731
731.
ON THE BINOMIAL EQUATION af-1-0; TRISECTION AND
QUARTISECTION.
[From the Proceedings of the London Mathematical Society, vol. xi. (1880), pp. 417.
Read November 13, 1879.]
THE solution of the binomial equation **-l = 0, p a prime number, or, say rather,
the equation %
a;*- 1 + a:*- 2 +...+X + 1=0,
depends upon the Jacobian function
Fa =0? + aafl + ... + a?
where g is a prime root of p, any root whatever of the equation '^-J--?- Tftkin g
e a factor of p-l, and / for the complementary factor (that is, p-
a we write a/ or, what is the same thing, taking a/, =/3, a root of -
Fft = X, + 0Z, + . . . + /fr-'Z.-,,
p 1
where Z., Z,, .... Z^., denote each of them a period or sum of /, = , r
X, =(1, flf, ...,sr </ - 1 ")-
(read ^0 = ^ +^' + ... + *<' </ " 1|e , and so for the other functions).
We have, of course, F(l), "Z. + Z. + .-.+Z,.,, the sum of all the roots . -lj
and, further, the general property that any rational and integral function
periods is expressible as a sum
with known coefficients
, a,,
731] ON THE BINOMIAL EQUATION 0^1=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( \) = X Y;
e = 3, three periods, X, Y, Z, and Fy = X + yY+ ^-Z, if 7 is a root of M 3 -l = 0;
e = 4, four periods, X, Y, Z, W, and FS = X + SY+S*Z+ B 3 W, if 8 be a root of t( 4 -l=0.
It is sufficient to attend to the prime roots 7 and B of the equations
u 3 - 1 = 0, it 4 - 1 = 0,
respectively; for, if 7 or 8 be =1, we have simply F(l), = 1; and if S be = 1,
then the function is F(-l), =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
iY-Z-iW,
respectively.
p-i
As regards the bisection, it is known that (X y) 2 = ( ) * 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 </, a prime root at pleasure of p ; and, in what
follows, I select the prime root used in Reuschle's Tafeln complexer Primzahlen
welche aus Wurzeln der Einheit gebildet sind (4to, Berlin, 1875) : viz. these are
;> = 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89, 97,
(7 = 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 i; , 17,, ^ 2 , and
X, Y, Z, W his t) , 77,, 7j 2 , % : 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, j)=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 x" 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, ^=13, quartisection (see p. 13), he has
'h > = '?i + 27a, 170171 = 1 -ij*, %k = 3 + 7?, + 77,, %;, = 1 -17,,
(where observe that here and in every case the value of t) i) 3 is at once obtained
from that of Jjo^i by a mere cyclical interchange of the suffixes, so that the last
equation is in fact superfluous) ; the other equations, using T;,, + 77, + 17., + / = 1 to
eliminate any constant term which occurs, give my values
X* = ( 0, 1, 2, 0)(Z, 7, Z, W),
XY=( 1, 1, 0, 1)( ),
XZ = (-3, -2, -3, -2)( ).
Similarly, in the case of a trisection, the equation for 17, ij, is superfluous, and the
other equations give my values of X* and X Y.
Reuschle gives also, and I take from him, the cubic and the quartic equations (such
as p=l'3, if -\-if- 4?; + 1 =0, if* 4 if + 2^ 4r) + 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. 166189; [Ges. Werke, t. vi. pp.
254274].
Trisection, e = 3; p = l (mod. 3).
We have three periods X, Y, Z; and we theuce obtain
X* =(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 8 =a, b, c,
Y' = c, a, b,
Z- = b, c, a,
=f, ff , h,
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 a^ 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-iy terms,
each a power of x, and the sum XY+YZ+ZX contains %(p - 1) 2 such terms, being
in fact the sum X+Y+Z taken (p 1) times; whence the relation in question.
Hence also
From the equation X+Y+Z=-l, multiplying by X, and for X\ XY, XZ
substituting their values, we obtain an expression
(a +f+ g
which must identically vanish; viz. the three coefficients must be each of them =0; or
we must have
0--/-0-1,
b = - g - h,
c = -),-/
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 X\
XY, &c. their values,
h(a, b, c)= g(f, g, h)
+f(f,g,h) +h(c, a, b)
+9(9, h,f) +f(h,f,g);
that is,
ah+f* +g-=gf +c h+fh,
bh +fg + gh = g* + ah +/,
ch+fh+fg=gh + bh +fg,
equations which reduce themselves to the single equation
gh + hf+fg + h =f* + g* + h* ;
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
= (f<j-h\ fg-h\ fg-h*);
that is,
XYZ=h*-fg.
We have
c-f-g-h)(X+Y+Z)
88 ON THE BINOMIAL EQUATION 0?" 1 = 0. [731
that is,
jy.jy-p.
We have, moreover,
[(a, b, c)
+ 7 [(b, c, o) + 2(/ >fl r, h)]
+ y*[(c, o, 6) + 2(<7, A./)],
which is
= {(a + 2A) + 7 (6 + 2/> + r (c + 2oO
as is at once seen by comparing the coefficients of X, T, Z respectively.
Hence, writing
a + 2A + y (b + 2/) + 7 2 (c + 2$r)
we have
.4 = a + 2/i - c - 2g = 3A - 3# - 1,
We have
and thence, writing <y* for 7 ,
equations which give
^ 7 .F 7
or, say p = A AB + B 1 ', viz. p has the complex factor
A + By, =3A-3 5 r-l+ 7 (3/-%).
Hence also
and, as before,
F 7 .fy-p;
which equations determine .F 7 , ^ 7 % and from these and F (!) = ! 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 = -
the equation thus is
As already remarked, the values of a, b, c ; f, g, h, and the equations in 77, are in effect
given in Reuschle ; the complex factors of p, as given p. 1 (7 = 2 7 3 7 2 , &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 X p 1 = 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 f, g, h by the
foregoing formulae. It will be seen that all the values are consistent with the theory.
TABLE FOR THE TRISECTION.
p
a, b, c
/. g, h
Tp + i? +
y v
A B
Page in
Reuschle
7
2 - 1 -2
- 2 - 1
2 3
p. 6
1 1
13
4 3 2
4 1
- 4 -3
P- 15
1 2 1
19
4 -5-4
6 -7
2 -3
p. 26
1 2 3
31
- 7 - 6 -8
- 10 8
5 6
P- 45
424
37
8 - 10 7
- 12 11
- 4 3
P- 54
543
43
- 11 8 - 10
- 14 8
- 1 6
p. 69
644
61
- 14 - 13 - 15
-20 - 9
4 -9
P- 97
587
67
- 16 - 13 - 16
- 22 5
2 9
p. 105
967
73
- 16 - 18 - 15
- 24 - 27
- 1 -9
p. 128
699
79
-20 -17 -16
-26 41
- 10 -3
p. 138
9 10 7
97
- 20 - 23 - 22
- 32 - 79
11 3
p. 1 68
10 9 13
C. XI.
12
90
ON THE BINOMIAL EQUATION a?- 1 = 0.
[731
Quurtisection, e = 4 ; p = 1 (mod. 4).
We have four periods X, Y, Z, W; and we obtain
X* =(a, b, c, d)(X, Y, Z, W),
XY=(f,g,h,k)( ),
XZ=(l, m, l,m)( ),
the coefficients being determinate integers. It can be shown that l + m = (p 1) or
l) according as p = 1 or 5 (mod. 8). And then, by cyclical interchanges,
Z' = a, b, c, d,
Y* = d, a, b, c,
Z 2 = c, d, a, b,
W* = b, c, d, a,
XY=f, g, h, k,
YZ =k,f,g, h,
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
is
and, in virtue of the foregoing value of I + 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 - I,
b= g h m,
c= -h k-l,
d= kfm,
XW+YZ+YW + ZW
and thence
and
+ d = -l-2 (f+g + h + k)-
a - b + c - d = - 1 + 2 (m - 1).
m),
731] ON THE BINOMIAL EQUATION X? I = 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
A; (a, 6, c, d) = I (f, g, h, k) = f(l, m, I, m)
+/(/. 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, 1) +k(h, k, f, g),
that is,
ka +f* + gl + gh = If + md + Ik + m" = If + gk + ch + kh,
kb +fg + gm + h- = Ig + am + If + ml =fm + fg + hd + k\
kc+fh+gl + hk = lh + mb+lg + m 3 =fl + g* + ah + kf,
kd +fk+ gm + fh = kl + me +lh +lm =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 + l) +fk + gl + hk = lf+g'-h(f+g + l+l) + kf, .
or, reducing,
which gives I.
Again, another equation is
- =fm +fg +
that is,
-k(g + k +'m) +fg+ gm + h 1 =fm +fg -h(k+f+m
or, reducing,
m(g + h-f- k) = k~- h* + gk-hf,
which gives m.
And we have also
md + Ik + m 2 = gk + ch + kh,
that is,
- m (k +f+ m) + lk + m 1 = gk + kh -h(h + k + I),
or, reducing,
I (k +h)-m (/+ k)=gk- fc.
Substituting herein for I, m their values, we have
(k + h)[g* + k*- 2hf- hg + kf- h] - (f+ k) [If - h* + gk - hf] + (h> - gk) [g + h -f- k] = 0.
122
92 ON THE BINOMIAL EQUATION 0^1 = 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,
ff - + k* - 2hf- h+(h- +f'-2gk-k) = ;
and the foregoing values of I, m are
_ tf + l?-ytf-h)-(gk-kf)
y+h-k-f
_k*-h* + gk-hf.
-g+h-k-f
and by means of these three equations all the foregoing equations are satisfied.
We have
FiFi* = (X - ZJ + ( T - WY
= Z 2 + F 2 + Z* + W 2 - 2 (XZ+ YW)
= - (a + b + c + d) + 2 (I + m) ;
or, substituting for a, b, c, d, this is
= 1 + 2 (f+g + h + k) + 4,(l + m),
viz. it is
-id + l)+4(J + );
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
FiFi* = (-)* p.
Again, we have
= X* - F 2 + Z* - W* - 2XZ+ 2YW + 2i(XY- YZ + ZW- WX)
= [a-b + c-d + 2 (m -
where
A=a-b+c-d + 2(m-l), =-1 + 4 (m - I),
B = 2(f- g + h-k);
or, since X - Y+ Z - W = F(-l), this equation is
.
and similarly
731] ON THE BINOMIAL EQUATION 3? 1=0. 93
Moreover
and we have therefore
that is,
4 + .B=p;
or the expression A + 5i determined as above is a complex factor of p.
We may investigate the quartic equation for the determination of the periods X, T,
Z, W. The values of X + Y + Z + W and X Y+ XZ + X W + YZ + YW + ZW are already
known: for the next coefficient XYZ + XYW + XZW+ YZW, we have XYZ=(a, 0, 7, 8),
where each of the coefficients a, /3, 7, is given under three different forms : the values
of YZW, ZWX, WXY are ($, a, 0, 7), (7, S, a, /3), (0, 7, B, a) ; and the required sum
therefore is
W), = -( + + 7+ g).
Taking the first expressions of these coefficients respectively, we have
+ m)
+y + h + k),
= k {- 1 -*(/> - 1) - -2 (l + m)} +(/+ h) [i(p - 1)1 + 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 + m 2 ) (f+g + h + k),
= Im {1 + 2 (/+ g + h + k)} - (I* + m 2 ) (/+ g + h+k);
or, substituting forf+g + h + k its value \ (p\), this is
Im -i(l- ,). ( p -i) >= i [ ( l + m -y -(I- TO ) Sp j.
Hence the required equation, having roots X, Y, Z, W, is
= 0,
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^=l or 5 (mod. 8).
s
94 ON THE BINOMIAL EQUATION a? 1 = 0.
We thus have the following:
TABLE FOR THE QUARTISECTION.
[731
p
abed
f 9 >> *
; m
V + V +
rf V ,
A B
Page in
Reuschle
5
0100
0001
1 1
1 1 1
-1 -2
p. 2
13
0120
1101
3 2
243
3 -2
P- '3
17
4 2 3 . - 4
2011
1 1
6 1 1
- 1 4
p. I 9
29
2302
1123
5 6
4 20 23
-5 -2
P- 3 6
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. 61
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
4 8 r. 5
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 0^ 1 = 0. 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
i i
i i
i i
n
1
2
3
4
5
225223
44449
... 3 6 8 8 10
2 5 3 13
3 10 6 5
2 21
4 4
8 15
16 16
13 14
2 3
4 9
8 27
16 19
3 26
2 6
4 36
8 11
16 25
32 27
3 10
9 6
27 13
38 36
28 31
2 2
4 4
8 8
16 16
32 32
2 2
4 4
8 8
16 16
32 32
62 5
10 25
52 52
29 41
23 59
3 2
9 4
27 8
2 16
6 32
30 10
10 3
33 30
11 9
63 90
1
2
3
4
5
6
7
8
9
10
11
9 12 15
7 11 11
3 9 16
6 5 14
10 8
77
7 18
14 10
9 3
18 17
17 12
15 22
6 16
12 17
24 20
19 29
9 25
18 13
27 39
17 29
34 10
31 19
25 32
13 28
41 28
37 45
25 27
32 35
10 21
30 22
11 5
22 10
44 20
35 40
17 21
34 42
3 64
6 61
12 55
24 43
48 19
35 38
6 3
17 15
60 2
28 10
32 50
67 31
18 64
54 45
4 7
12 14
36 28
29 56
21 27
7 76
32 81
70 34
53 49
77 5
6
7
8
9
10
11
12
13
14
15
4
12
2
. ... . . 6
11 2
3 19
6 8
12 7
7 8
14 24
28 10
27 30
26 4
15 24
30 21
23 3
4 32
12 38
36 4
22 40
15 25
30 50
7 41
14 23
9 9
18 18
36 36
11 5
36 9
31 45
5 6
26 30
8 29
24 58
72 33
58 66
85 50
58 15
49 53
46 45
12
13
14
15
16
17
5 9
10 5
25 28
21 22
9 18
18 26
23 24
26 5
28 46
3 33
22 10
44 20
50 4
47 20
16 49
48 15
45 62
15 H8
16
17
18
19
20
?1
13
20
6
... 11
13 4
26 12
23 5
17 15
36 33
35 34
33 40
29 35
35 3
19 30
14 18
42 39
6 7
12 14
24 28
48 56
27 40
54 13
47 26
33 52
3 27
44 62
30 18
14 17
65 30
37 60
32 37
17 74
5 89
61 17
50 73
76 51
18
19
20
ttl
22
23
24
25
26
7
5 14
10 11
20 2
11 6
22 18
15 23
21 5
5 30
10 16
20 14
3 2
6 12
40 14
34 46
16 37
5 41
15 34
2 11
43 53
33 47
13 35
26 11
52 22
51 44
5 37
10 7
20 14
40 28
19 56
38 45
16 12
69 60
18 8
51 40
38 54
13 51
51 65
74 47
64 11
34 22
23 44
69 5
55 25
48 56
16 75
35 71
71 31
83 19
22
23
24
25
26
97
28
'W
7
. . 21
12 31
24 22
6 16
18 19
49 29
45 58
15 23
30 46
25 36
59 34
48 10
68 20
87 93
29 57
28
'9
30
31
32
33
34
H5
11 9
22 13
7 37
14 17
28 20
19 38
11 2
33 20
13 12
39 26
31 25
7 15
37 57
21 55
42 51
31 43
9 27
18 54
60 25
59 50
57 33
53 66
45 65
29 63
37 24
22 47
15 16
7 7
8 35
70 29
46 40
59 80
19 77
57 71
13 59
39 35
69 85
23 74
67 61
52 28
47 86
75 84
30
31
32
33
34
S 1 )
36
37
38
3t
23
15
8
. ... 7
21 9
20 43
17 7
8 23
36 49
19 39
38 19
23 38
58 59
55 51
49 35
37 3
9 72
61 68
19 48
42 21
38 70
35 57
26 31
78 62
25 64
38 58
72 95
24 77
36
37
38
S9
40
41
24 42
29 44
46 17
39 34
13 6
26 12
48 32
65 14
76 41
70 82
8 91
62 37
40
41
42
43
44
15
11
29
8
... 33
25 9
50 18
47 36
41 13
52 24
43 28
25 29
50 58
45 70
11 58
43 71
39 63
52 81
77 79
73 75
61 67
80 79
86 14
88 43
59 42
42
43
44
45
46
47
48
49
50
29 26
5 52
10 45
20 31
40 3
39 49
17 31
34 62
7 57
14 47
4 23
35 42
40 64
66 28
45 67
25 51
75 19
67 38
43 76
50 69
79 32
56 29
78 96
26 87
68 94
46
47
48
49
50
96
ON THE BINOMIAL EQUATION a? 1 = 0.
[731
TABLE (continued).
53 69 61 67 71 73 79 83 89 97
N
51
27 6 28 27
?1
43
71
55
81?
67
51
52
53
64
55
56
57
12 56 54
24 51 41
48 41 1.5
37 21 30
15 42 60
30 23 53
24
68
27
41
57
55
69
53
46
11
55
56
55
7
21
63
31
14
27
54
25
50
17
HI
57
19
36
12
4
31
88
7
70
21
16
63
52
53
54
55
56
57
58
59
46 39
31 11
2
53
61
13
42
47
68
53
40
43
48
<W
58
59
60
61
62
63
64
65
22
44
21
42
17
34
20
33
58
46
12
84
65
33
19
22
37
39
62
28
5
15
45
56
23
46
9
18
36
79
44
74
84
28
39
13
47
82
44
52
35
n
60
61
62
63
64
65
66
67
68
69
49
56
64
63
49
26
57
fir.
10
30
11
33
61
39
78
73
34
41
73
54
8
80
24
46
66
67
68
69
70
71
38
44
20
60
63
43
18
6
72
41
70
71
72
73
74
75
76
77
X
22
26
40
41
44
53
3
6
12
24
48
13
2
60
20
66
22
37
22
26
66
78
4
40
72
73
74
75
76
77
78
79
80
81
26
52
21
43
42
14
64
51
12
23
36
69
78
79
80
81
82
83
84
85
86
87
17
65
81
27
9
|
11
13
33
39
2
'0
82
83
84
85
86
87
88
89
90
91
92
93
!H
95
6
60
18
83
54
55
65
r.R
88
89
90
91
92
93
94
95
732] 97
732.
A THEOREM IN SPHERICAL TRIGONOMETRY.
[From the Proceedings of the London Mathematical Society, vol. xi. (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 \l> sin (A B) = tan ^b sin A tan \a sin B,
tan \c [1 tan^a tan \l> cos (A B)} = tan 6 cos .4 + tan ^a cos B;
which are both included in the form
n . . m tan Ac tan i 6 (cos A +i sin J.)
tan ia (cos 5-i suijB)= Vr-r T-, ~.
1 + tan^c tan o(cos A + ismA)
For the first of the two identities : from
cos A + cos B cos G
cos a =
cos b =
sin sin C
cos + cos A cos C
sin A sin (7
we deduce
1 /cos A cos B\ cos C /cos cos .AN
COS COS 6 = - -, -: - i, s + TV - i, -- ;
sin V \sm B smAJ sin C \sm B smAJ
^ (sin 24 -sin 2.B) cos C sin Q4 -
sin C sin .A sin B sin G' sin .4 sin B
= _.J^L-4). {cos (A+B) + cos (7}
sm(7sm.d sin B l
sin (^1 - B) .
= ri HOMO 1);
sin t;
c. xi. 13
98 A THEOREM IN SPHERICAL TRIGONOMETRY. [732
that is,
/ A T>\ s i n @ ,
sin (A - B) = r- - (cos a cos b)
1 cos c
sin C sin c
. = - (cos a cos 6) ;
sin c 1 cos c
or, what is the same tiling,
- tan ^c sin (A - B) = -. (cos a cos b).
sin c
Here cos a cos b is = ( 1 + cos a) ( 1 + cos b) ; substituting for - '-. successively =
sin c J sin a
and - , the right-hand side is
sin b
\
I + cos a . 1 + cos b .
= = sm A . j sm B,
sin a sin b
= cot ^a sin A cot ^b sin B ;
whence, multiplying each side by tan $a tan 6, we have the relation in question.
For the second identity which is
tan \c {1 tan \ a tan 6 cos (.4 B)} = tan \ b cos A + tan a cos B;
if on the right-hand side we substitute for cos .4, cos B their values
cos a cos b cos c , cos b cos a cos c
: i : and : ; - ,
sin b sin c sin a sin c
the right-hand side becomes
1 (cos a cos b cos c cos b cos a cos c)
sin c { 1+ cos b 1+ cos a j '
whence, multiplying the whole equation by sinc(l +cosa)(l +cos&), it becomes
(1 cos c) j(l + cos a) (1 + cos b) sin a sin b cos ( A B)}
= (1 -f cos a) (cos a cos 6 cos c) + (1 + cos b) (cos b cos c cos a).
We have here
. . (cos a cos b cos c) (cos b cos c cos a) + D
cos (A - B) = cos A cos B + sin A sin B =
sin 2 c sin a sin 6
by substituting for cos A, coaB their foregoing values, and for sin .4, sin B their values
VD \/D
. , . , . . , where
sin b sin c sin a sin c
D = 1 cos 2 a cos 2 6 cos 2 c + 2 cos a cos i cos c.
732] A THEOREM IN SPHERICAL TRIGONOMETRY. 99
The numerator is
cos a cos b cos c (cos 2 a + cos 2 6) + cos a cos b cos 2 c
-f 1 cos 2 c (cos 2 a + cos 2 b) + cos a cos 6 . 2 cos c ;
viz. this is
= cos a cos b(l + cos c) 2 (cos 2 a + cos 2 6) (1 + cos c) + 1 cos 2 c,
having the factor 1 + cos c, which is also a factor of sin 2 c, = 1 cos 2 c, in the
denominator. We have, therefore,
, . R , _ cos a cos 6(1+ cos c) (cos 2 a + cos 2 b) + I cos c
COS (-a. /3 ) 7^ r i ; _ I
(1 cos c) sin a sm b
and the equation thus is
(1 cos c) (1 + cos a) (1 + cos b) {cos a cos b(l + cos c) (cos 2 a + cos 2 b) + 1 - cos cj
= (1 + cos a) (cos a cos b cos c) + (1 + cos 6) (cos b cos c cos a),
where each side is in fact
= cos a + cos 2 a + cos b + cos 2 b cos c (cos a + cos b) 2 cos a cos 6 cos c ;
and the second identity is thus proved.
132
100 [733
733.
ON A FORMULA OF ELIMINATION.
[From the Proceedings of the London Mathematical Society, vol. xi. (1880), pp. 139 141.
Read June 10, 1880.]
CONSIDER the equations
(a, ...\6, 1)"=0,
(A,.. .19, 1) = 0,
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, I) 3 = 0,
(A, B, CIO, 1)' = 0;
giving rise to an equation
v, =(o, b, c, dy(A, B, cy, =0.
And it is required to perform the elimination so as to put in evidence the nodal
line of this surface.
Take 0,, # 2 the roots of the second equation, or write
(A, B, C$0, \Y- = A (6- 6^(6-6,);
that is,
733] ON A FORMULA OF ELIMINATION. 101
then, if
! = (a, 6, c, d$ft, I) 3 ,
2 = (a, b, c, d%0,, I) 3 ,
we have
V =
viz. on the right-hand side, replacing the symmetrical functions of ft, 0., by their
values in terms of A, B, C, we have the expression of V in its known form
V = a-C 3 + &c.
Form now the expressions
,-0.,, fl,!-^,, 0/0! - ft 3 .,, ft 3 ! - ft 3 .,,
each divided by ft ft. These are evidently symmetrical functions of ft, ft, the
values being given by the successive lines of the expression
0, 1,
-i; o,
(0i + 0<>), - 0A, o,
+ ftft + ft 2 ), ftft (ft + ft), ft a ft 3 ,
ft + ft, ft 2 + ftft + 0fd, 3c, 36, a) ;
ftft, ftft (ft + ft)
ft 2 ft 2
o
and, consequently, these same quantities, each multiplied by A 1 , are given by the
successive lines of
( 0, A', -'LAB, - AC + 4B^d, 3c, 3b, a).
-A-, 0, AC, -2BC
2AB, -AC, 0, C-
AC-1&, 2BC, -C\
Calling these X, Y, Z, W, that is, writing
X = 3^1 2 c - 6^56 + (- AC+ 45 s ) a, &c.,
then ^T, F, ^, W are the values of
,-,, ftQj-ft,, ft^j
each multiplied by A 2 + (ft ft) ; and the functions all four of them vanish if only
! = 0, a = 0; or, what is the same thing, the equations X = Q, Y=0, Z=0, W=0
constitute only a twofold system.
The functions
( X, Y, Z )
Y, Z, W
102 ON A FORMULA OF ELIMINATION. [733
contain each of them the factor B,0 3 , that is, V ; they, in feet, each of them vanish
if 6i=0, and they also vanish if B 2 =0; or, by a direct substitution, we have
XZ - F* = -^- .-(*.- 0=) 2 @A, =
XW-YZ = - (0, - 0,y
y \\--Z- = -(8,-0if
Or, what is the same thing, these are =--4V, 2fiV, -CV, respectively; thus the
fii-st equation is
(3-4'c - 6ABb +(-AC + 4&) a} {2ABd - SACc + C'a]
_ (_ A*d + 3ACb- ZBCaY = -A (AW + Sec.), = - A V ;
and similarly for the other two equations. The nodal curve is thus given by the
twofold system ^ = 0, F=0, Z=0, W=0.
The method may be extended to the case where, instead of the quadric equation
(A, B, CQ0, 1) ! = 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 moving
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. i.
It *
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 0,a;i, 0,y,; then the position of the axes O l x l y l may be determined, say by
104 ON THE KINEMATICS OF A PLANE. [734
a, ft, the coordinates of 0, in regard to Oxy; and by 6, the inclination of O t x, to
Ox, And denoting by x, y, a?,, y l the coordinates of a point P in regard to the two
sets of axes respectively, then
x = a + x t cos 6 y, sin 0,
y = ft + #, sin + y, cos & ;
or, what is the same thing,
#, = (x a) cos 6 + (y ft) sin #,
y, = (x a) sin + (y ft) cos ,
or, as these last equations may be written,
x l = ttt+x cos (- 0) y sin (- 0),
y, = ft + a* sin (- 0) + y cos (- 0),
where at,, &, = acosfl- /Ssin d, asm ft cos 0, are the coordinates of referred to
the axes O^y,, and 6 is the inclination of Ox to 0,3:,.
When the motion is given, a, , are given functions of a single variable
parameter, say of t* ; or, if we please, a, are given functions of 0.
The velocities of a given point (*, y) are determined by the equations
x = a' (x t sin + y, cos #) #',
%
y' = ft' + (, cos # y : sin 0) & ;
that is,
or, as these equations may also be written,
- (x' - a.'} sin + (?/' - /3') cos =
- (x - a') cos - (y' - &) sin = yfl.
Hence if x =0, y' = 0, we have
x,0' = a' sin - ft' cos 0, or a' = (y - ft) 0',
y,0' = a.' cos + ft' sin 0, -ft' = (x-a)0',
which equations determine in terms of t, ar, and y l the coordinates in regard to the
axes OiX^i, and x and y the coordinates in regard to the axes Oxy, of 7, the centre
of instantaneous rotation.
If from the expressions of #,, y l we eliminate t, we obtain an equation between
(x lt y,), 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
' . 8' 8
x l = ,.1 sm 6 QJ cos 0, x = a. -& ,
I O' f,'
y l = Q, cos 6 + p sin 0, y = 8 + j, ;
or, if we take 6 as the independent variable,
x l = a' sin 0-8' cos 0, x = a - 8',
y 1 = of cos + 8' sin 0, y=8 + a'.
To find the variations of 7, we have
Xi = a" sin - 8" cos + a cos 6 + 8' sin 6, = a" sin - 8" cos + y lt
yi = a" cos + 8" sin - a sin + 8' cos 0, = a" cos + 8" sin - x l ,
y' =& + *',
x' = a' - 8".
Hence
i = x cos + y' sin 0, or x' = ,' cos y/ sin 0,
i = x' sin + y' cos 0, y' = #/ sin + y/ cos 0,
values which give *' 2 + y' 2 = a;,'- + y/ 2 , 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, Aj (fig. 2) were originally in contact, then the arcs AI, A-^I are equal, and,
calling each of these s, and X, Y, X lt Y t the coordinates of 7 in regard to the two
Fig. 2.
sets of axes respectively, we have X, Y, X lt F, given functions of s, such that
X'< t +Y'- = l, X^+Y^ l, the accents now denoting differentiation in regard to s.
We have, from the figure,
-' - -' Fl
C. XI.
14
106
ON THE KINEMATICS OF A PLANE.
[734
or, what is the same thing,
tan 6 = ( Y'X,' - F/A') -r (A"AY + Y' I'/),
say
sin 0,cos0 = F'Z,' - Y,'X, X'X t ' + F 1',' ;
and then, as before,
x a. + #1 cos d y-i sin 6,
y = @ + x 1 suiff + y 1 cos 8 ;
or, what is the same thing,
x X = cos (#, X t ) sin 8 (y l F,),
y - F = sin 5 (#, - A",) + cos (y l - F,),
where X, F, AT,, Fi, and therefore also 0, denote given functions of s. The formulae
will be of a like form if X, Y, X t , F, 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, C (fig. 3) describing the lines OA ,
OC a , which meet in 0; through A, C, describe a circle, centre 0,, and with centre
Fig. 3.
\
and radius =200,, describe a circle touching the first circle in a point /; and suppose
that A,,, C 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
IOjC 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 IOC a ; and therefore (the radius of the small circle
being half that of the large circle) the arcs 1C, IC a are equal, so that the rolling
motion will carry the point G along the radius OC , and will, in like manner, cany
the point A along the radius OA , 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 OB ; in particular, taking AB a diameter, the angle A OB will be
a right angle ; and the motion is determined by means of the two points A, B
describing respectively the two lines OA , OB Q 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 Oi of the small circle describes a circle). Considering any point Q
of the moving plane, let the line Q0 l 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 OB , (L1 at right
angles to each other, these lines may be taken for the axes Ox, Oy; the point O l
for the origin of the coordinates a;,, y,, the axes 0,^ being in the direction O t B
and 0^! at right angles to it; calling the length AB=2c, we have O^A = 0^ = 0,
and the angle ABO may be called 6 (but this angle was previously taken with a
contrary sign). We have then for the point P, having in regard to O l x 1 and 0,^ the
coordinates (#,, y^,
x = a + x 1 cosO y l sin 01
y = /9 x l sin 6 y l cos 0} '
where the sign of y, has been changed, and a = ccos#, /3 = csin#: the equations thus
become
x = (c + #1) cos 6 y l sin 8,
y = (c #,) sin 6 y cos 0,
where observe that c + x lt c-a.\ are the distances M^A, M } B respectively. And we
have, conversely,
#1 = x cos y sin 6 c cos 20,
y l = x sin 6 y cos + c sin 20.
If, in particular, y, = 0, then
1) cos 6, (c - #,) sin ;
or we have
a?
142
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 + x 1 ),
(c x 1 ), each taken positively; if a;, 1 + yS = c*, viz. if P be on the circle having AB
for its diameter, then y, 1 = (c + x t ) (c a;,), and we have
y+x = -(c-x 1 )(siu0 -- yj cos 6 } -- y, ( sin 6 - - ^<x9\, =- (c -a:,) -=-y,,
\ c Xi j \ y\
viz. as mentioned above, the curve on the fixed plane is a right line.
In the general case, we have
x(c- a:,) + yy l = (c 2 - a;, 2 - y, 2 ) cos 0,
oyi + y (c + ^0 = (c 2 - x i ~ 3/i 2 ) sin 6,
and thence
{a; (c - x,) + yy,}' + {xy 1 + y(c + a;,)! 2 = (c 2 ~ , 2 - 2/i 2 ) 2 ;
or, what is the same thing,
x 3 {(c - ,) + yf] + 4arycy, + y> {(c + x,)' + y*} = (c 2 - x? - yff.
Considering (a;,, 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 (,, 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 x^ write
x l GI ; the equation thus becomes
a? {(x, - 2cy + y, 2 } + 4cy,a;y + y 2 (a;,' + y, 2 ) = (*,' + yr -
or, what is the same thing,
(*i 2 + y? - Sea;,) 2 - (x 1 + y 2 ) (ar, 8 + y, 2 ) + 4,cx (xx, - yyO - V^ = ;
and if we suppose herein x 0, it becomes
(i* + yi 2 - Zcxtf - y 2 (x? + y, 2 ) = ;
or, writing a 1 , = i\ cos 0, , y, = r 1 sin^ 1 , where BI = angle QAB, this is
or say it is
r, = 2c cos 0, - y,
which is the polar equation of the curve described on the moveable plane by the
point S, 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
S whatever of the fixed plane, if we draw 08 meeting the small circle in A, and
734] ON THE KINEMATICS OF A PLANE. 109
through 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 OA,,, OB ; 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 OA 0> OB,, thereof (at right angles to
each other) pass always through the points A and B respectively, and the curve is
that described by the point S on the line OA ; the point describes the circle on
the diameter AB (the small circle), equation r^ = 2c cos O-i ; and OQ having a given
constant value =y, we have for the curve described by the point S the foregoing
equation i\ = 2c cos 0, y ; or writing y=f, that is, taking S on the other side of
at a distance OS =/, the equation is ^ = 2ccos 0^+f; viz. this is a nodal Cartesian
or Linden, the origin being an acnode or a crunode according as f> or <2c; and
if /=2c, then we have the cuspidal curve or cardioid n = 2c (1 + cos 0J, =4ccos 2 # 1 .
The general conclusion is that the centre of the large circle describes on the
moving plane a small circle (centre OJ, and that every other point of the fixed plane
describes on the moving plane a Lima9on 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 (x lt y^, (x, y), a second
point, coordinates (X lt F,), (X, Y), in regard to the two sets of axes respectively,
we have
x = (c + Xi) cos yi sin 0, X = (c + XJ cos d Fi sin 0,
y = (c HI) sin y 1 cos 0, Y = (c X t ) sin Y t cos d ;
from the first two equations we have
cos : sin : 1 = * (c ]) + yy l : xy-^ + y (c + a^) : c- x^ yf ;
and substituting these values in the second set, we find
X : Y : 1
= x {c 2 + c (X, - x,) - X,x, -Y iyi }+y{ c (y, - F,) + y,X, - ^ F,|
: x { c (y, - F,) - y,Z, + x, Y,} +y{c*-c (X, - x,) - X& - F.y,}
or the points (x, y), (X, F), 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(a\, 2/0 = of the curve in the moving plane
to substitute for x lt y^ their values in terms of x, y, 0, and then considering 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 + #1) cos 6 y t sin 0,
y = (c- a-j) sin 6 y, cos 6 ;
or conversely
#1 = x cos 6 y sin 6 c cos 2#,
y, = a: sin 6 y cos + c sin 20 ;
the envelope on the moving plane of the line
Ax + By + =
of the fixed plane is given as the envelope of the line
[A (c + #,) - By,} cos0+{-A + B(c- ,)} sin 6 + C = ;
viz. this is
{ A (c + ,) - By,}"- + {A yi -B(c- *,)!" - C* = ;
that is,
(A* + &) (x* + y, 1 + c") + 2 (A 2 - B 2 ) ex, - 4.ABcy, = 0,
a circle.
But the envelope on the fixed plane of the line
Ax, + By 1 + C=0
of the moving plane is given as the envelope of the line
C + (Ax + By) cos - (Ay + Bx) sin - AC cos 20 + EG sin 20 = 0,
which can be obtained by equating to zero the discriminant of a quartic 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. 109112.]
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
(x, y, z), (x', y', z'), (X, Y, Z), (X', T, 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, U' = 0) al , ->/, z' proportional to d x U, d y U, d z U, and x, y, z proportional to
d-d U', d^ U', d? U' ; or, what is the same thing, we must have
x'dx + y'dy + z'dz = and xdx' + ydy + zdz = ;
one of these is implied in the other, as appears at once by differentiating the equation
xx' + yy +zz' -1 = 0.
N
112 NOTE ON THE THEORY OF AP8IDAL SURFACES. [735
The other two surfaces will therefore be reciprocal if only we have the like
relations between the coordinates (X, Y, Z) and (X', Y', Z'); that is, if
XX' + YY' +ZZ' -1=0,
X'dX+ Y'dY + Z'dZ = 0,
XdX'+
To find the apsidal surface, we consider an arbitrary section x cos a + y cos /9 4- z cos 7 =
of the surface U = 0, and seek to determine the apsidal radii thereof, that is, the
maximum or minimum values of R* = a? + y 1 + z* when x, y, z vary subject to these
two conditions. Writing x', y', z' to denote functions proportional to d x U, d y U, d z U.
we thus have the set of equations
x + \x 4- p cos o=0,
y+\y' + /*cos/9=0,
z + \z' + fj. 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 If = X* + Y 3 + Z-,
X Y Z
and cos a, cos /9, cos 7 = -p , -5 , -5 ; substituting these values in the equation
/ i / * it
a; cos a + y cos /9 4- z cos 7 = 0,
we have Xx + Yy + Zz Q, and substituting in the other equations, and instead of
X, /j, introducing the new indeterminate coefficients p, <r, we obtain
X, Y, Z = px + ax', py + ay, pz + az'.
Hence these last equations, together with
-R 2 = X 2 +
and
Xx+Yy + Zz = l,
contain the solution of the problem. If for convenience we introduce R' 2 to denote
x' 2 + y' 3 + z'", and imagine the absolute values of x', y , z 1 determined so that xx + yy' + zz' = 1,
then substituting for X, Y, Z their values in the equations X 1 + Y" + Z 2 = R- and
Xx+Yy + Zz= 1, we find
1p<
and thence
or, finally assuming
p ~
we have
X, Y, Z = x- R>x', y - R*y', z -
each divided by
- 1),
735] NOTE ON THE THEORY OF APSIDAL SURFACES. 113
where I recall that x', y', z' are proportional to d x ll, d y U, d z U, and are such that
xx' + yy' + zz' = 1 : they in fact denote
d x ll, d y U, d 2 U, each divided by xd x U + yd y ll + zd,U ;
and that R 2 and It'- denote a? + y- + z* and <c' 2 + y'- + z'- respectively. The coordinates
X, T, 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
X', T, Z' = x'-R'*x, y'-R*y, z'-R-z,
each divided by
"> - 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-, and ^(R 2 R 2 1) = V(pp' 1), = o>,
then
X'dX +Y'dY+ Z'dZ
= x'-xp' x--*p
CO CO
' xp (dx pdx' x'dp (x x'p) dot
- - &c.
ft) ft)
= { x'dx + y'dy + z'dz
- p (x'dx' + y'dy' + z'dz')
p (xdx + ydy + zdz )
+ pp' (xdx + ydy' + zdz' )
+ p'(xx' +yy' + zz' )dp]
- p (x" 1 + y'- + z'' 2 )
+ pp (xx +yy' + zz' )},
or, since the terms in { } are
- p .%dp - p'dp - p . %dp + + p'dp, = -\ (pdp + p'dp),
and
1 - pp' pp + pp', = 1 pp', = co",
this is
= ,{-% (pdp' + p'dp) +coda>], =0,
in virtue of co- = pp' 1. And similarly the other equation XdX' + Yd Y' + ZdZ' =
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. 179185.]
I CONSIDER only the most simple case, that of a quadric equation a? = n t , where
n 2 is a given imaginary quantity, having the square roots n, and n; starting from
an assumed approximate (imaginary) value x = a, we have (a + hy = n 2 , that is,
[2 yfi QJl J_ ^2
a 2 + 2a& = ri*, h = _ , and a + h = = ;
'-it "
that is, the successive values are
_ a 2 + n- _ U! 3 + w 2
~2a~' ~2a7'-'
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 a t is to n, means mod. (a n) < mod. (Oj n), and so in
other cases; in the course of the approximation a, Oi, a^, ... to n, any step, for
instance a to a,, is regular if a : 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-FOUBIER METHOD.
115
if mod. (a n) < mod. n, viz. this condition is a sufficient one*; the first step a to a^
will moreover be regular under a less stringent condition imposed upon a ; and it would
seem that, without the condition mod. (a ?i)<f mod.?i 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, A lt &c.
to represent the quantities a, a,,...; we have then
AN = mod. (a n), A 1 N=mod.(a l n)...,
so that the approximation is measured by the approach of the points A, A l to N.
The Hue NN' joining the points N, N' passes through, and is bisected at, the origin
; drawing then QQ 1 through at right angles to NN' the condition
mod. (a re) < mod. (a + n)
means that the point A, which represents the imaginary quantity a, lies on the
.ZV-side of QQf, 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 7 , 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-n) <<jmod. n ; then if Oj= -, , 0,,= 1 h , &c., show that the terms a, a lt a. 2 ,... will converge
-" !-</]
to the limit n." This is strictly true, but it would have been better to say "will converge regularly."
152
116 APPLICATION OF THE NEWTON-FOURIER METHOD [736
unfitness, AjN 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; and, further, that when A is
outside the segment of unfitness, then the next point A t , 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 t N is less than AN, but that it does not thence follow
that A 3 N is less than A^N, A 3 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 .4 lies within the circle centre N and radius NM, and
is consequently outside the segment of unfitness ; AjN being less than AN, the point
A! is a fortiori outside the segment of unfitness, and the like for all the subsequent
points A.;, A,,..., that is, in this case, the approximation is regular throughout. The
circle, centre N, and radius NM, = 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, A lt A 3 , ... (or, as it will be convenient to call them,
A lt AI, ...) may be obtained each from the preceding one by a simple geometrical
construction.
X
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 .AT-side of Qty, 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 .4,, for which this ratio A t N : A t N' has a
given value, suppose A 1 N = kA ] N', we describe a first circle (passing of course through
A t ) for each point of which this ratio has the value k; let the diameter of this
circle be FjTPi, V l being the extremity between and N, W l (not shown in the
figure), that beyond N '; we then describe a second circle, for which the ratio is
= &"; let its diameter be F 2 T7 3 , F a being the extremity between and N (or say
between F, and N), TF 2 , that beyond N (or say between N and TTj); the point
AS lies on this second circle, and is determined as the single intersection of the line
F..A! with the second circle. And of course drawing a third circle, for which the
ratio is =A^, on the diameter V,W 3 , then A 3 lies on the third circle, and is the
intersection with it of the line F*4 S , 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 A lt A,, A,, ..., 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 A l be within the segment of unfitness, then also some of the
subsequent points may be within the segment of unfitness, and for each point A p ,
736] TO AN IMAGINAKY BOOT OF AN EQUATION. 117
for which this is the case, the next point A p+t is at a greater distance, so that
NA P+1 >NA P ; it is, however, clear that we always arrive at a point A q , such that
< $ON, and so soon as such a point is arrived at the approximation becomes regular.
The point A^ determined from A lt as above, is a point such that the subtended
angle NA,N' is = twice the subtended angle NA^'; or calling the latter angle <f>,
the former is = 2<. It is, in fact, this property which gives rise to the construction ;
for let the values of A^N, A r N', regarded as imaginary quantities, be called for a
moment
p l (cos 61 + i sin 0j), // (cos 0/ + i sin #/) ;
and, similarly, those of A*N, AJf' be called
p (cos 2 + i sin 2 ), p 2 ' (cos #/ + i sin #,') ;
then these are the values of o^ n, c^ + n, a, n, a., + n respectively, or we have
<^ n = Pl_ { cos (^ _ ^') + i s i n (^ _ #/)) = k (cos <f> + i sin 0),
Ctj T" 71 pi
= ^ {cos (0., - 0,') + i sin (0, - 0,')} = 2 (cos 20 + { sin 20),
Gt-2 T fl
that is,
a 2 - n
which relation between a 2 , Oj is in fact the original relation
a? + n 2
Oj = - 2^ ;
and, conversely, Oj, a being thus connected, then the representative A z is obtained
from the representative point A 1 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 a; 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, d> which are such that
_
x + ly + 1
X it/ 1
?
x ly + 1
equations which may also be written
- i) 2 = e-** [a? + (y +
or, what is the same thing,
X- + f - 1 - ly cot </> = 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 0,
taken in the second form, shows that <j> is the angle subtended at the point by N, N'.
It is sometimes convenient to write /re**, ke~* l +=p, q respectively; we then have
1 + l
,
.
Suppose for a moment that we have (p l , q^), (p 3 , q 3 ), (p 3 , g,) 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-J,' 1-q'
pr, what is the same thing, it is
l-pq+p-q, l-pq-p + q, l+ pq -p- q ,
pq-l, p-q, p + q-2,
or, again
viz. the condition is
N p,q 3 -I, p 3 - q 3 , p 3 + q 3 -'<
Suppose the (k, </>) coordinates of the three points are (I, a), (m, /9), (n, 7) respectively ;
then this equation is
Z 2 1, I sin a, I cos a 1 =0,
m 2 1, ?/tsin/9, mcos/9 1
n s 1, n sin 7, n cos 7 - 1
I s 1, I sin a, 1 =0,
m- 1, m sin /3, 1
n 2 1, ?i sin 7, 1
viz. it is
Z 2 1, I sin a, I cos a
m? I, msin/3, mcos/3
w* 1, n sin 7, n cos 7
or, what is the same thing, it is
[(f - 1) mn sin (0 - 7) + (m 2 - 1) nl sin (7 - a) + (n 1 - 1) Im sin (a - /3)]
+ [(?>t 2 - n") I sin a + (n 2 - P) m sin ^ + (P - m 2 ) n sin 7] = 0.
If in this equation 7 is put = IT, and ft = 2a, so that sin (a - /3) = - sin a, the equation
will contain only terms in sin a, and sin 2a, viz. it will be
that is,
[ (m 1 -n^l + ^-lJnl- (n 2 - 1) Im] sin a
+ [- (P - l)mn + m (n 2 - 1-) ] sin 2a = 0,
I (m - l)(n + 1) (m - n) sin a + m (m + 1) (w - P) sin 2a = 0,
736] TO AN IMAGINARY BOOT OF AN EQUATION. 119
or, what is the same thing,
(m + 1) sin a. {I (n + 1) (m - n) + 2m (n - P) 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 = I 1 or, writing
k instead of I, say if I = k, m = n = k a ; that is, writing also <f> in place of a, the
three points
(k, f), (If, 2<f>) and (*, IT)
are in a right line; viz. the point A lt circle k, subtended angle </>; the point A 2 ,
circle k 1 , subtended angle 2<; and the point V 3 , same circle, subtended angle TT;
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 p it q l refer to the point A l : then we have
(A.N)* = (x, - I) 2 + y, = (, + ty, -!)(*,- iy, - 1), =
that is,
, , .
PI L VI
Similarly, if p. 2 , q 2 refer to the point A^, then
since jj 2 , q i =pi i , qf. The two are equal if
that is,
Writing for a moment x l + iy, = , ^ iyj = ?;, we have
"
and the equation is
that is,
or substituting for f, T; their values, the equation is
that is,
120 APPLICATION OF THE NEWTON-FOUK1ER METHOD [736
the equation of a circle on the diameter N'M, which is, in fact, the before-mentioned
circle of unfitness; viz. .A, being on the circumference of this circle, or say on the
boundary of the segment of unfitness, then A 1 N=A t N; whence also, according as
AI is inside or outside the segment, A^N<AtN or
Suppose A l to be on the circle, that is, p l + q t + 1 = ; it is easy to show that
the locus of A is also a circle. We have in fact (p, + q,)" 1 = 0, that is,
or say
1^1+^1 + 2^-1 = 0,
viz. this is
>
that is,
or finally
3 2A- 2
Measuring off from in the direction of ON, a distance OS= ,, (always >^,
since k*<I), 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
A! on the line QQ 1 ) 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
N-side of Qty, that is, .4, being on the boundary of the segment of unfitness A is
never on this boundary ; and it thus appears that A l being inside the segment, A., is
always outside the segment.
It is to be further noticed, that we have
or
_ _
that is,
_
where T is the tangential distance of A l from the circle of unfitness; there should,
it appears to me, be some more elegant formula for the ratio A^N-r-AJf 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 n) < f mod. n,
present themselves in the real theory. Making the usual construction by means of
the parabola y = a?, the first condition means that the point A must be taken on
the JV-side of (fig. 2); the second that, in order to the regularity of the approxi-
Fig. 2.
matiou, A must be taken at a distance from >$ON; in fact, if (as in the figure)
OA = \ON, then AN = NA lt or the point A^ is at an equal distance with A from
N; and thence, according as OA is greater or less than $ON, the point A^ is
nearer or further than A to or from N.
C. XI.
16
122
[737
737.
N
ON A COVARIANT FORMULA.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XVI. (1879),
pp. 224226.]
STARTING from the equation
fa
-
which presents itself in the Newton-Fourier problem, it is easy to see that, if a be
a root of the equation fa = 0, then
(x - a)f'x-fx
Xi a, = ,,, ,
J x
contains the factor (a; -a) 2 , that is, the equation (x-x l )f'x-fo = 0, considered as an
equation in x containing the parameter ar,, will have a twofold root, if <c, is equal
to any root a of the equation fx = Q; and, consequently, the discriminant in regard
to x of the function (x-xjf'x-fx will contain the factor fa,. But if fa be of the
order n, then the discriminant is of the order 2n-2 in x lt and there is consequently
a remaining factor #, of the order n 2.
The like theorem applies to the homogeneous form
, = 1 ; or, changing
which reduces itself to the foregoing on writing a = l, /9 = 0, y
the notation, say to the form
737]
ON A COVARIANT FORMULA.
123
viz. the discriminant hereof in regard to f, 77, being a function, homogeneous of the
order 2n 2 in regard to x, y, to a, /3, and to the coefficients of f(j^, 17), will
contain the factor f(x, y), and there will be consequently a remaining factor of the
order n 2 in (x, y), 2w 2 in (a, ) and 2?i 3 in the coefficients of /(f, 17).
The most simple case is when /(, rj) is the quadric function (a, b,
The form here is
a, 6,
, 7,)' = (a, b, c
where the coefficients are
a= 2y (aa + 6/3) - a (ay - #B), = a/3a; + (aa + 26/3)y,
b = y (6a + c/3) - a; (oa + 6/3) - 6 (ay - #),
= aa# + c/3y ,
c = - 2# (6a + c/3) - c (ay - /8), = - (26a + c/3) - cay ;
and we then have
ac - b 2 = - (26a/3 + c^ 2 ) aa?
- {2a6a 2 + (2ac + 46 2 ) a/3 + 26C/3 2 } xy - (aa 2 + 26a/S) cy"
0,3? . 00? {- 2aca/9j xy c/3 2 . cy 2 ,
which is
= - (aa 2 + 26a/3 + c/3 2 ) (a# 2 + 26*y + cy-).
The discriminant is in this case
= -(a, 6, c$a, /3) s .(a, b, c$ar, y) 2 .
In the case of the cubic function (a, b, c, dQ%, ijf, the form is
= (a, b, c, dj[f,
t>) 2 ,
-(ay-/3^)(a, b, c,
the values of the coefficients being
a=
b = aaa; + ( ba + 2c/3) y,
c = -(26a+ c/3)+ d/3 y,
d = - (3ca + 2dy3) a; - da y.
Attending only to the terms in of, we have
ac - b 2 = - (aa z + 26ay3 + c/3 2 ) aa?,
ad - be = - 2 (6a 2 + 2ca + d/3 2 ) oaf,
bd-c 2 =
162
124
ON A COVARIANT FORMULA.
[737
x
And hence, in
ad' + 4ac + 4b*d - 3b J c J - 6abcd, = (ad - be)" - 4 (ac - b a ) (bd - c'),
we have the term
4euc . x [a (bo* + 2coy3 + d/3 2 ) 2 + (aa 3 + 26a/8 + c/9 5 ) {(Sac - 46=) a 2 + (2ad - 46c) a - c'/S 2 }] ;
then, forming the analogous term in y 4 , and assuming that the whole divides by
(a, b, c, dQx, yY, and also expanding the a^- functions within the square brackets, we
find
Discriminant = 4 (a, b, c, dQx, y) s multiplied by
3a s c -Sab 1
Za'd + 6abc - Sb 3 6abd - 6b-c
6a&d+6ac" -
Qacd 6bc"
ad* c*
Writing down the Hessian of (a, b, c, d$ct, $) 8 ,
H = (ac b", ad -be, bd c^a, /9) 2 ,
and the cubicovariant
a 2 d - Sabc + '.
abd - 2oc 2
- acd + 2b-d - b<?
it is easy to see that the coefficient of x is
= 3 (a, 6, c$a, @y.(H
hence also that of y is
= 3(6, c, d$a, /3) 2 . (H + <!>),
and the final result is that the discriminant = 4 (a, b, c, dj[x, y) s multiplied by
{3 (a, b, c, d*$a, &Y(x, y) H + (ay - fa) *}.
It would be interesting to calculate the result for the quartic (a, 6, c, d,
March 14, 1879.
738]
125
738.
NOTE ON A HYPERGEOMETRIC SERIES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879),
pp. 268270.]
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
*y \-l* dy -&
da? x.lxdx x.lx y
which belongs to the hypergeometric series F(%, ^5, f, ), is algebraically integrable,
having in fact the two particular integrals
y = ^(a - a 5 **) + V(- 5 + #*),
where a is a prime sixth root of 1, a 6 +l = 0, or say a 4 a 2 +l=0 (see p. 326,
a being for greater simplicity written instead of 8'-, and the form being somewhat
simplified).
It is interesting to verify this directly ; writing first y = ^(Y) and then x = X s ,
the equation between Y, X is easily found to be
fj dY /dr
l-X* dX f \dX
and the theorem in effect is that that equation has the two particular integrals
Y= V(P) V(Q),
P and Q being linear functions of X : in fact,
P= a -ofX,
Q = - a 5 + a X.
126 NOTE ON A HYPERGEOMETRIC SERIES. [738
Starting say from the equation
r= V(P)+V(Q).
or, as it is convenient to write it,
F = P* + Q^,
where P and Q are assumed to be linear functions of X, we have
dY _
dX~
and thence
where P', Q' are written to denote the derived functions of P, Q respectively.
Substituting these yalues, the resulting equation contains on the left-hand side
a rational part, and a part with the factor P~$Q~*, and it is clear the equation
can only be true if these two parts are separately = 0. We have thus two equations
which ought to be verified ; viz. after a slight reduction these are found to be
9X- Y
j (P' + Q') - (P + Q) = 0,
+ <?P' 2 + PQP'Q' + 1 3 PQ (PQ' + P'Q) - i- 3 P 1 = 0,
and it is very interesting to observe the manner in which these equations are, in
fact, verified by the foregoing values of P, Q.
We have
and hence
or, in the first equation, the second part
738] NOTE OX A HYPERGEOMETRIC SERIES. 127
viz. this is
We have
QP' 2 + PQ' 2 = a 10 (- a 5 + aX) + a- (a - a*X),
= a. 3 - a 15 - (a 7 - a 11 ) X, = (a - a 5 ) X ;
and
PQ = - of + (a 2 + a 10 ) X - a e X-, = l+X+X 3 ;
hence
* i pcfi\- ( a - a ) z
* '
and the sum of the two parts is = 0.
Similarly as regards the second equation, the second part
o y
'
s
Here PQ' + P'Q is a (a - a'Z) - a 5 (- a 5 + aX), which is =1+2Z; and PQ being
= 1 + X + X-, the term in { j is
hence, outside the { } writing for PQ its value = 1 + X + J? 2 , the term is
which is the value of the second part in question,; the first part is
(PQ' + QPJ - PQP'Q', = (1 + 2X Y- - (1 + X + X*), = 3X (1 + X) ;
and the sum of the two terms is thus = 0.
128
[739
739.
NOTE ON THE OCTAHEDRON FUNCTION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XVI. (1879),
pp. 280, 281.]
A SEXTIC function
\
such that its fourth derivative
(U,
= (a, b, c, d, e,f, #$>, y)*,
+ (ag - dee +
+ 2 (bg - 3c/ + 2de) xy 3
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 = ; and then assuming, as is allowable,
b = -/=!,
we have his canonical form xy (x* y*) of the octahedron function.
But the equations may be satisfied in a different manner; viz. the first and last
equations give
739] NOTE ON THE OCTAHEDRON FUNCTION. 129
and, substituting these in the remaining equations, they become
-j(-9ce + 8d 2 ) = 0, -9ce + 8d- = Q, ^ (- 9ce + 8d 2 ) = 0,
4a 4
all satisfied if only 9ce + 8rf 2 =0. Assuming b=f=2, the values are
b, c, d, e,/=2, 2V(2), 3, 2V(2), 2,
and the form is
3
= a*/ (a? + r xy + f {of + ^(2) a;y + y 2 },
} ( x + 7Sy y ) {x+ y V(2)1 (* + vfsi
This is, in fact, a linear transformation of the foregoing form XY(X i T i ); for
writing
we have
X> = a? + (1 + 1) V(2) ay + if,
and therefore
or finally
and the two forms are thus identical.
C. XI. 17
130 [740
740.
ON CERTAIN ALGEBRAICAL IDENTITIES.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XVI. (1879),
pp. 281, 282.]
IF P , PI, P 3 are points on a circle, say the circle a? + y a = l, then it is possible
to find functions of (P , Pj) and of (P lt P 2 ) respectively, which are really independent
of P,, and consequently x functions of only P and P: 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 (P , P,) and of (P 1; P 3 ) being
#<#!-##, and XtX, + y^, x^-x^, 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 M, y = sin u ; and the identity written down is obtained by expressing the
cosine of M 2 u , = (tt a M,) + (w, u,), in terms of the cosines and sines of ,, ,
and w, M O .
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 /(M), is meant a function such that f(u + v) is
an algebraical function of /(M), f(v) ; the sine and cosine are each of them an additive
function, because
sin (u + v) = sin u V(l sin 2 v) + sin v V(l sin 2 u),
and, similarly, for the cosine. But it is convenient to consider pairs or groups f(u),
<f>(u),..., where f(u + v), <f>(u + v),... are each of them an algebraical (rational) function
of f(u), $ (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
P 0> PJ, P 2 , points on the curve, and (#, y , z a ), (a;,, y lt z^, (x^, y^, z?), the coordinates of
these points respectively, we have in the same way, from w 2 w = (u 2 Wj) + (MI u ),
three equations, of which the first is
1 - texfx? (1 - te V) 2 (1 -
The form of the right-hand side is
A + Bx^z,
C + Dx^y^ '
where A, B, C, D are each of them rational as regards &?; and it is easy to see
that the equation can only subsist under the condition that we have separately
xiyoZo x y^z ^L_B_
"i-yfVar,* ~~C~ D'
implying of course the identity AD BG=0. The values of B and D are found
without difficulty; we, in fact, have
B =
D = 2
so that, comparing the left-hand side with B + D, we have the identity
W. 1 - **W*f = (*? ~ *o 2 ) (1 - texfxf),
which is right. The comparison with A -=- G would be somewhat more difficult to effect.
172
132 [741
741.
ON A THEOKEM OF ABEL'S RELATING TO A QUINTIC
EQUATION.
[From the Proceedings of the Canibridge Philosophical Society, vol. in. (1880),
pp. 155159.]
THE theorem in question is given, (Euvres Completes, [Christiania, 1881], t. n.,
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 :
aa + A
ou
a = m + n
a, = m - w V(l + e 2 ) + >J[h (1 + e 2 - V(l + e 2 ))],
a, = m + w V(l + e-) - V|> (1 + e 2 + V(l + e 2 ))],
o 3 = m - n V(l + e 2 ) - >J\h (1+ e 2 - V(l + e 2 ))],
A- + K'a + K"a, + K'"aa,, A 1 = K + K'a 1 + K"a 3 + K'"a
"a 4- K'"aa, , A 3 = K + K'a, + K"^ +
Les quantit^s c, h, e, m, n, K, K', K" , K'" sont des nombres rationnels. Mais de
cette maniere liquation a? + ax + b = n'est pas resoluble tant que a et b sont des
quantites quelconques. J'ai trouve de pareils thdoremes pour les equations du 7 i3Ue ,
It is easy to see that x is the root of a quintic equation, the coefficients of
which are rational and integral functions of a, a,, Oj, a 3 : these coefficients are not
symmetrical functions of a, a,, a,, 03, 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 <z I( (t, into a 2 , a. 3 into a 3 , a 3 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, a lt a.,, a 3 , 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 aaid.ja 3 , a^aM^a,
o-ja^aaj, a-jfl.a^a.2. 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, y functions of e, h, such that y' + y'' 1 is a rational
function of e, h, but that f fy'", yy' are rational multiples of the same quadric radical
\fff. Assume that we have
7 2 -7 /2 =2/>V0, yy'=q^0;
then
that 7 2 + 7' 2 may be rational, we must have p- + q- = \-0, or say p* + q* = li?6 ; hence,
p2 qi
6 = + * must be a sum of two squares, or, assuming one of these equal to unity
/fr fl
and the other of them equal to e 2 , say 6 = 1 + e", we satisfy the required equation by
taking p = h, q = he: viz. we thus have
7 2 - y- = 2h Vl + e 2 , 77' = he Vl~+~e 2 , 7 2 + y' 3 = 2A (1 + e 2 ) ;
and thence also
y- = h (1 + e 2 + Vl + e 2 ), 7 /2 = h (1 + e- - vT+e 2 ),
the roots of these expressions, or values of 7, 7', being such that
77' = fte Vl + e 2 .
Taking now a rational, =m suppose, and /3 a rational multiple of
Vl + e 2 , = h Vl+e 2 ,
suppose ; it is easy to see that the quartic equation which has for its roots
a, a,, M 2 , a 3 = a + /S + 7, a- + y', a + p-y, a-fil-y,
has the property in question, viz. that every rational function of the roots unchangeable
by the cyclical change a into a lt a t into a*,, a into a 3 , a s 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
a 2 , a 3 + <,, a 2 , a 3 , a + <a 2l a,, a, a,+</>a :1 , a,
134 ON A THEOREM OF ABEL'S RELATING TO A QUINTIC EQUATION. [741
and if $(o, a,, a,, a,) contains a term a m /9V7 >9 . then the other three functions will
contain respectively the terms
"'(-/9)V(-7) 9 . -(- 7V (-7>- m (-/9)"(-7') 1 '(7) ? ;
viz. the sum of the four terms is
- flC/9- [{1 + (-)+ 1} 7*7' + !(-)"*" 1 + (-)"+' 1} TV]-
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, p and q odd. Writing, for greater distinctness,
2 or 2n + 1 for n, according as n is even or odd, and similarly for p and q, the
term is, in the four cases respectively,
= 2O9* (7* 7'* +7* 7'*),
= 2O9"- 1 - 1 (7*- 7'" -7* 7'*),
1 7 /1J " H ).
The second, third, and fourth expressions contain the factors
-l"), 77' (7 s -7 s ),
respectively; and the first expression as it stands, and the other three divested of
these factors respectively are rational functions of a, /S 2 , 7", 7'", that is, they are
rational functions of m, n, e, h. But the omitted factors 0( r f-y' i ), 77' (7 s 7 /2 ),
$77', = 2nA(l + 6 2 ), 2h-e(lJ- e 3 ), 71/16(1 + 6*) are rational functions of , /<, e; hence
each of the original four expressions is a rational function of m, n, h, e; and the
entire function
<f>(a, a,, a,, a,)+^(a 1 , a,, a,, a) + </>(a,, a a , a, O 1 ) + <^(a 3 , a, a,, a.)
is a rational function of m, n, A, e.
Replacing o, ft 7, 7' by their values, the roots of the quartic equation are
m + n V(l + e>) + V[A (1 + 6 s + V(l + e 2 ))],
m - V(l + *) + V[* (1 + e 8 - V(l + e 2 ))],
m + n V(l + e 2 ) - V[A (1 + 6 s + V(l + e 2 ))],
m - n V(l + ) - V[A (1 + e> - V(l + e 2 ))].
And I stop to remark that taking in, n, e, h = -{, +{, 2, -^ respectively, the
roots are
741] ON A THEOREM OF ABEI/S RELATING TO A QUINTIC EQUATION. 135
viz. these are the imaginary fifth roots of unity, or roots r, r 3 , r 4 , r 3 of the quartic
equation a^ + as 3 + ar ! +#+l=0; which equation, as is well known, has the group
rrW, r-Wr, rVVr 2 ,
Reverting to Abel's expression for x, and writing this for a moment in the
form
x c+p
the quintic equation in x is
= (x - c) 5
s'-r)
+ (x c) . 5 (p 3 q + <fr + r 3 * + srp) + 5 (p-r* + q-s~) - 5pqrs
+ 5 (p 3 rs + q 3 sp + r'pq + s?qr)
5 (p 3 q*r + q^s + r's-p + s'-p 2 q).
If we substitute herein for p, >j, r, s their values, then, altering the order of the
terms, the final result is found to be
0=(a;-c) 5
+ (x c) 3 . 5 (AAs + AfA,) aa l a* ! a.. i
+ (x cf . 5 (A^A^a, + A^A^a.^1 + A-fA^ui^ +
+ (x c) . 5 (
+ 5 (
+ (x c) . (A^aJaJ + A^a^a^ + Aja./tfaj' + A^aa^af) aa^M^
+ A^A^A^a + A^A^ao! + A
viz. considering herein A, A lt A,, A-., as standing for their values
'"aa,, &c.
respectively, each coefficient is a function of a, a lt a 2 , a 3 , 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'".
L8
[742
742.
ON THE TRANSFORMATION OF COORDINATES.
[From the Proceedings of the Cambridge Philosophical Society, vol. in. (1880),
pp. 178184.]
THE formulas 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 )-
', ff, 7'
// rt// "
a , P , 7
see my "Memoir on the Theory of Matrices," Phil. Trans, t. CXLVIII. (1858), pp. 17
37, [152]; moreover (a, /9, 7$ar, y, z) denotes ax + fly + yz, and so
( a , /3 , 7 $a, y, z)
', ft 1 , V
denotes
and again
", ft", 7"
y"z),
(
denotes
Consequently
( a >
a',
", 7"
7
i
i"
, ', a"
O & Q't
P, p , p
7, 7', 7"
742]
ON THE TRANSFOEMATION OF COORDINATES.
137
In the case of a symmetrical matrix
( a, h, g ),
h, b, f
9' f' c
), =( a, h, g
h, b, f
g, /. o
x, y, z),
the equal expressions
(a, h, g $#, y,
h, b, f
g, f, c
are also written
(a, b, c, /, g, h~$x, y
In particular, if
then
( a, /(, g ~$x, y, z)- is written (a, b, c, f, g, h\x, y, z)-.
h, b, f
9^ /- c
Two matrices are compounded together according to the law
f, i), f), or (a, ...$ TI, t&x, y, z).
(f V, ) = (. y, z),
(a, a', a"), (& P, /3"), (y, y, 7").
a , b , c
a', b', c'
a", b", c"
5
a, ft , 7
.' G' '
Gc , D t ^j
a", ft", 7"
) = (o, b, c)
(a 1 , b', c')
(a", b", c")
>? )
)) >)
>
viz. in the compound matrix, the top-line is
(a, b, c$a, a', a"), (a, b, c$/3, /fr, /8"), (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, b, 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.
( , ft, 7 )-> = ^ ( /9'7" - "/ , /9"7 ~ #/', W ~ /9'7 ),
' , P , y
a", /8", 7"
where V is the determinant
O'/S" - a"/3', a"^ - o/3", a/8' - a'/3
a, /8, 7
a", 0", 7"
C. XI.
18
138 ON THE TRANSFORMATION OF COORDINATES.
Coming now to the question of transformation, write
x v t i .Vi *i y *
[742
a:,
a a'
a" =<c
fi iy
$" y
n
IT
7 7
7" *
1 v,
/*, *,
"i 1
x, y,
F
n,
/*! X,
1 z,
viz. the axes of x, y, z are inclined to each other at angles the cosines whereof arc
X, ft,, v: those of JT,, y,, 2, are inclined to each other at angles the cosines whereof
are X, , ^, , K, : and the cosines of the inclinations of the two sets of axes to each
other are a, ft, 7; a', ff, 7'; a", ft", 7": as is more clearly indicated in the diagram,
the top-line showing that cosine-inclinations of a; to
are
x, y, z, #j, y,, *,,
1, v, ft, a, a, a",
respectively, and the like for the other lines of the diagram. The letters ft, ft,, V,
W are used to denote matrices, viz. as appearing by the diagram, these are
(1, v, ft ), ( 1 , v lt /*, ), ( o ,
v, 1,
ft, X,
Ah,
7 ), (a, a', a" ),
', P, V
a", ft", y"
ft, ft', ft"
I //
7, 7. 7
respectively.
The coordinates (a;, y, s) and (a;,, y,, z t ) 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 j. y\> z\ respectively, we have two sets, each of three equations, which may be
written
y, z) (ftiji^ij yi> *i/j
where of course each set implies the other set.
We have
, y ,
the first giving in two forms (x, y, z) as linear functions of (#,, y,, .*,), and the
second giving in two forms (,, y,, .*,) as linear functions of (x, y, z); comparing
the two forms for each set, we have
n- w = F-> n, ,
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
(a, b, c, f, g, h) = (l X 2 , 1 fj,-, 1 v-, fj.v \, V\ IJL > \/j, v),
and
we have
-' = -p ( a, h, g ).
h, b, f
g. f, c
The first equation, written in the form
V( a, h, g ) W =
h, b, f
g> f. c
denotes the six equations
(a, b, c, f, g, h)(, /9 , 7 ) 2
K ,
K ,
(a", p", yy
(a 1 , ft',
(a". /3",
And, similarly, writing
(a,, bj, c,, f,, g,, h,) = (l-X 1 2 , l-/ij 2 , 1-;
and
then
1
fV*-J< a,, fa,
hi, b,
gi. f i
and the second equation, written in the form
W( a,, h,, g,
hi, b,, fj
gi, f,, c,
gl
182
140
ON THE TRANSFORMATION OF COORDINATES. [742
denotes the six equations
(a,, b,, c,, f lt g lt h,$, ', a")'
09, ff, PJ
(7, 7. 7")'
</9, #, "$7, 7- 7")
(7- 7. 7"$> ' ")
(a, a', a"$, fr ") = *,*.
The two seta 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 7, a', i?, y', a", 0", 7"),
and the cosines (X, /*, <) and (X,, /*,, i>,). 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 (an. y lt z t ) and adding, we have
(fl $a; , y , z Y = ( W $#!, y lt z$x , y , z),
, y , z\x lt y s , z,).
But
(TFfta;,, y,, z&x , y , z)
and
, y , z^x,, y lt z,)
denote one and the same function ; hence
(n$#, y, zY
that is,
(1, 1, 1, X, ^, vfo, y, z) i =(l, 1, I, Xj, /*,, !$ y lt ztf,
or the linear relations between (x, y, z) and (x lt y lt z t ) 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
(<fi, yi, *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
J fiT = (a, b, c, f, g, h$, ft 7 ) 2 ,
is a relation between X, /i, i, the cosines of the sides of a spherical triangle, and
(a, 0, 7) the cosines of the distances of a point P from the three vertices : it can
be at once verified by means of the relation A+B+C=^7r, and thence
1 -cos 2 .4 cos 2 B cos 2 C + 2 cos .A cos B cos 0=0,
which connects the angles A, B, C which the sides subtend at P. Writing a, b, c
for X, fji, v, and f, g, h for at, ft 7, the relation is
- c*) h?
+ 2(ca-b)hf+2(ab-c)fg,
viz. this is
1 - a 2 - b" - c 2 -/ 2 - (f - h* + 2aic + 2a$r& + 2bhf+ 2cfg
- a 3 / 3 - fcy - c 2 A 2
where (a, b, c, f, g, h) are the cosines of the sides of a spherical quadrangle ;
(a, 6, 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, h) are sides meeting in a vertex.
The equation
#1/1 = (a, b, c, f, g, h$a, ft 7) (a', ft, 7')
is a relation between \, p, v, the cosines of the sides of a spherical triangle ; a, ft 7, the
cosines of the distances of a point P from the three vertices ; a', ft, 7', the cosines of
the distances of a point Q from the three vertices; and v lt the cosine of the distance
PQ.
Drawing a figure, it is at once seen that
i/! = ao' + Vl - a j Vl a' ! cos (d - ff),
where
cos 6 = --- , ,
and therefore
vv
S1H0 =-- ;
also
cos 6' = jt===-
and therefore
Vl - '' Vl - ^
142 ON THE TRANSFORMATION OF COORDINATES. [742
the values of V, V being
V = l-a-/8 s -v* + 2a/3i/,
V = 1 - a' 1 - ff 1 - 1/ 5 + 2a'y9V ;
the resulting value of i>, is therefore
/
The equations
Jf = (a, b, c, f, g, h$a, /8, y)\
give
and we therefore have
(ga + fft + c 7 $ga' + f/3' + c 7 ') = K V V V ' ;
recollecting that 1 v* = c, the formula thus is
t - aj/5/3' - a'v) + g (ga + fft + c 7 ga' + f/3' + OyO| ,
or say,
Kv l = Kaa.' + -{K(ft- avQft' - &'v) + (ga + f/3$ga' + f/3')} + g (a-/ + a' 7 ) + f (/3 7 ' + Py) + c 77 ' ,
c
The sum of the first and second terms is readily found to be
and the equation thus becomes
Ki>i (a, b, c, f, g, h][a, ft, 7 ^a', ft', 7 '),
as it should do.
743] 143
743.
ON THE NEWTON-FOURIEK 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'(u) = 0, consists in deriving from an assumed approximate root f a new value
fft\
, = fTTf. , 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 f, to be each of them real, Fourier investigated the conditions under which
1 is in fact a closer approximation. But the question may be looked at in a more
general manner: f may be any real or imaginary value, and we have to inquire in
what cases the series of derived values
fc,*_/<0 f-f /(&
/'()' /'<.)"
converge to a root, real or imaginary, of the equation f(u) = 0. Representing as usual
the imaginary value f, =x + iy, by means of the point whose coordinates are x, y,
and in like manner , =#, + iy lt &c., then we have a problem relating to an infinite
plane; the roots of the equation are represented by points A, B, C,...; the value
f is represented by an arbitrary point P; and from this by a determinate geometrical
construction we obtain the point P lt and thence in like manner the points P t , P 3) ...
which represent the values ,, f 2 , ,,... respectively. And the problem is to divide
the plane into regions, such that, starting with a point P, anywhere in one region,
we arrive ultimately at the root A ; anywhere in another region we arrive ultimately
at the root B ; 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 M O n - II (m) UP TO m = n = 2Q.
[From the Transactions of the Cambridge Philosophical Society, vol. xni. Part I. (1881),
pp. 14. Read October 27, 1879.]
THE differences of the powers of zero, A CT 0", 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 A 10 1C) is
given in Herschel's Examples (Camb. 1820), and is reproduced in the treatise on
Finite Differences (1843) in the Encyclopaedia Metropolitana. But, as is known, the
successive differences AO", A'O", A'O", ... are divisible by 1, 1.2, 1.2.3,... and
generally A m O n is divisible by 1.2.3...m, =II(m); these quotients are much smaller
numbers, and it is therefore desirable to tabulate them rather than the undivided
differences A'"0 n : moreover, it is easier to calculate them. A table of the quotients
A0 n -T- II (m), up to m = n=12 is in fact given by Gnmert, Crelle, t. xxv. (1843),
p. 279, but without any explanation in the heading of the meaning of the tabulated
numbers C^, = A"0* -s- II (n), and without using for their determination the convenient
formula C n *+' = nC n k + C^f given by Bjorling in a paper, Crelle, t. xxvm. (1844),
p. 284. The formula in question, say
is given in the second edition (by Moulton) of Boole's Calculus of Finite Differences,
(London, 1872), p. 28, under the form
A0 n
m
It occurred to me that it would be desirable to extend the table of the quotients
-T- II (m), up to m = n = 20. The calculation is effected very readily by means
744]
TABLE OF A'"0"-f-n(m) UP TO w = n =
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, ... 2B + A,
C, ... 3(7+5,
D, ... W+G,
E, ... 5E + 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 + 6E.
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 A">0" -=- II (m).
<
d
J3
O 1
0"
s
O 5
O 7
O 8
Q10
O 11
O 1 ' 2
O 13
0"
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
3
7
15
31
63
127
255
511
1 023
2047
4095
8 191
3
1
6
25
90
301
966
3025
9330
28501
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
2646
22827
179 487
1 323 652
9 321 312
63 436 373
7
1
28
462
5880
63987
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
2431
66066
12
1
78
3 367
13
1
91
14
1
15
16
17
18
19
20
C. XI.
19
146
TABLE OF A m O" + II
UP TO m = n = 20.
[744
<
1
0*
1?
<)'"
1
1
1
1
1
1
1
1
2
16383
32767
65535
131 071
262 143
524 287
2
3
2 375 101
7 141 686
21 457 825
64 439 010
193 448 101
580 606 446
3
4
42 355 950
171 798 901
694 337 290
2 798 806 985 1 11 259 666 950
45 232 115 901
4
5
210 766 920
1 096 190 550
5 652 751 651
28 958 095 545
147 589 284 710
749 206 090 500
5
6
420 693 273
2 734 926 558
17 505 749 898
110687 251 039
693 081 601 779
4 306 078 895 384
6
7
408 741 333
3 281 882 604
25 708 104 786
197 462 483 400
1 492 924 634 839
11 143554045 652
7
8
216 627 840
2 141 764 053
20 415 995 028
189 036 065 010
1 709 751 003 480
15 170 932 662 679
8
9
67 128 490
820 784 250
9 528 822 303
106 175 395 755
1 144 614 626 805
12 Oil 282 644 725
9
10
12 662 650
193 754 990
2 758 334 150
37 112 163 803
477 297 033 785
5 917 584 964 655 10
11
1 479 478
28 936 908
512 060 978
8 391 004 908
129 413 217 791
1 900 842 429 486
11
u
106 470
2 757 118
62 022 324
1 256 328 866
23 466 951 300
411 016 633 391
12
13
4550
165 620
4 910 178
125 854 638
2 892 439 160
61 068 660 380 13
14
105
6020
249 900
8 408 778
243 577 530
6 302 524 580
14
15
1
120
7 820
367 200
13 916 778
452 329 200
15
16
1
136
9 996
527 136
22 350 954
16
17
1
153
12 597
741 285
17
18
1
171
15 675
18
19
1
190
19
20
1
20
Writing down the sloping lines as columns thus:
123-4 5 6
(0) (2) (4) (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
2646
6951
7 770
3025
1
45
750
5880
22 827
42525
34 105
1
65
1 155
11 880
63 987
179 487
246 730
1
66
1 705
22275
159 027
627 396
1 323 652
1
78
2 431
39325
359 502
1 899 612
5 715 424
1
91
3367
66066
752 752
5 135 130
20 912320
1
105
4550
106 470
1 479 478
12 662 650
67 128 490
1
120
6020
165 620
2 757 118
28 936 908
193 754 990
1
136
7 820
249 900
4 910 178
62 022 324
512060978
1
153
9996
367 200
8 408 778
125 854 638
1 256 328 866
1
171
12597
527 136
13 916 778
243 577 530
2 892 439 160
1
190
15675
741 285
22 350 954
452 329 200
6 302 524 580
20
19
18
17
16
15
14
13 etc.
744]
TABLE OF A"*0" -H II (m) UP TO m = n = 20.
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 infrti).
d
1
2
3
4
5
6
7
3
1
1
1
1
1
1
1
1
2
6
14
30
62
126
2
1
12
61
240
841
2 772
3
10
124
890
5060
25410
4
3
131
1 830
16990
127 953
5
70
2 226
35 216
401 436
6
15
1 600
47062
836 976
7
630
40796
1 196 532
8
105
21 225
1 182 195
9
10930
795 718
10
945
349 020
11
90090
12
10395
We have, by means of this Table, the general expressions of A r Q r , A^O 1 ", A r ~ 2 r ,
up to & r ~ > r , viz. the formulas are
A r (K -=- n (r) = 1,
A-'O'- n (r- 1) r ~ 2
1 + 2
~ 2 )'
1
&c., &c.,
where the numerical coefficients are the numbers in the successive columns of the
(V 7TL\
, is written to denote the binomial coefficient
rC /
IT* vfL I
"
i ns >tance, r=10, we have
A0 10 + 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 A 1 ^ 20 -=- II (14), = 6 302 524 580 as above. The
Table shows that we have A r ~'"0 r -=- II (i 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 r = 1, A 2 r -=-1.2 = 2 1 "" 1 1, &c., are not algebraical functions of r.
22 October, 1879.
192
148
[745
745.
ON THE SCHWARZIAN DERIVATIVE, AND THE POLYHEDRAL
FUNCTIONS.
[From the Transactions of the Cambridge Philosophical Society, vol. xm. Part i. (1881),
pp. 568. Read March 8, 1880.]
THE quotient s of any two solutions of a linear partial differential equation of
//"-'// (lit
the SeCOnd Order ~ -I- 4- = is Hpfprmmpfl Vw a Hiflfprpntvial Amin.t.irm nf t.Vm
third order
+ qy-, ls determined by a differential equation of the
ds /d 3 *
da? , / do?
dx
where the function on the left-hand is what I call the Schwarzian Derivative; or
say this derivative is
'" /e"\ 2
* I s \
l. J. -7-H 7 J
where the accents denote differentiations in regard to the second variable x of the
symbol.
Writing in general (a, b, c .'.^X, Y, Zf to denote a quadric function
(a, b, c, ^(a-b-c), (_ a + b-c), \ (- a - b + c)$X, Y, Zf,
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
{, a] = (a, b, c .-.)
x a ' x b' x 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.
Kummer, Ueber die hypergeometrische Reihe l+^-x+... Crelle, t. xv. (1836),
pp. 3983 and 127172.
Schwarz, Ueber einige Abbildungsaufgaben. Crelle-Borchardt, t. LXX. (1869), pp. 105 120.
Ueber diejenigen Falle in welchen die Gaitssische hypergeometrische Reihe
eine algebraische Function ihres vierten Elementes darstellt. Do. t. LXXV. (1873),
pp. 292335.
Cayley, Notes on Polyhedra. Quart. Math. Jour. t. vn. (1866), pp. 304316; [375].
- On the Regular Solids. Do. t. xv. (1878), pp. 127131; [679].
Fuchs, Ueber diejenigen Differentialgleichungen zweiter Ordnung welche algebraische
Integralen besitzen, und eine Anwendung der Invariantentheorie. Crelle-Borchardt,
t. LXXXI. (1875), pp. 97 142.
Klein, Ueber binare Formeri mit linearen Transformationen in sich selbst. Math. Ann.
t. ix. (1875), pp. 183209.
Brioschi, Extrait d'une lettre a M. Klein. Math. Ann. t. xi. (1877), pp. Ill 114.
Klein, Ueber lineare Differentialgleichungen. Math. Ann. t. xi. (1877), pp. 115 118.
Brioschi, La the'orie des formes dans I'mtdgration des Equations diffe'rentielles lineaires
du second ordre. Math. Ann. t. xi. (1877), pp. 401 411.
Gordan, Ueber endliche Gruppen linearer Transformationen einer Veranderlichen.
Math. Ann. t. xn. (1877), pp. 2346.
Binare Formen mit verschwindenden Covarianten. Math. Ann. t. xii. (1877),
pp. 147166.
[* Schwarz, Ge. Werke, t. n. , p. 351, remarks that the Derivative occurs implicitly in a memoir by
Lagrange, " Sur la construction des cartes ge'ographiques," (1779), (Euvres, t. iv., p. 651.]
150 ON THE 8CHWARZIAN DERIVATIVE [745
Klein, Ueber lineare Differentialgleichungen. Math. Ann. t. xn. (1877), pp. 167179.
Weitere Unterauchungen tiber das Icosaeder. Math. Ann. t. xn. (1877),
pp. 503 560.
Cayley, On the Correspondence of Homographies and Rotations. Math. Ann. t. XV.
(1879), pp. 238240; [660].
On the finite Groups of linear transformations of a Variable. Maih. Ann.
t xvi. (1880), pp. 260263, and pp. 439440 ; [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
r > , v, N/ 1 l l V
Is, x\ = (a, b, c .'.) , r , ,
' \x a x b x cj
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
, (&*{ i J l l ! V
{x, z +( j- ( a > DI c .-.) --r- "
\az] v '\x-a x b x cl
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 = RF - R'P = Pq - FQ.
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
, / 1 1 1 V
{s, x = (a, b, c .-.) - - , , , ,
\x-a x b x c)
although the other form in {x, z] may equally well be regarded as the fundamental
form.
We consider in the theory:
A. The Derivative {s, x}, meaning as above explained.
1
B. Quadric functions of any three or more inverts
x I '
C. Rational and integral functions P, Q, R having a sum =0, and which are
such that QR-Q'R, =RF-RP, =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.
s" d f. ds\ dp
t T) I \r\rt 1 rrmrk Jo /m *
J.. A! jj . ,
2. The derivative {s, x} may be transformed in regard to either or both of the
variablea
Suppose, first, that s is a function of the new variable S, (hence also 8 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
whence, differentiating the logarithms,
s"
152 ON THE SCHWARZIAN DERIVATIVE [745
Again differentiating, we have
'"'
Bat
and consequently
that is,
the required formula.
In a very similar manner, taking x a function of X, it is shown that
3. If in this formula we write S for s, and substitute the resulting value of
[S, x] in the former formula, we have
dS*
which is the formula for the change of both variables. It, in fact, includes the other
two: viz. writing X = x, or S = s, and observing that {*, } = = {x, x}, we have the
other two formulae.
4. By putting in the first formula X = s, we obtain
/<b
M (
a formula for the interchange of the variables.
5. Writing S= -
cs
in regard to *, we have
5. Writing S= - ,, and using for a moment the accents to denote differentiation
cs T a
y_od-6c g"_ -2c
(cs + d)'' ^"
and thence
' 2C 3
' ~(c+d)"
(cs
Consequently {S, ) = 0, whence also {, /S} = 0.
745] AND THE POLYHEDRAL FUNCTIONS.
Hence in the first formula {S, x} = {s, x}, that is,
(as + b
153
viz. we may, in the derivative {s, x}, write for s any homographic function (as + b) + (cs + d)
of s.
6. Again, if X = - s , then from the second formula
<
that is,
s. x} -
-
ax + fi
S
and here, changing s into (as + b) + (cs + d), we have finally
(as + b ouc + P} _ (yx + S) 4 ,
\cs+d' yx + s}~(*S-/3yr lS
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 {S, x} = {s, x} is a differential
equation of the third order in S, and by what precedes, its general integral is S = -^.
S" s" 2cs'
The direct process is as follows : we have a first integral -- = -- '- j ; a second
o S CS "T" CL
integral log S' = log s 2 log (cs + d) + const., that is, S' = r r, 2 ; and thence a final
\CS T d)"
integral S = B -- . , which is equivalent to the foregoing value of S.
CS ~r Clt
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-a' x-0'
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
, product terms ~ , and linear terms , where the sum of the
(x a.y x a.x p x a.
coefficients A is =0. Any product term ; -= is expressible in the form of a
h 1 h 1
difference
- ^ ----
o /8 x a a- & x
C. XL
of two linear terms, and (the coefficients of these
20
154 ON THE 8CHWARZIAN 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
A B
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
x a
reduced form : this will be the case if a = 0, that is, if the original form contains no
term in , and if also .4=0. An invert thus disappearing from the reduced
(x-af
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 (a; a)* (as )* (a; 7)* ... of the 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 decree. But this numerator will for any non-essential invert contain the
a; 7
factor (x yY, 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 B C
11. It is to be remarked that the linear terms -- -+ - ^H ---- h .. , where
x o. x p x <y
A+B+C+...=0, can be (and that in a variety of ways) expressed as a sum of
differences ---- _ , that is, as a sum of product-terms Hence the
x o. x-fj tcd.te ft
quadric function can be (and that in a variety of ways) expressed as a homogeneous
function (a, ...0- -3,...) ; we must have in the form all the essential inverts,
\ Aa; 3 x p I
and we need have these only. Supposing that this is so, and that the number of
the essential inverts is =n, then the number of constants is = w(w + l), 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
0-7 , 7~ - .
'
745] AND THE POLYHEDKAL FUNCTIONS. 155
In particular, if the number of essential inverts is = 3, then the quadric function is
of the form
(a, b.c.f.g, \ x 1 -, ^g, ^J,
which contains one superfluous constant, and equivalent functions differ only by a
multiple of
ff-Y + 7- + ?JL__
x ft . x y x y . x ct x fit. x p
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
1 V
, Y 1 1 IV
a, b, c, f, e, h - - , ~ , -
Axax fix-'y/
may be curtate, are easily found to be
a + b + c 4- 2f + 2g -f 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 4- h + g, h + b + f, g-ff+c each = ; and
this being so the values of f, g, h are = (a b - c), ( a -f b c), ( a b -f c)
respectively. Hence the form is
(a, b, c, i(a-b-c), H-a + b-c), J(-a- b +
which, as already mentioned, we denote by
( a ' b> C -''JU^a' a, -ft' x-y)
We have thus the theorem that a curtate function of any number of inverts, but with
only the three essential inverts
1 1 1
x a' x f}' x y'
is always expressible in the foregoing form
> b< c "
13. It may be remarked that the function (a, b, c .'.QX, Y, Zf is a function of
the differences of the variables X, 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, Wy, for which
a+h+g + 1, h + b+f+m, g+f+c + n, 1+m + n + d,
202
ON THE 8CHWARZIAN 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
_JL_ _A_ -...}
(a -a) 1 (x-0y (*-7> s )
where the coefficients a, b, c, ... satisfy the relation a + b + c + ... = -2, is diaphoric,
and therefore curtate. In fact, forming the sum, coeff. _ +^coeff. ^_ g g ,_p+ >
this is -a-ia'-iab-fcac-..., -- ia(2 + a + b + c+ ...), which is =0; and similarly
the other conditions are satisfied.
15. The function
regarded as a function of the inverts
11
x-tt' aj-d' '' a;-/9'
where
..., =k suppose,
is diaphoric, and therefore curtate. In fact, the condition in regard to is
a (a 1 + ao, + oa, + . . . ) + (- a + b - c) (ab + o^ + . . . ) + (- a - b + c) (oc + oc, + . . . ) = ;
that is,
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 = :
hence, using the accent to denote differentiation in regard to z, we have also F + Q? + R' = ;
and therefore QR - &R = RP' - R'P = PQ' - P'Q, =0 suppose: and we require to find
P, Q, R such that the function 6 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 + ft) -7- (yz + S), 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
-m)i, R = HH(z-n) r ,
where (z I)* 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 = 2p, = "2.q, = 2r.
19. It is at once seen that is of the degree 282, and moreover that it
contains the factors II (z J)*" 1 , II (z m)^ 1 , II (z n) r ~' ; hence it contains the factor
n (z - 1)"- 1 (z - m)*- 1 (z - n) r - 1 .
Suppose the number of distinct indices p is =<r lt that of distinct indices q is a. 2 , and
that of distinct indices r is <r 3 ; then the degree of the factor is = 38 <r, a-, <r 3 ;
and if this be = 28 2, then can have no other variable factor : viz. if the numbers
<ji, o- 2 , <7 3 of the distinct indices p, q, r respectively are such that a- } + r 2 + <r 3 = 8+ 2,
a relation which is henceforth taken to be satisfied, then we have
= Kli (z - I)"- 1 (z -
(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 = here implies that two functions,
say P, Q, are of the same degree, and the third function R of an inferior degree ;
but, this being so, we have only to regard R as containing the factor II 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-Table : in which, where required,
the proper factor fl 1 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 8CHWARZIAN DERIVATIVE
[745
B
^s.
"18 |8 .
""i^, 2r ^Ts" ^
.2 =
C
i i '"""^
|H '^"^ ' ~?
^ o
7 V vx
> ^ X "I* o + X ^r x ^zr'
l\
ij 5 tVi|*^|
S
J
: ,a f ^ =+
U - ^ , i S
: i Ji > oT" J
,5 J o> 2s. -7 J5 :
3 "
S oc
*p 5 I S $
S. i. w : g
i * | J If* I a
o a
- 'c
M |
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v^x -'+-
i i i. 9
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+ 2- *N -^- S
M OD O 00 W 00
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i S i i 111 i
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m
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x* v
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flfi fc,
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p(
<; i-H
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O
$ x Nig ^.
O
2^ |8 N
N 1 8 "' "18 +
>" S
!* 1
+ rt 2s
1 f-t
c
^- * r ^
^"^ ^^ ^^ ^^s- ? i
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II
e;
^H ^* O* "I*"*"
+ ' ' 00 3
*, & i 25
^ ^ """^ *""' 4 "~* 00 OD
i e^, ^* t- "* O t CO
B 9 M CO i^ <N i-t
M
i i ii
2- CQ I 1 III +
X g
w
Cl_^ ^
8
^7
O S
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+
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S
^
rH i t
o "5
Cfl
= 2
Cl
JT ** 7^
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M| 8 "* e"
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W5 CO O
* ^
^ + i 1
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Ofl 00 _j_ to 1C o
It:
o X
7 i 7 g
i? s- ^ 3 i ||
ftf g
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X + -? ? ! +
HI! Iii r
ft
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iii ii
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en *a
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"V i -2- 5 tit f
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&
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^F H M B 9 S ^
BH q x x x
M
M
p *
><
745]
AND THE POLYHEDRAL FUNCTIONS.
159
a
13
a
i
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, <a
I I !!!
1 1 11 Ii
S s .s J r S
O rg rg
f -a a 21
flcaHgcc- - ~ * *
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o
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iH ^H ^^ IH|W rH
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a
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8 Q 8 8 ^c 8
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^^ f^ 3 OV^
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sj \^*J s^~^ \fj. ^~^ y^V S^i^ V~^ \^--J
f--^\ f^3*> /^^S ^ l /^^S /^^ r^ /^^ /^^S
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160 ON THE 8CHWARZIAN DERIVATIVK [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 = XP , P = X^P t ,
where P, is written for II (z - /)*-', the G.C.M. of P and P ; and X is consequently
= F multiplied by the product n (z l) of the several factors taken each with the index
unity; and so for Q and R: viz. we write
P, Q, R = XP,, YQ,, ZR ,
P, Q', R^X.P,, Y,Q,, ZA,
and the foregoing value of O 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
22. Calculation of the first term {x, z}.
P \ f P
* ( P \ f P \ (P )
We have # = a function (a ^ + /3J H- fy p + SJ , and thence {x, ^} = jp, *r, = {, z}
for a moment; then
f=( F \= RP '- RP
? \R) ' R*
Substituting the values
P, = n(s-l)r- 1 , Q<,= U(z-m)i-\ R = H (z - ny-\ Z=Il(z-n),
we have
f z I z m z-n'
and thence
{x, z} = \- 2 7j^ - 2 7-^ + 2 .
__LI _ S
2 t ^ m z n
or say
_if P" 1 | ff'" 1 + . + g- 1 .. gi-1 . r+1 r, + l y
z \ 2 / z ii ^-m z m l z-n ' z n^ ' '") '
745] AND THE POLYHEDRAL FUNCTIONS. 161
where it is to be observed that
2 (p - 1) + 2 (q - 1) - 2 (r + 1), = 8 - o-, + 8 - 0-3 - (8 + a,) = 8 - <r, - -., - ov, = - 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
*
we have
f(x-a), g(x-b), h(x-c)=SlP, SlQ,
where fl is a determinate function of z\ hence
a
____ _ _ _
x- a ' x-b' x-c~ P Q,' Q SI' R + ~fi'
Then substituting these values, by reason that the function is diaphoric, the terms
in fr disappear, and we have
Wa, b, c ..$, V
\ \x-a x b xc)
= (&, b, c /.jj-y, -Q, f) ,
-(a, b, c .:faP 2-2- S - r -Y.
X * z m z nj
which is
We have 2p = 2g = 2?', = 8 : and hence by what precedes, this function, considered as
a function of the inverts . , &c., is diaphoric, and therefore curtate.
Z 6
24. We have therefore
*/ X 1 ! ! V
{a;, f )+*(, b, c .-.(I , -7, 1 =
\ ^s>*' " ^ C mt L-/
{^ JJ V 2^ _L^
\Z ~~ W^" (^ fflJT \Z fir) J
z m
-2
p ^ 1 2 r
~~ ? > " >
r t ^ m ^ n
where the whole function on the right-hand side is curtate.
C. XI.
21
162 ON THE 8CHWARZIAN DERIVATIVE [745
25. We have to bring the function on the right-hand side into the reduced form
A.
(2-*Y *z-a
for the purpose of getting rid of the non-essential inverts (if any).
We write
v p- l -PrL + pi- 1 +
*(z-lY (z-l? + ( s -l# +
. P ~ l . v P*- 1
viz. zl here denotes any particular factor, and z I, represents any other factor of
the same set; and so in other like cases.
26. The whole coefficient of ~r is
(z If
an expression which, regarded as a function of a and p, is represented by (a/>): 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
-l 2 l-_l_ s r _l
* I \ z t, z m z n
+ *{2a2'-- +(-a-b + c)2- 9L + (-a + b-c)2'- },
zl { z l z -m z n)
viz. each term such as - =- is to be replaced by T ( -- --- ), giving
z l . z (i t l\\z I z ]/
rise to the term . =- - -., or contributing the term r to the coefficient of -
' ^i z L t tj ^ l
The whole coefficient thus is
m
h
i n
28. Suppose first that z - 1 is a multiple factor of P, viz. a factor with an index p
greater than 1: then, for z = l, we have Q + R = 0, Q' + R' = 0, and thence ^ = 4'.
(jj H
(I t*
that is, 2 , * = 2 . . We have therefore
I m l n
- -._ -,-
m ln
-n
745] AND THE POLYHEDRAL FUNCTIONS. 163
moreover, in the top line, the terms 2 5^ and 2 ; destroy each other. The
I m l n
whole coefficient of - , , when z I is a multiple factor of P, thus is
z t
_ ^ : _ j __i
I m l n
^L + 2
I m
a form which is now symmetrical in regard to the inverts , and ,
I m ln
29. The value just obtained must be equal to
- 7
m ln
12 - + 2 r - - 2'
V I
viz. comparing the two forms and reducing, they will be identical if only
l-
m
and it can be shown that the function inside the { } is in fact =0.
30. We have, as before, 2 v-_ - = 2 y ; or writing each of these quantities = <J>,
the equation to be verified is
2'^, -^ = (p + 1) <I> - p2 . jp2 -. .
{ I, I in ln
We have
that is,
' Pi __ P for -/
- ' *'
_ rz.^-Q-pZ-i
.Y(^-0
The first derived function of the numerator is X{(z l) + X l pX', which for
z = I is X l pX', which is = ; and, for the denominator, it is X' (z l) + X, which
is also = 0. Passing to the second derived functions, we find
O V ' . V" V ' \ ~* V"
?' P I - l ~P JL _ ~*~~tf~-
* z-l,~ 2X' X'
From the equation
X z I zl
212
164 ON THE 8CHWARZIAN DERIVATIVE [745
we find in like manner
" J - *, ~ X'
and we thence obtain (* being always =1)
so that the equation to be verified becomes
31. But from the equation , =PQf-FQ, =KP,Q<>R, we find XY.-X.Y--
and then, differentiating, XY t ' + X'Y, X t 'Y X t Y' =KR a ': writing in these equations
z = I, they become
so that, dividing the second by the first,
Y ff
or, recollecting that X^=pX' and ^ = 7) ' we
that is,
the required relation.
32. The result is that, z I being a multiple factor of P, the coefficient of the
term . is
33. In the case where ^-/ is a simple factor of P we have />=!, and the
coefficient is
-m
745]
AND THE POLYHEDRAL FUNCTIONS.
1
34. Of course the formulae for the coefficients of
-,
\z -~
=-, and
z ~*
165
give at once,
by a mere change of letters, those for the coefficients of . ;-, , and
; and the function in question,
(z - riy z n
f Y 1 1 IN 2
+x'*(*, b, c :JQ-^ t - ,, ] ,
V A - a x b x c/'
is now obtained in the required form
(a|) (bg)
(z-lf" (z-mf" ^
(cr) _A_ B
z-rif" *-l" z-m
" "*"
C
*-
where (ap) denotes ^(1 p 2 )+ap 2 , and the like for (bq) and (cr); and where, z I
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 = ; we have
al, = a, which does not vanish; that is, z I being a simple factor of P, the
expression contains -. .-^ , or the invert - . is essential : and similarly, z in being
{Z ^ vJT Z L
t -i
a simple factor of Q, or z n a simple factor of R, the inverts
z m
and
zn
1
are essential. But for z I a multiple factor of P, the coefficient (ap) of the term
may vanish, viz. this will be the case if a = i(l ; and, when this is so, the
V p-/
pV
coefficient A of the corresponding term ; also vanishes; that is, , is a non-
z ' z ~~ f/
essential invert. And similarly for any multiple factor z-m of Q or z n of R, the
invert - - or - - may be non-essential.
zm
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)) p , G (U (z m))i , H(U(z n)Y, the result contains only the six terms
written down: and then, if a, b, c are = (l - j , J (l - - J , fl - -J respectively
the result is = : viz. we then have
or we in fact have, for the values in question of a, b, c, a solution
f(x-a) : g(x-b) : k(x-c)=P : Q : R
of this differential equation of the third order.
166 ON THE 8CHWARZIAN DERIVATIVE [745
37. The reasoning applies directly to lines 2, 3, 4, 5 of the PQE-Table: and
with a slight variation to line 1 ; viz. here the factors of R (= - 1 + z n ) are all simple
factors, but in virtue of c = 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-TabIe give each of 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) : A (a - c) = (*< + 2 \^3* J + I) 3 : -12 V-~3 (-*)> : -(- 2 V-
satisfies
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 - ,
z ctj
, , belonging to the three functions P, Q, R respectively, or it may be
2 01 Z Cj
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
i, bj, Cj /.
where a,, b,, c, are the values of the three which do not vanish in the series of
expressions (&p), (b<?), (cr).
The remaining lines (III, V, VII, VIII) and IX to XV of the PQR-fable give
such values of P, Q, R, the values of (a, b, c); and the calculation of the values of
(a,, bj, GI) is shown by the corresponding lines of the Annex. And we have thus
values of x determined by the equations
/(*-o) : g(x-b) : h(x-c) = P : Q : R,
and giving
(, .| + W.. b, . ...j. j,. -a,, b,, c, ...
39. For instance, from line IX we have
f(x-a) : g(x-b) : h(x-c} = (z- *>?
4 3 12
the values of (a, b, c) are g, -, -^5 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
~~ j ~Q > ) etc * c rt respectively,
o zo ou
3 12 21 1 1
Hence writing a l , b,, c^-, , - , the corresponding inverts are -, ,
o *5 oU z 1 ,2 GO
-; and the result is
, /4 3 12 Y 1 1 1 V /3 12 21 Y 1 1 IV
I M '. _1_ Of* I * ( 1 . . ., 1 I II ,
\9' 8' 25"X*-a' x-b' x-c)~\S' 25' SO"**-!' z-<x>' z)'
40. It is hardly necessary to remark that an expression
)\Z ttj ' Z b t ' 2
in fact denotes
a Vv Q ^_ l-t I
<*1 Wj <*j Llj T^ v^
/T _ \*> ~"~ / i vi " 7
- a,) (z - 60 '
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 (a,, b lt GI)
arbitrary ; or we can give to the a, , 6j , 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 ~ 20 Z ~~ X
not worth while to effect it.
41. The conclusion is that lines (III, V, VII, VIII) and IX to XV of the
PQE-Table, give, for determinate values of (a, b, c) and (a^ b 1; c,), solutions
f(x-a) : g(x-b) : h(x-c) = P : Q : R
of the equation
{*, ,} + *"(a, b, c ,-!_, ^ ^'-(a,, b,, c, ...Jk-L, ^, ^J,
where a, b, c, a,, 6,, c a 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 PQR-Table, but for
convenience writing s instead of z, we have
f(x a) : g (x b) : h (x c) = P : Q : R,
where P, Q, R are now functions of s, a solution of
V
61 I ~~ "
T f* I
But we have
,'dsV .
*J (3 {x ' s} '
168 ON THB 8CHWARZIAN DEKIVATIVE [745
and the foregoing is therefore a solution of
{.,}- a, b, *-'-~> -, ~
a differential equation of the third order. This is the Differential Equation {, a;}.
43. From the Roman lines, if we assume
f(x- a) : g(x-b) : h(x-c) = ^ : O : ,
when- 'Is G, 3i 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,
I*, ,) + ffiU b, c ,.fl , l ., J-Y - (a,, b,, c, 4-J- , 1 , -J-Y
V<k/\ X*-a-6a-c/ \ Xz-a^ z-h' 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
'l*/ If.. > I Q 1^
y Jl I I*' X \\ & 1> "l>
or since
this is
the corresponding relation between s, z being of course obtained by the elimination
of x from the two sets of equations
f(x-a) : g(x-b) : h(x-c) = P : Q : R, and/(e-a) : g(x-b) : A(*-c) = $ : d : 91;
vix. the required relation is
P : Q : R = $ : O, : 91,
where P, Q, R are functions of s ; ty, d, 9t functions of z ; and, in virtue of
the relations are equivalent to a single equation between z and *. And writing
finally x in place of z, that is, now considering $, Q,, 9t as functions of #, we have
^ : Q, : 9i=P : Q : R
as a solution of
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,, b,, c,) 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 *.
1
J \X ft) ', ff \X 6) : ft (a/ C)
P:Q:S(l)
Polygon
I
JJ
(2)
Double Pyramid
II
(3)
Tetrahedron
III
4a; : -(x+lf : (a;- I) 2
(3)
IV
f(x-a) g(x-b) h(x-c)
(4)
Cube and Octahedron
V
(x-iy -(x+iy 4x
W
VI
ft ft\ / r*i 7i\ Jt f W /\
(5)
Dodecahedron and Icosahedron
VII
4a; (x + 1) J (* I) 2
(5)
VIII
(a; -I) 2 -(x+iy 4x
(5)
IX
P Q R (IX)
(5)
,,
X
(X)
(5)
XI
(XI)
(5)
XII
(XII)
(5)
i
XIII
(XIII)
(5)
XIV
(XIV)
(5)
XV
(XV)
(5)
"
The values of the P, Q, R as functions of x, or of s, are taken out of the
PQE-Table: only in the lines III, V, VII, VIII, where P, Q, R are given as
and where, as regards V and VIII, there is a transposition of P and R, I have
inserted the actual values of the ^-functions. (See as to this the foot-note referring
to the Annex.)
The Schivarzian Theory. Art. Nos. 46 to 62.
46. Considering the foregoing equation
/ . Y 1 1 1
{s, ai} = (& 1 , b,, 01, /.ft -, - r-, -
\ A# Oi x o-i x Ci
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.
c. XL 22
170 ON THE 8CHWARZIAN DERIVATIVE [745
Consider, in general, a solution of this form, x=F(s) a rational function of s: then
a is an irrational function of x, and if ,, , are any two of its values, {,, x} = R(x),
{,, x}**R(x); that is, {,,*}- {i, *}, and therefore (ante, No. 7) * = hfrf- And
then * = JP(s,) = .F(^ L ^-5), =.F(*,) : viz. .F(*) is a rational function of , transform-
able into itself by the transformation s into -%: and it is moreover clear that
CS T Cfc
between any two roots a 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 a; to be a rational function of s, it is only in the
case x = F(a), 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 b 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 , 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 t 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, djS ; 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 a 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 (a;, a), the relation at a critical point between the corresponding elements dx,
ds being of the form da = (a + bi) (dx)*, (\ 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 = \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'ffe ie ,
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
0' 0'J
and thence
{s, x} = {p, x} + {0, x} + \ff* - P ^ff- i ^~
Putting this = R (x), and assuming that x is real, we have
The last equation gives p"ff = 0, that is, 0' = 0, which gives s = 0, and may be
disregarded ; or else p" = 0, therefore p, a real constant, = 7 suppose, and {p, x} = :
hence for the solution of the equation {s, x}=R (x), we have s = iy0'e ie , a real
quantity determined by {0, x} + ^0"* = R(x): and then, integrating the equation for s',
we have s = a. + /9i + ye' 9 , a, y9, 7 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 + fti + ye
(0, and the constants a, /9, y, being real), that is, the point s moves in a circle,
coordinates of the centre a, ft, and radius =7.
50. Suppose a, b, c are any real values of x representing points a, b, c on the
-line; and A, B, C any given imaginary values of s representing points A, B, C
222
172 ON THE 8CHWARZIAN DERIVATIVE [745
in the -plane : since {s, a:} = R (*) 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 ar=a, b, c shall correspond the values s=A, B, C respectively.
If there is not on the jr-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 da
at the corresponding points A, B, C ore equal to multiples of (dxf, (dxf, (dx)" re-
spectively, then to the flat angles ir at a, b, c correspond in the path of the
angles XTT, pir, inr at the points A, B, C respectively: and, assuming that a, b, c
are the only critical points on the ar-line, the path of s is made up of the three
circular arcs CA, AB, EC meeting at angles XTT, fiftr, inr respectively. The arcs are
completely determined by these conditions; for supposing the arc EG to make with
the chord EC, 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, /wr, vir = A + g + h, Z.B + h+f, /^.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 EC is the arc on the chord EC meeting it at angles /, f:
and the like as regards the arcs CA and AB respectively.
52. The foregoing equation
where a, b, c have the values (1 - X s ), (1 - ft 1 ), (1 - K"), and X, /*, v are real and
positive, has x a, b, c for critical points of the kind in question. In fact, writing
xa = h, the equation is of the form
- X 2
, =-ftr~
which is satisfied by
1
we thence obtain an integral of the form
s = kh.-* (1 + k^h + k 3 h" + ...), = k<f> for shortness.
This is a particular integral, but we have from it the general integral
a +
Since there is no critical point on the x-line there can be no abrupt change of direction in the path
of i, that is, the path of cannot consist of circular arcs meeting at an angle: but it is in the text
further assumed that the path of cannot consist of different arcs of circle, the one continuing the other
without any abrupt change of direction.
745] AND THE POLYHEDRAL FUNCTIONS.
If A be the value of s corresponding to h = 0, then /3 = 8 A, and we find
173
viz. reducing -r to its principal term A x , and then writing ds, dx for s A, and h(=x a)
respectively, we have ds = K (dx)*; or x = a is a critical point with the exponent X ;
and similarly x = b and x = c are critical points with the exponents fj, and v respectively.
53. Hence in the equation
/ , Y 1 1 IV
{s, a; = a, b, c .'.(I - , r ,
\ \x a x b x cj
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 a-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 a; 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, pir, vrr 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 \, /j., v, giving for a, b, c, =^(1 \ a ), |(1 /t 2 ), (1 z> 2 ), the series of values
mentioned in the Annex of the PQS-Table : and thus showed a priori that the equation
\s, }=(a, b, c ,'.0- i r, )
V, Xx ax bx cJ
is algebraically integrable for these values of a, b, c ; and only for these values, or
for values reducible to them.
174 ON THE BCHWARZIAN DERIVATIVE [745
56. As an instance, take the double pyramid form : the integral equation is
or say
or if, for greater simplicity, we assume a, b, c = l, 0, oo , this is * =
or say
1
), that is, s n = - -.-
.-, a solution of the differential equation
- a solution of
In particular, if n = 3, we have *
8 4 3 VI 1 1
, 9, 8 ). w _ a . x-
57. We have here the spherical surface divided by the equator and three meridians
into twelve triangles, each with the angles ^TT, TT, TT: and then, projecting from the
South pole on the plane of the equator, we have the annexed figure of the s-plane,
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 avplane is by the r-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, s 3 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, s 3 is real and negative, and we have the infinite
half-lines at the inclinations 60, 180, 300. If x is real and negative, then s 3 is of
1 ki
the form - - ,. , =cos# + isin#; whence s is of the same form, or the locus of the
,. ,
KI
1 _
_
point s is a circle radius unity. Writing s 3 = =. , and supposing that the point x
1 + \'x
moves along the online from b through a to c at oo , and then from c at + oo to b,
the point s describes the sides BA, AG, 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 = 2*% (K small), then taking for 'Jx the value
1 K ( 1 t)
*(1 i), we have s 3 = ^ ^, = 1 2*(1 i) nearly, and hence a value of s is
X ~T~ K ^ J. 1)
= 1 l/e + f/w, 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 = 2/c% 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
that is,
\s, x} = (a, b, c .'.A , T, I ,
\ X# a x-b x cj
_ (b c) (c a) (a b) f a b c \
x a.x b .x c \b c.x a c a.xb 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 a; 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, /8, 7, #, lt 0. 2 , such that a, b, c are = a + fji + ye^, a + /
a + /3i-\- ye* 1 ' (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 = at + @i + ye ei ,
6 a variable parameter, the equation which expresses that the point x is situate on
the circle in question.
We have x a = 7 (e* e~ 9 ')> = ye^ (e+e ' } [ei<- fl o> e -4(-o)j ; the second factor is
tsin(0 # ), = iP suppose, or the equation is x a = iPye* (e+e <> >i , say
x a = iPy expi (0 + # ).
Similarly x b = iQy expi (# + #,), and x c = illy expi (6 + # 2 ) ; where P, Q, R denote
sin (6 ), sin ^ (0 0j), sin (8 # 2 ) respectively. In like manner, we have b c, c a,
a - b, = iFy expi \ (# + a ), iGfy expi \ (0 a + ), iffy expi (0 + ^), where F, G, H denote
d t \ sin (#., ), sin(0 dj respectively.
176 ON THE 8CHWARZIAN DERIVATIVE [745
\\ V have
x-a.x b.x c
i + ffi _
with the like values for - r and - . Hence the right-hand side of
c a.x o a b.xc
the equation is
-Ji
62. Considering now the left-hand side of the equation, we have
/ CLX \
m
substituting for x its value =a + @i + ye ei , this becomes
*, *}-*),
that is,
Assume s = + J/i + JVe*, Z, J/, and N constants ; then using the accent to denote
differentiation in regard to 6, we find without difficulty {*, 0} = {, 0}+l s & 3 , and the
value of {, #} becomes
Hence, substituting the values of the two sides of the equation, the imaginary
factor expi ( 26) divides out, and the equation becomes
an equation, in which everything is real and which thus determines as a real
function of B : 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
dry
745]
AND THE POLYHEDRAL FUNCTIONS.
177
again P, Q given functions of z, and v a function of z determined by the equation
and assume
d*v
d^"" 1 dz
y = -
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 -5- 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 a; ; and if we assume z = x,
P, Q will become =p, q respectively : the given equation in v will be
dv
and w will thus denote the quotient of any two solutions of the equation
d*y dy
viz. writing X = p- + 2 j- _ 4q, then, by what precedes, the equation for w will be
d &
{w, x}= %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 = wv, and taking any other two solutions
y v
we have yi = wv lt so that = ; calling each of these equal quantities s, we have s
denoting the quotient of two solutions of the equation in y, and also the quotient
(if)
of two solutions of the equation in v ; whence, writing as before X =p^+ 2-f- 4:q,
i 7}
and similarly Z = P" + 2 3- 4Q, we have
ctz
and since in general
{s,z}=-\Z,
C. XI.
23
178 ON THE SCHWARZIAN DERIVATIVE [745
we obtain
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.
v ? = CeS pa *-l> >d *+T.
ax
If z, x are regarded each of them as a function of the new independent variable
6, then the equation is
66. Jacobi's differential equation of the third order for the transformed modulus \,
Fund. Nova, p. 78, [Ges. Werke, t. I, p. 132], is
2 * = 0,
where the accents denote differentiations in regard to an independent variable 6 : viz.
dividing by 2jfc'*X' s , 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
l)a! dy _
_
did' x.\x dx x.\x
d*v + y-(a'+ff + l)z dv *pv_ =Q
dz* z.l z dz z.\ z~
Hence
^ |
_
x.l-x x \ x x.l x'
and hence
viz. writing
745] AND THE POLYHEDRAL FUNCTIONS.
and putting in the formula x 1, = (!), we have
179
a
~
- a+b c
x.x-l '
= a, b,
with a like formula for
y = wv,
w 3 =
A* ' X 00 ' X 1
2 - - - *Q . We then have
dx
and the differential equation of the third order for the determination of z is
(z, *l + L*ii bj, Cj .'.()-, =] IT] fa, b, c .'.()-, T) =0,
V X * B z\l\dxl \ A.X x -oo x-lj
where a!, bi, c, are the same functions of a', ft', y' which a, b, c are of a, /3, y.
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
Yl 1 1 \
}, x = a, b, c ..()-, - - , -- ,) .
\ AX x x x \J
PART II. THE POLYHEDRAL FUNCTIONS.
Origin and Properties. Art. Nos. 69 to 80.
69. The functions in lines 1,...,5 of the PQB-Table are connected with the
geometrical forms :
fl. 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.
232
180 ON THE 8CHWARZIAN 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 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. 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=sin0coaf, sin sin/, cos#;
Y
y = - = tan $8 sin/,
and conversely,
X, Y, Z=2x, 2y, \-tf- f, +(l +a *+jf).
We represent the point (X, Y, Z) on the spherical surface by means of the
magnitude x + iy, = tan $0 (cos/+ 1 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 II {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 stereographic 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 -5- {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
ITT . . 2-n- (n-\)1ir . (n-\)2-jr
1, cos -- Msm ,...,cos -+tsm-
n n n n
The corresponding function of s is s" 1.
745]
AND THE POLYHEDRAL FUNCTIONS.
181
The values of x + iy for the mid-points of the sides are
TT . . -n- Sir . . Sir (2n.-l)7r . (2n,-l)7r
cos M sin . cos \- i sin - , .... cos - ' h t sin ^
ft n n n n n
The corresponding function of s is s n +l.
The North and South Poles, which form with the n points a double pyramid of
n+2 summits, correspond to the values s = and s=x>. We have thus
_ .1) (,_!)
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 D ,
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 4 2i \/3 s 2 + 1.
In fact, the equation s* 2i v/3 s 2 + 1 = gives s 2 = 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
# + iy
4.^
1 1
i + i
V3
135 -
225 -
315 +
/O /Q
yO v "
V3-1
-l+i
v/3 + 1
-1 -i
V3-1
l+i
V3 + 1
Corresponding to the centres of the faces, or summits of the opposite tetrahedron,
we have the function s*+ 21^3 s* + 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=i on the equator at longitudes 0, 90, 180, 270
respectively.
182 ON THE 8CHWARZIAN DERIVATIVE [745
7M. (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 are, as in the last case, 0, ao , 1, i, and the function is
The function for the centres of the faces, or summits of the cube, is 8 +14s 4 + l.
The function for the mid-points of the sides of the octahedron or of the cube is
s 15 -33s 8 - 33s 4 +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 --
11*- 1).
The function for the centres of the faces of the leosahedron, or summits of the
Dodecahedron, is a* - 228s 1 " + 494s' + 228s 5 - 1.
The function for the mid-points of the sides of the Icosahedron or the Dodecahedron
is
s - 522s 25 + 10005s 20 + Os 15 - 10005s 10 + 522s 5 + 1.
I give for the present these results without demonstration.
75. Writing - for s so as to obtain homogeneous functions (*$#, y) n , it will be
u
recollected that the x, y of these functions have nothing to do with the x, y of
the foregoing values a; + iy the forms which have thus presented themselves may be
denoted as follows :
(3): /3 = (1, -2tV3, 1%*?, yj,
(4): f
/i4 = (l, 14,
4-(l, -33, -33, 1$^, yO",
(5): /5 = sy(l, 11, - 1$< y 5 ) 2 ,
/io = (l, -228, +494, +228, -
= (!, -522, 10005, 0, -10005, 522, 1$V, /),
where observe that /4 is the same function as <3. In each set of functions / h, t,
we have h and t covariants of /, viz. disregarding numerical factors,
// is the Hrssun, or derivative (/,/)*, and t is the derivative (/, h).
745] AND THE POLYHEDRAL FUNCTIONS. 183
76. Since /4 is the same function as tS, we have of course /4, A4 and <4
themselves covariants of f3 : but it is convenient to separate the two systems.
77. It is to be observed that /3 is a quartic function having its quadrinvariant
{/) = ; 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 /3, A3 and t3. In particular, A4 and
H are rational and integral functions of f'3, A3 and t3 ; but inasmuch as /3 and
A3 are not covariants of /4, this is not a property of A4 and tA considered as
covariants ofy4, and the relation in question need not be attended to.
78. It has just been stated that fS qua quartic function has (in the sense
explained) only the covariants A3 and <3 : f4> qua special sextic function and fa 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 f3, f^, 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, UVU'V
and (U, V) differ only by a numerical factor: and further that, writing as before
fjt
*=-, and in the expression UVU'V regarding U, V as functions (*]s, 1), and
J
the accent as denoting differentiation in regard to s, we have UVU'V and (U, V)
differing by a numerical factor only. We have in the PQR-T&ble, lines 3, 4, 5,
P, Q, R equal to given numerical multiples of hP, tf, f", the indices a, /3, 7 being
such as to make these to be functions of the same degree: hence, neglecting
numerical multipliers, PQ 7 P'Q is equal to a function (A 3 , <*), which is = h?~ l tf~ l (A, t) :
and the theorem that Pty - 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, f) 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 = (f, ff and t = (A, f), equations giving the functions A and t,
upon which P and Q depend.
ON THE SCHWAKZIAN DERIVATIVE [745
Covariantive Formulae. Art. Nos. 81 to 84.
81. The various covariantive formulas will be given with their proper numerical
coefficients.
Tetrahedron function. /, h, t stand for the before-mentioned values,
/3, A3, t3 (P, Q, R = h>, -12iV3.F, -f 3 ).
For /3.
(a, b, c, d, e) = l, 0, =, 0, 1.
i (/, fy = - 96i V3 . A, 4 (A, A) s = 96t V3 ./, 4 (t, (f = - 25/A,
(/ A)= 32iV3.*, (//)= 5767-0, (/, A)* = 1152/ = 1152.^,
/A = (l, 14,
It is convenient to remark that t s , f 3 , h 3 being of the same order we have
f (f 3 , h 3 ) +/" (A 3 , f) + h 3 (f, f 3 ) = 0,
that is,
2 . 3 . 3/W (/, A) +/ s . 3 . 2h-t (h, t) + h 3 . 2 . 3</ 2 (, /) = 0,
an equation which, substituting for (/, h), (h, t), (t, /) their values, reduces itself to
the before-mentioned relation h 3 f 3 12z-v / 3< 2 = 0; and we have thus a verification of
the values of (f, h), (h, t) and (t, f). 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 = h 3 , -V, -
For /4.
(a, I, c, d, e,f, 0) = (0, i, 0, 0, 0, - J, 0).
t (/. /) 2 = - 25A, 4 (/ fy = o, i (/, fy = (720)* . i,
(/, = - 12A", 4 (, )' = 2 4 . 3 s . IP ./ 2 A,
(A, )--1728/,
A 8 -< 2 -108/ = 0.
745] AND THE POLYHEDRAL FUNCTIONS. 185
83. Dodecahedron function. /, /;, t stand for the before-mentioned values
/5, h5, t5 (P, Q, R = h 3 , -t\ - 1728/ 5 ).
For /5.
(a, b, c, d, e,f, g, h, i, j, k, I, m) = (0, -fr, 0, 0, 0, 0, , 0, 0, 0, 0, -^, 0).
- 12U, i (/ fY = 0, i (/, /)' = i (924)=
= 0, i</,/) M =0,
(/, h) = - 20*, | (h, A)* = 173280/ 3 ,
(/ i) = -30A s , J(, <) 2 = 9082800/ 3 A,
(A, t) = -86400/ 5 ,
A 3 -f--1728/ 5 = 0.
84. We have
t = (a?" + y w ) (1, 522, - 10006, - 522, l~$a?, y*)*.
Write
= (* + y).(l, 2, 6, -2, l$ar, y),
then
= (!, -10, 45$f,/).
Or putting
_f = (^ + y g )(l. 2, 6, -2. l$g. y y
'
that is, % = p>Jf, then
+ 45^ = - . (Klein.)
Investigation of the forms fo and ho. Art. Nos. 85 and 86.
jfy _ j
85. Writing for shortnessf i = tana= ^ , and g = cos 36 + i sin 36, then the
31
values of x + iy corresponding to the summits of the Icosahedron are
0,
k, kg 1 , kg*, kg*, kg*,
and the function fb is thus
* The numerical coefficients - \ and fj 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 (/, /) 8 contains the factor \-, and (/ containing the factor X) the form is
4(/./> 6
which is linear as regards X. We have also
gay ^=JiX', B=--ff\; or 84*=A. Of course in the case of a general dodecadic function /, we have
(/, /)*, an irreducible covariant, not breaking up into factors.
t a is the a, 7 is the y, and / the a-/3 of the Table, No. 99.
c. xi. 24
186 ON THE 8CHWARZIAN DERIVATIVE [745
where the product of the last two factors is P + (fir* -&)*-!. We have
k~> _ A (80 V5 + 170), = | (5 V5 + 11),
i- = ^ (80 V5 - 176), =i(5V5-H),
and consequently &-*- = 11; or the function is
86. Similarly, writing for shortness* I - tan 7, i'=tan^y' ( where
5 + 2V5 10-2V5 cos 7 3 + V5
C08i<y= 15 ' 7 --- 15 ' and theref re sin 7 4
5-2V5 , , 10 + 2^5 cosy 3-^5
__ -_ __
,
ry =
____, __ - r _ 7 __
and <JT = cos 36 + t sin 36 as before, then the values of x + iy for the summits of the
dodecahedron are
Ig, If, If, If, lg\
eg, I'g 3 , ly. fy 7 . ^y.
r-s v-y, i'-y, r-y, r-y,
f-', z-y, i-y, i-y, f-y
The function h5 is therefore
= s 10 + s 5 (/" - i~ 5 ) + 1 . s 10 + s 5 (/' 5 - I'-") - 1.
We have
(1 +C087) 5 (1 COS 7) 2C087-.
j[- Z" = l I - -tij ^ IL = . ' (5 + 10 cos 2 7 + cos 4 7)
sin 5 7 sin 8 7
>
sm7 4o > 45 sm 5 7 v
viz. this last identity depends on
H(3 + V5) (6 + V5) = (114 + 50 V5) sin 4 7,
that is,
160(3 + V5)(6 + V5) = (H4 + 50 V5) (120- 40
or
2 (3 + Vo) (6 + V5) = (57 + 25 V5) (3 - V5),
or finally
(7 + 3V5)(6 + V5)= 57 + 25V5,
which is right.
Similarly
and observing that the sum and product of 114 + 50\/5, 114 50\/5 are =228 and
496 respectively, the required function of s is
(s 10 - 1 y - 228 (s 18 - s 5 ) + 496s 10 ,
= s 20 - 228s 15 + 494s 10 + 228s 8 + 1,
which is the required value of Ao.
* a is the o, 7 is the 7, and y' the a - ft 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 6 and azimuth f, and
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 (cos /+ i sin /), and tan \ff (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
or, what is the same thing,
or, observing that
,_ s tan^c
1 + s tan c '
tan c (1 + ss) s s' ;
ss' = tan tan \ff (cos (/+/') + i sin (/+/')},
we have the two equations
tan \c {1 + tan tan 0' cos (/+/')} = tan cos/- tan 0' cos/',
tan ^c [ tan tan 0' sin (/+/')} = tan sin/ tan ^ff 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 0, 0' = b, a; / f' = A, ir B, whence
s, s' = tan 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 \b cos (A B)\ = tan $b cos A + tan ^a cos B,
tan ^c { tan a tan ^b sin (A B)\ = tan %b sin A tan |a sin B;
242
188 OX THE 8CHWAEZIAN DERIVATIVE [745
equations which may be verified by means of the ordinary formulae of Spherical
Trigonometry.
89. But it ia interesting to give the proof with rectangular coordinates.
Taking (X, Y, Z), (X lt Y lt Z,) for the coordinates, referred to two different sets
of axes, of a point on the spherical surface: also x, y, x,, y, for the coordinates of
the corresponding stereographic projections, we have
(X lt F,, *,) = ( a, ft, 7
a', ft 1 , V
", &', 7"
X : Y : Z : l = 2x : 2y : I - of -y 2 : l+ar>
X, : Y, : Z t : l = 2x, : 2y, : 1-af-yf : 1+xf
and thence
a* : y, : 1= 2or + 2/9y +7 (I -a? -if)
: 2o.'x + 2/9'y + 7' (1 - of - y 2 )
: 1 + a? + y* + 2a"x + 2/9"y + /'(!-- y>).
90. Introducing z, z l for homogeneity, or writing - , - and , * j n place of
z i z \
x, y and x 1 , y lt respectively, we have
x l = 2*x + 2/3y +7 (z*-tf-f\ =( -7 , -7 , 7 , ft , * , 0$, y, zf,
y,= MX +*ft'y+J (*-<*-?), =(-7- -7- 7',/3',a',0$ ),
*, + * + + 20-a; + 2|8"y + 7" (^ - ^ - y), = (1 - 7", 1 - 7"
and thence without difficulty
" " '
* ~
+7
1 - 7") * + (- " + tf") (* + ty)},
^ - W = ^Zy> KI - 7") * - (<*" + i/8") (* - y)} {(1 + 7") * + ( " - /8") (x
viz. the form is z^ : x l + iy t : x t iyi = MN : NL : LM (L, M, N linear functions of
z, x + 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 Mb'bius.
91. The actual values are
l+V (l-J')g-( e f-ift)( ai +iy)
7 + 7 ' (1 +7) * + (" ~ *ft") ( x + y) '
1 + 7" (l-7")g-(q" + ift")Jfe -iy)
7 - 7 ' (1 + 7") * + (" + /8") (* - iy) '
745 J AND THE POLYHEDRAL FUNCTIONS. 189
X \ 1 1/
viz. attending only to the former of these, we have - a homographic function of
- , which is the before-mentioned theorem.
z
92. Supposing that the transformation from (X, Y, Z) to (X lt Y lt Z^ is made by
a rotation, the coordinates of which are X, /*, v : that is, if f, g, h are the inclinations
of the resultant axis to the axes of x, y, z respectively, and the angle of rotation,
putting X, p., i> = tan0cos/, tan J0 cos g, tan \d cos h: then the coefficients of trans-
formation are
v) , 2(Xi,- M ) )-(
:', ff, i 2(/*X-i>) , 1 - X 2 + /i 2 - z/ 2 ,
i", y8", y'' 2 (j/X -+- fi) , 2 (// X) ,
Substituting these values, the formulae become, after an easy reduction,
i + iy\ _ (v+ i) (x + iy) + (X + i/*) z
(X i (x + iy) + (v-i)
attending to the former of these, and writing for greater simplicity
respectively, we have
*'] s~ . \
or writing this
then
_As + B
SI ~W+D'
A : B : G : D = v i : X + ifj, : \-ifj, : v i.
93. I call to mind that the condition, in order that the homographic transformation
s l = (As+ B) + (Cs+ D) may be periodic of the order n, is
(A + D)* - 4 (AD - BC) cos 3 = 0,
m being an integer different from zero and prime to n. In particular, when n = 2, it
is A+D = 0: ft = 3, it is A*+ AD + & + BC= 0: n = 4, it is ^-f-D 2 + 25(7= 0: and
n=5, it is (A+Dy>-$(3
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 8CHWARZIAN DERIVATIVE [745
(As + B) -r (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 abcde.
95. I call to uiind that a group of functional symbols 1, a, /3, ... can always
be expressed in the equivalent form 1, &a&-', ^/3%~', ... where ^ is any functional
symbol whatever : clearly, o, ft, ... being homographic transformations, then, S- being
any homographic transformation whatever, the new symbols ^aS-" 1 , S-jSS-" 1 , ... will also
be homographic transformations ; and thus the group of homographic transformation*
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 6, and 60 points 4>. 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 <I>
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 <1> are points lying by threes on the faces
of the icosahedron, each point 4> of the face being given as the intersection of a
perpendicular A<& of the face by a line BB joining the centres of two adjacent
faces and which intersects .4 at right angles.
97. The points <I> 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 :
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 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 40 = a, 0=ft AB = y, B,B t = x, ^B 1 B t B = 0, B t =g,
From the triangle A@B, the angles of which are 36, 90, 60 and the opposite
sides ft 7, a, we find the values cf a, f), 7, and these are such that at + /3 + 7 = \ir.
From the triangle B t BB lt where the sides B t B, BB^ and the included angle are
2ft 2/3, 120, we have the opposite side x, and the other two angles each =6.
From the triangle B t B, where the sides B t B, 0, and the included angle are
2ft ft 120', we find the opposite side g, the angle BBi, =$, and the angle
B 4 <P)B, =45.
Hence each of the angles B t B, #,05,, being =45, the angle 4^ is =90:
in this triangle the hypothenuse B^B t is =#, and each of the other two sides is
= (j: whence we have cos x = cos 2 g, as is in fact the case, and moreover the values
give x + 2g = 180. Also each of the other angles is found to be =60; that is, we
have Z B 2 B t = 60", or the whole angle at B t being = 120, the sum of the remaining
angles B,B t B 3 and BB t is =60 C : that is, <? + </> = 60.
From the triangle 05,0' where the two sides and the included angle are
/3, ft 120', we find 00' =36.
192
ON THE 8CHWARZIAN DERIVATIVE
[745
And from the triangle QB<&", where the two sides and the included angle are
, g and (120- 2<=)20, we find e" = 60.
99. We thus arrive at the following Table:
sin cos
Ql JO'
/5-75
/5 + V5
AS
a
ol 4o
OA KK'
V 10
75-1
V 10
V&+1
JiS
P
JU 00
273
2^/3
A J)
070 i)9'
/10-2V5
/5 + 2 75
AD
(BB)
<*)
y
X
9
70* 32'
54 44
17 4fi'
V 15
272
3
V2
>/3
N/8
V 15
1
3
1
V3
s/5
Jiao
/Y.I /'
6
22 14
2 ^2
V3(V5-1)
2^2
s/5 + 3
<p
2a
2/3
9-u
63 26
41 50
74 44
rji
2
V5
2
3
2(^/5 + 1)
4J2
1
^5
^5
3
4-^/5
J y ^
3^/5
3^5
a
/5-2V5
/lQ + 2^/5
a p
V 15
V 15
18
75-1
/5 + 7S
4
V 8
<-"->
36
/5-V5
75 + 1
V 8
4
where as above
a + + 7 = 90,
+2# =180,
6+<f> =60.
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.
745]
AND THE POLYHEDRAL FUNCTIONS.
193
101. Fig. 1. Pole at
N.P.D.'s
/
Longitudes.
2A
o= 31 43'
0, 180
2A
90 - o = 58 17
90, 270
iA
90
( 0, 180) + a = 31 43'
2A
90 + a= 121 43
90 , 270
2A
180 - a = 148 17
0, 180
2B
P= 20 55'
90, 270
\B
g = 54 44
45 , 135, 225, 315
2B
90 - ft = 69 5
0, 180
IB
90
(90, 270) + = 20 55'
2B
90 +/3= 110 55
0, 180 "
45
180-^ = 125 16
45 , 135, 225, 315
2B
180 -y3= 159 5
90 , 270
10
40
36
(90, 270)+ a = 31 43'
40
60
( , 180 ) + /8 = 20 55
40
72
(90, 270)+a = 31 43
40
90
0, 90, 180, 270
40
108
(90 , 270)+a=31 43
40
120
( 0, 180) + 0=20 55
40
144
(90, 270) + a =31 43
10
180
C. XI.
25
194 ON THE SCHWARXIAN DERIVATIVK
102. Fiff. 2. Pole at A.
[745
x N.P.D.'s
Longitudes.
A
5A
2a= 63" 26'
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
5B
90 - a + ft = 79 12
36 108 180 252 324
5B
90 + a - = 100 48
72 144 216 288
SB
180 - y = 142 38
72 144 216 288
J
a= 31 43
72 144 216 288
59
90 - a = 58 17
36 108 180 252 324
ioe
90
(36 108 180 252 324) + 18
58
90 + a = 121 43
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.
N.P.D.'s
Longitudes.
ZA
y= 37 22'
30" 150 270
3A
90 -a + /3= 79 12
90 210 330
3A
90 + a - /3 = 100 48
30 150 270
3A
180 - y = 142 38
90 210 330
B
i
ZB
2/3= 41 50
90 210 330
65
x= 70 32
(30 150 270) + 3 = 37 46'
6/y
180- x= 109 28
(90 210 330) + 5 = 37 46
35
180 -2/3=138 10
30 150 270
K
180
30
/3 = 20 55
90 210 330
60
^ = 54 44
(90 210 330) + <f> = 22 14'
30
90 -/3= 69 5
30 150 270
60
90
60 120 180" 240 300
30
90 +/3 = 110 55
90 210 330
60
180-0 = 125 16
(30 150 270) + $ = 22 14'
30
180 -/3= 159 5
30 150 270
252
196 ON THE 8CHWARZIAN 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 8, 6 axes each through a pair of opposite
points A, and 10 axes each through a pair of opposite points B; or say 15 0-axes,
10 .B-axes and 6 .A-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 J.-axis 4 positions, and in
respect of each .B-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 cosy, cos*/, cos A of
the cosine-inclination of an axis of rotation to the axes of coordinates, and thence
calculate the values of
X, p t v = tan \^i cosy tanj^cosjr, tan J^ cos h,
and thence the values of
A, S, C, D = v i, \ + ifi, \-ifi, v i;
viz. in the case of a 0-axis, ^ is = 180, (so that here tan ^ = 00, or the values of
A, B, C, D are = v, \+ift,, \ ip, v, that is, cosA, cosf+icosg, cosf-icosg, cosh);
in the case of a .B-axis, the values are ^ = 120, 240, and therefore tan^=V3;
and in the case of an 4-axis, they are ^ = 72, 144, 216, 288, and therefore
V10 + 2V5 V10-2V5
tani=+ r= 5 , 7^ .
V5 - 1 \A> + 1
105. The 0-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) -H (Cx +D)
2
3
4
5
6
7
8
9
10
11
11
13
14
15
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
2
2
2
2
2
2
2
2
2
2
2
2
-i
-1
1
-i
i
1
-1
i
3 + J5)
1-^5)
3 + N /5)
l-v/5)
3 + v/S)
3 + v/S)
1-J5)
-l-v/5 + i(
1-^/5)
l + v/5 + i
1--/5)
3+v/5)
3 + v/5)
oi> . e<J
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
aeb
adc
acd
adb
abd
bed
bdc
aec
ace
bed
bde
bee
bee
aed
ade
cde
ced
aeb
abe
abcde
acebd
adbec
aedcb
adceb
acbde
aedbc
abecd
acbed
L98
ON THE 8CHWARZIAN DERIVATIVE
[745
9
9
3
2
2
9
2
1-V6)
-9
-2
-2
-2
-2
-2
3+^/5)
1-VS)
abdce
aecdb
adebc
aecbd
acdeb
abedc
adbce
aebdc
abced
adecb
acdbe
abdec
adebe
ached
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.
(Ax
Groups of 12 and 24. Pole at .
+ B) -T- (Cx +D)
1
1
1
I
-1
o.
1
3
1
1
4
1
-1
6
-i
i
1
1
6
-1
i
1
t
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
t
1
14
- i
1
16
t
1
16
t
-1
17
1
-1
1
1
18
- 1
-1
1
i
19
i
1
1
t
20
1
1
1
-1
21
-1
-1
1
-1
>>
i
-1
1
-t
23
1*
1
1
-i
24
-1
1
1
1
1
aft . nl
ac.bd
ad .be
abc
acb
ode
acd
Hill,
abd
bed
bdc
adbc
acbd
cd
ab
acdb
bd
abed
be
abdc
ac
adcb
ad
745]
AND THE POLYHEDRAL FUNCTIONS.
199
108. The group of 60 was obtained in the .A-form by Gordan in his paper.
The passage from the -form to the A-form is made as follows: let X, Y, Z be
the coordinates of a point when the axes are as in the 0-form, X lt Y^, Z l the
coordinates of the same point when the axes are as in the A -form : we may write
where
X, Y, Z = bX l -aZ 1 : Y, :
/5
' V
a, b =
O
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
aX. + bZ^L : M : N;
or taking these to be
we may write
these values are such that
+ a^, : Y,
X, : Y,
L lt M lt N^
A* + M?
: M, : N lt
, M, -ai +
and hence, \, /i, v and Xj, /*,, i>, being the rotations, we may write
where ^ has the same value in each set of equations. From the equations
A : B : C : D = v i : X + ifj, : \-ijj, : v i,
B + C : B-C : D-A : D + A = \ : i/j. : v : -i
we have
and similarly
D i /"
Jj\ i" ^
Hence we may write
l -C,
= L : iM : N : -
, + A, = Z, : iM, : N t :
B-C,
A
D + A;
or say,
ei(D-A)-(B-C),
which are the values for a transformation (A lt B r , C lt D,) in the .4 -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 + 1 sin 72, we have
V5 -
5 + V5
*
where the upper signs belong to e, e 3 and the lower to e 4 , e 3 . It may be remarked
that
/5 + V5 1_ /5 - V5 b _ V5 + 1
"' b~V" 2 ' a~ 2
1
a
a
b :
1
For instance, we have in the -group (A, B, C, D) = (-l, 0, 0, 1); ab.cd: and thence
in the 4 -group A lt B lt C,, A = (-2b, 2a, 2a, 2b); ab.cd: or say this is
(-1, |, , l), =(-1,
, 1);
which in the Table is given as (- e 3 , r' + e 4 , r'+e 4 , *); 06. 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.
6No. (At +B) -HC* +D)
1
1
1
1
1
2
4
-1
1
(( . lii-
3
13
-6*
1
ac . be
4
9
->
1
ae.cd
5
10
-e 2
1
ab.de
6
14
e
1
bd .ce
7
6
t+t*
t*
1
-(+)
ae .be
8
5
t+<?
1
e
-(, + a
bc.de
9
16
f + r 1
e 3
- (*')
ac .de
10
3
f + e 3
e 2
t 2
-(f + e 3 )
ac .bd
11
15
f + t>
"
e
-(e + e 3 )
ae .bd
12
12
-1
e+f"
e a + <
1
ab .ce
13
11
-e
* + !
r + e
c
be.cd
14
7
- J
1+e 3
r' + e 4
e s
ad. ce
15
2
->
r>+e
e a +e<
e 3
ab .cd
16
8
-t*
*+
e^+e*
e 4
ad. be
745]
AND THE POLYHEDRAL FUNCTIONS.
201
17
21
e 3 + l
e
1
-(e + e 3 )
18
35
e 3 +l
e 2
e<
- (e + e 3 )
19
30
e>+l
e 3
e 3
-(e + e 3 )
20
34
e 3 + l
e 4
e 2
-(e + e 3 )
21
19
e 3 +l
1
e
- (e + e 3 )
22
33
e + e 4
2
1
-(e + e 3 )
23
20
e + e*
e 3
e 4
-(e + e 3 )
24
22
e + e 4
e 4
e 3
-(e + e 3 )
25
36
e + e 4
1
e 2
-(e + e 3 )
26
29
e + e 4
e
e
-(e + e 3 )
27
31
e
e 2 + e*
e 2 + e 4
1
28
17
-e 2
e 4 + e
e 3 +e 4
e
29
27
-e 3
e+e 3
e 2 + e 4
e 2
30
25
-e 4
e 3 + l
e 2 + e 4
e 3
31
23
-1
1 +e 2
e 2 + e 4
e 4
32
24
-e 4
1+e 2
e 2 +e 4
1
33
32
-1
eS+e 4
e 2 + e*
e
34
18
e
e< + e
e 2 + e 4
e 2
35
28
-e 2
e +e5
e 2 +e 4
e 3
36
26
-f
e* + l
e^ + e 4
e 4
37
44
e
1
38
43
e 2
1
39
42
e 3
1
40
41
e 4
1
41
38
e 3 +e 4
1
1
-(e + e 3 )
42
46
e 2 +e 4
e
e 4
- (e + e 3 )
43
58
e 3 +e
e 2
e 3
-(e + e 3 )
44
55
e^e 4
e 3
e 2
-(e + e 3 )
45
50
e>+t*
t 4
c
-(e + e 3 )
46
51
1+e 3
e 3
1
-(e + e 3 )
47
39
1+e 3
e 4
e 4
-(e + e 3 )
48
47
1+e 3
1
e 3
-(e + e 3 )
49
59
1+e 3
e
e>
-(e + e 3 )
50
54
1 +e a
e'-:
e
-(e + e 3 )
51
56
-e 3
e 3 + l
e 2 +e 4
1
52
49
-e 3
1+e 2
e 2 + e 4
e
53
37
-e 4
e- + e 4
e 2 + e 4
e 2
54
45
-1
4 +
e 2 +e 4
e 3
55
67
-e
e+f>
e s +e 4
e 4
56
48
-e 3
e 4 + e
e 2 +e*
1
57
60
-e 4
e+e 3
e 2 +e
e
58
53
-1
e 3 + l
e 2 + e 4
e 2
59
52
-e
1+e 3
e 2 + e 4
e 3
60
40
-e 2
.s+e 4
e 2 + e 4
e 4
aeb
bee
ced
ode
cde
aed
abd
dbe
bee
aed
abc
bed
aec
bed
bdc
ode
acb
bde
ace
abecd
aedbc
acbde
adceb
acebd
abdce
adcbe
adecb
acdeb
abedc
adbec
aecdb
aebcd
abced
acdbe
aecbd
abcde
acbed
abdec
adebc
acedb
aebdc
adbce
aedcb
C. XI.
26
20-2
ON THE SCHWARZIAN DERIVATIVE
[745
1 1 1. Selecting the transformations which correspond to the positive substitutions
/. and completing the group of 24 we have
Homographic Transformations. The groups of 12 and 24. Pole at A.
(A* +B) -HC +D)
I
1
1
1
9
-1
1
ad . be
3
+ >
'
f
-(< + *)
in- . lul
1
-e
" + *
>+
e 3
ab.cd
5
->
e +e
+ ^
e
a l>c
6
-t
f+e 4
e'+e*
( 5
acb
7
e+e
*
t*
-(e + ^J
acd
8
c+l
1
e
-( + )
adc
9
t+e*
t 4
e 3
-(e + ^)
abd
10
e+l
6
1
-(+*)
adb
11
-1
l + c-
* +
^
bed
12
-e 4
1+f 2
r' + e 4
1
bdt-
18
1
l + 2e<
1-fft
-1
alt
14
-= +
l + f + 3e*
-l-3-<
-
cd
15
?-(*
3 + t + r 1
-l-Se-t 3
-* + (*
ac
16
-l+e
-l-f 3 +2 4
l + e-2e
!-<
bd
17
2+ 3 + 2 4
-2-2e 2 -e 3
26 + ^+26*
2 + 2+ 3
ad
18
2+2e"+ s
2 + e + 2t 4
-2e-2fi>-
2e + t+2 4
be
19
-2 + e + r"
- + e>
-f + e 3
6 + ^-2^
abed
20
1
-1
1
1
abdc
21
1
1
-1
1
acdb
22
1 + e + Se 4
e 2 -r
*-
l + 3 + e 4
acbd
23
l + 2e 4
-1
-1
-l-2e
adbc
24
3 + e + e 3
-r' + t 4
-^tt 1
l + 3f + 3
adfb
As an example of the calculation we have (A, B, C, D) = (0, i, I, 0); ab. Hence
a a
The second and third coefficients are
V5 + 1 _. /5 + V5 V5
which, in virtue of the values of e and e 4 , are =l + 2e* and 1 + 2e respectively: or
the result is as above (1, 1 + 2* 4 , 1 + 2e, -1).
745]
AND THE POLYHEDRAL FUNCTIONS.
203
112. In like manner for the passage from the 0-form to the .B-form, if X, Y, Z
be the coordinates of a point on the spherical surface in regard to the B-axes,
X.,, Fj, Z? those of the same point in regard to the S-axes, we may write
where
X : Y : Z=
a, b =
o - 1 V5 + 1
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
X, : bF 2 + a^ 2 : -&Y t + \>Z,=*L : M : N,
F 2 : Z. 2 = L* : M, : N,,
, = L : WI-aN :
or calling these
X,
we have
Z, 2) M,,
these values are such that
Lf + M* + N 3 * = L- + M* + N-,
or X, fj., v, Xj, fj,. 2 , v., being the rotations, we have
L, M, N=**\, V. *v', L, M a , N 3 = *>\.,
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 : -i^,
B* + C, : B, - C, : D, - A, : D, + A t = L, : 2M, : N, : - to,
and hence
and thence
B, + C, = B + C,
B i -C i = b(B-C)-ai(D-A),
A-A = -ai(-B-(7) + b (D - A),
A + A,, = D + A;
A,= ai(5-6')-b (D-A) + (D + A),
2 = b (B-C)- &i(D-A) + (B + C),
then
113. As an example of the transformation, take
(A, B, C, D)= ^2, -3 + V5 + i(l-V5), -3 + V5 + i(-
B-C, B+C, D-A, D + A=i(l-J5), -3
5, -2, 0;
[bc.de]:
262
204
and thence
ON THE St'HWARZIAN DERIVATIVE
[745
, t(6-2 V5) + 2
viz. multiplying by 2V3, these are
that is,
or since
2 + V3 = - 2iw and - 2 + V3 = 2iar,
dividing by 4 these are
as in the table.
114. Homographic Transformations. The group of 60. Pole at B.
(At
+B)
-MC
+D)
-8,
1
1
1
1
9
1
1
ac . bd
8
u
1
ae.bd
4
U*
1
bd.ce
6
2
M 3-V5)
i( -3 + ^/5)
-2
ab .cd
6
2
M-3-V5)
t( 3 + v/S)
-2
ad. be
7
2
( S-^/oJw
i( -3 + j5)ur
-2
be ., it-
8
2
i(~3-J5)u
i( 3 + ^/5)^
-2
be .cd
9
2
i( Z-J5)v'
t( -3 + V5)"
-2
ad.be
10
2
(-S-VS)" 2
t( 3 + ^/5) w
-2
ab . de
11
2
<-^/3-tV5)<-
(-x/S + iVS)^
-2
ab .ce
12
2
-v/3-tV5
-x/3 + tV5
-2
ac. lie
IS
2
(-v/S-t^SJw 2
(-v/3 + tV5)ft>
-2
at .be
14
2
X/3-JV5
v/3 + i^S
-2
ac .de
15
2
( V3-'V5)
( v/S + iVSJw 2
-2
ad. ce
16
2
( VS-'VS)" 2
( J3 + iJ5)u
-2
ae .cd
745]
AND THE POLYHEDRAL FUNCTIONS.
205
17
H
1
ace
18
ur
1
aee
19
x/3-iV5
2
-2
V3 + f V
6ed
20
-Jt-ijl
2
-2
-^3 + iJS
Me
21
-^3-tVS
2o> 2
-2u>
-v/3 + V5
bdc
22
x/3-iVS
2 3
-2u
x/3 + i^/o
bed
23
-v/3-iV5
IM
-2u-
-V 3 + tN/ 5
abd
24
tfl-iJS
in
-2u-
v/3 + iVS
adb
25
2ur
-v-w
-x/3 + iv/o
-2w
abc
26
2u>
-x/3-1^5
-V3 + i v '5
-2w 2
acb
27
2a"
-V3-W5
(-V/3 + 1V5)" 11
-2
abe
28
2
-x/3-i^/S
(-v/3 + iV5)w 2
-2U 2
aeb
29
2w
v/3-iVo
V3 + tV 5
-2i^
acd
30
2ur
v/3-,V5
,/8+V
-2w
adc
31
2or
^3-iV5
( Vs+WS)^
-2
ode
32
2
x/3 - i ^5
( ^3 + <V5)u-
-2w=
aed
33
2
-v/3-ix/S
(-x/3 + iV5)w
-2u
bee
34
2w
- ^3 - i ^/o
(-V 3 + V5)w
-2
bee
35
!
V3-W 5
( V 3 + W 5 ) W
-2
cde
36
2
v/3-.^/o
( V3 + \/ 5 ) w
-2u
ced
37
2
i( S-VSJu 2
i(-3 + V5)
-2U 2
adceb
H
-v/3-tV5
+ 2w a
-2
(-V 3 +V 5 )w 2
acbde
39
VS-iJS
2
-2w
( x/3 + 1 x/5) w
aedbc
40
2
( 3-V5)
i (-3 + ^5)0-
-2u
abecd
41
2
f( S-^/oJu
f(-3 + v/5)
-2w
aedcb
42
-VS-iVS
2u
-2
(-V3 + tV 5 ) u
adbec
43
V3-W 5
2
-2W 2
( VS+iJSJw 2
acebd
44
2
i( 3-^/5)
i(-3 + v /5)u-
-2u 2
abcde
45
2
( 3-^5)^
it-S + x/SJw 2
-2w
adebc
46
^3-iVS
2u=
-2u 2
( x/ 3 + *\/ 6 ) w
aecdb
47
-JS-i^S
IM
-2w
(-^a+t^/sju 2
abdce
48
2
i( 3-^5)^
i (-3 + ^/5)0)
-2w=
acbed
49
2
i (-3-V5)*
i( 3 + ^/5)0)
-2w s
acdeb
60
^3 - i ^5
2 U
-2o>
( v/3 + W 5 )^
adbce
51
-^S-iJS
2u=
-2w-
(-Vli + iv/Sju
aecbd
52
2
if-S-^SJfcT
i( 3+^/5)w-
-2w
abedc
53
2
i(-3-V5)ttf
i( 3 + v/5)
-2w
atibcd
54
-x/3-iV5
2u>
-2
(-x/S + tx/S)"
abdec
55
v/S-i^S
2
-2ur
( VS + 'VS)" 2
ucedb
56
2
it-S-^/S)
i( 3 + V5)w 2
- 2u-
adebe
57
2
i(-3-v/5)
'( 3 + ^5)01
-2u
adecb
58
-v/S-t^S
2
-2u
(-x/3 + iV5)w
aebdc
59
J3-IJ5
2ur
-2
( N / 3 + '\/ 5 )< tf2
acdbe
60
2
iC-S-^S) w*
i( 3 + ^/5)
-2U 2
abced
206 ON THE 8CHWABZIAN DERIVATIVE
115. We hence derive
Monographic Transformations. The groups of 12 and 24. Pole at B.
(A. +B) +(C* + D)
[745
1
1
1
1
>
2
i( S-,/5)
t(-3+V6)
-2
ill: . Cd
9
1
1
ac . bd
4
2
.(-3-^/5)
i( 3 + ^/6)
-2
ad . be
6
2*>'
-N/8-tVS
-N/3 + JX/5
-2u
abe
6
IM
-x/S-i^S
s/3 + iv/S
-2"
aeb
7
-^3-tVS
2
-2w
-^3 + iV5
abd
8
s/3-iVS
IH
-2w 3
v/3 + 1^5
adb
9
M
s/3-tVS
J3 + tV5
-2u
acd
10
2
s/3-1^5
J3 + rV5
-2w
ode
11
J-iJB
2w2
-2w
^3+iV 5
bed
12
- x/3 - 1 ^/5
2u
- 2w
-N/S + i^S
bdc
18
2
N /3( l + x /5)+ (-3-^5)
v /3( l + JSJ + tJ 3 + ^/5)
-2
ab
14
2
^(-1-^5)+ (-3-^/5)
^(-l-^/SJ + if 3 + v/S)
-2
cd
15
N/5
-i
t
-V*
ac
16
1
v/5 '
-<^8
-1
bd
17
2
^(-l + ^/SJ + tt 3-^5)
^(-I+VSJ+M-S+N/S)
-2
ad
18
2
V3( l-J6) + i( 8-s/S)
^3( l-J5) + i(-3 + J5)
-2
be
19
1
i
i
1
abed
30
1
-/
-i
1
adcb
21
^/3( l-V6) + ( 3 + ^5)
2
-2
V3( l-V5) + i( -8 + ^/6)
abdc
22
JS( l+v/oJ + tf-S + JS)
2
-2
^3( l + v /5) + i( 3 + ^/5)
acbd
23
s/Sf-l + ^ + 't 3-JS)
2
-2
^(-l + ^SJ + it-S + ^S)
acdb
24
Jt(-l-J5) + i(-9-J5)
2
-2
J3(-l- N /5) + i( 3+^/6)
adbc
116. I give also the group of 12, (abce), slightly modifying the form: viz. I
write first V3 + i Jo = 2 */2k, and therefore
.-r: then for x I write \x,
fc
and divide the A and B by X : the A and B then contain , and the C and I)
A,
X k \
contain j , and assuming - = i, we have j- = i. For instance, in the transformation
K A, K
corresponding to abc, the Ax + B and Cx + D,
and (-^3+ 2
become first 2o) s a; 2 V2A:, and 2 \/2 r x 2o>, and then (omitting also the factor 2)
K
k \ k
a>"a; V2 - and V2 T a; w, viz. when - = i, they are ufx i \/2 and a; . i V2 at ; that
An/ A,
is, the values of A, B, C, D are m 2 , i'V2, V2, - w. The group is
745]
AND THE POLYHEDRAL FUNCTIONS.
Group of 12. Pole at B.
207
1
1
1
u
1
ace
u 2
1
aec
1
-*/*
i<i N /2
-or
abc
1
-iuV 2
iwV 2
(0
act)
1
-i- WN / 2
V2
6)
abe
1
-W>
iV V 2
-U 2
aeb
1
-iuV'2
V2
-w 2
bee
1
-.V 2
) w ^2
- w
bee
1
-iuv/2
J'w V 2
-1
ab.ce
1
- ^/2
10) V 2
-1
ae . be
1
-W2
'V2
-1
ac .be
117. From the Table of the Groups of 12 and 24, -form, it appears that the
group of 12 is
l) -i(x-l) i(x+l) -i(x+l)
X ' x' X> x'
#+1 X 1 X 1
i x i (x + f) (x i)
' ei ' x+i '
xi' x + i'
and if we proceed to form the product of the twelve factors s x, s -- , s + x, &c.,
we have first the three products
-. --,.
a?
= s 4 + aw 3 + 1 ;
if for shortness
x+l\* /x + i\* /ic-tV
; s 5 - - ^ .s 3 - -- .)
x-lj \x-^] \x + ij
s 4 -f /9s 2 + 1 ; s 4 + 75" + 1 ;
Qa-'+l
The product of the three quartic functions is
= (s 4 + iy + (s 4 + ly s- (a 4- /3 + 7) + (s 4 + 1) s 4 (/3y + 7 a + a/3) + s 6 .
and we have
*(- 1)'
Hence the product is found to be
= (" - 33s 8 - 33s l + 1) - s 2 (s 4 - I? .
- 33s 8 - 33s 4 +1)
x- (a* - I) 2
- Wo* -
a? (a 4 -I) 2
208 ON THE SC'HWARZIAN DERIVATIVE [745
which is
f" - 38*- 33** + 1 _ <c"-33s'-38E' + l)
"(-!) of^-iy ['
We thus verify that the twelve transformations a: into x, into -, &c., give each of
them a transformation of the function
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, : \ i/j, : v i,
and thence
X :/t:i>:l = B + C : - i(B - C) : D - A :i(D + A),
that is,
\, M , v = -i(B+C), - (B-C), -i(D-A); +
The equations of the axis thus are
x ly
B + C~B-C~ D-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 8-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 a + ? - 1) - 2 (z + 1) (Lx + My + Nz) = 0,
and hence, writing z = 0, we find
N (a? + y- - 1 ) - 2 (Lx + My) =
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 B-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]
the equations are
AND THE POLYHEDEAL FUNCTIONS.
209
= 0,
= 0,
= 0,
(ab . cd)
(ac . bd)
(ad . be)
y-o,
(-1-
= 0,
= 0,
(ae . be) ft [( 3 \
(ab . ce) ft [( 1 \
(ad.be) and similarly for the other circles,
(at . de)
(ae . bd)
(ad . ce)
(ae . cd)
(ac . de)
(6c.de)
(6e . cd)
(bd . ce).
121. Observe that the arrangement is in sets of 3 planes, or circles, intersecting
at right angles. One of the circles is the circle ft, = a? + y' 2 1, =0 corresponding to
the equator, and two of them are the right lines x = and y = 0. The equations of
the remaining 12 circles may be written in the somewhat different form
ft + (V5 - 1) [y - i (V5 - 1) *] = 0,
ft - (V5 - 1) [y - i (\/5 + 3) x] = 0,
ft - (\/o + 3) [y + (V5 -!)#] = 0,
ft - (V5 - 1) [y - i (V5 - 1) *] = 0,
fl + (V5 - 1) [y - i (V5 + 3) *] = 0,
n + (V-5 + 3) [y + \ (V5 -!)] = 0,
H + (V5 - 1) [y + ^ (V5 -!)]= 0,
n - (V-5 - 1) [y + | (V5 + 3) ] = 0,
ft - (V5 + 3) [y - i (Vo - 1) ] = 0,
fl - (V5 - 1) [y + i (V5 - 1) *] = 0,
n + (V5 - 1) [y + i (^5 + 3) a;] = 0,
o\ r ^^ i / /e T \ "] /\
It hence appears that 4 and 4 circles have with O = the common chords y + J(\/5 1)# = 0,
y ^ (^5 1)^ = respectively: and that 2 and 2 circles have with H = the common
chords y + i ( V5 + 3) tc = 0, y - $ (V5 + 3) <c = respectively.
c. xi. 27
210
ON THE SCHWARZIAN DERIVATIVE [745
122. The equations of the 12 circles are, in fact,
n ( V5 - 1) (y * ( V5 - 1) *] = o, n (V5 + 3) [y j (V5 - 1) ] = o,
n (V5 - 1) [y i(V5 -t- s)] = 0:
hence the radii are = -JZ - 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 yHV5-l) = (AA in the figure), and through the
extremities of each describe 2 pairs of circles with the radii V - 1, ^5 + 1 respectively.
Lay down the diameters y \(<J$ + 3)a; = (BB in the figure), and through the
extremities of each describe a pair of circles with the radius 2.
123. For the A -form, the equations of the fifteen planes are at once found to be
y =o,
ad . be
X
+ (e + e 4 )2 = 0,
ac .bd
(e +e 4 )a;
+ * = o,
ab .cd
(e 3 - ) x
i (e- 4- e*) y =0,
ac .be
-(* + *)*
+ 1 (e 2 - e 3 ) y + 2 (e + e 4 ) 2 = 0,
ae .be
a;
+ i (e 2 + e 4 - e - e 3 ) y + 2s = 0,
ab .ce
(e-e 4 )*
-i(e+6 4 )y =0,
ab .de
-(e -He 4 ) a;
+ z(6 -e 4 )2/+2(e + e 4 )2 = 0,
ae .bd
+ (e 2 + e 3 + 2) a;
- i (e 2 - e 3 ) y + 22 = 0,
ad . be
(e - e 4 ) a;
+ I '(e+e 4 )y =0,
ae .cd
- (e + e 4 ) a:
- i (e - e 4 ) y + 2 (e + e 4 ) 2 = 0,
ac .de
(e 2 + e 3 + 2) a;
+ 1 (e 2 e 3 ) y + 22 = 0,
ad .ce
(e'-e 3 )*
+ i(6 2 + 3 )y =0,
bd .ce
-(< + )
- i (e- - e 3 ) y + 2 (e + 6 4 ) 2 = 0,
be .de
a;
- i (e 2 + e 4 - e - e 3 ) y + 22 = 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 x 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 a 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 + e t )z = Q,
(e + e t )x + z = Q respectively. The equations of the circles thus are (e + e 4 ) fl + 2x = 0,
745]
AND THE POLYHEDRAL FUNCTIONS.
211
fl 2 (e+ e*)x= 0, or recollecting that
the equations are
= V5 1 and therefore
--
x- + y 3 - (v/5 - 1) x - 1 = 0, a? + f + (V-5 + 1) x = ;
hence for the first circle the a'-coordinate of the centre is (V5 1) and the radius is
= J \/(10 2 V-5); for the second circle the ^-coordinate of the centre is = (V5 + 1),
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
ia <B 2 = iV3, the equations of the 15 planes become
x
= 0,
ac . 6d
(-3 + ,
/5)y+ 22 = 0,
aZ>. erf
(3 + A
/5)y+ 22 = 0,
rf. be
\
/3*+ A
/5 y + 22 = 0,
ac . be
(1 + \/5) \
'3 + ( 3 - ^
/5) y + 42 = 0,
ab . ce
( 1 + V5) \
f&c + (- 3 - A
/5) y + 42 = 0,
ae . be
* +
/3 y =0,
ae .bd
'3# -f
y + (3 + V5) 2 = 0,
ad. be
\
y + (3-V5) 2 = 0,
ab . de
'3# + '
/5 y+ 22 = 0,
ac . de
(1-V5K
'3* 4- (- 3 - *
/5)y+ 42 = 0,
ad. ce
(1 + V5) \
/3 4- (3 - A
/5)y+ 42 = 0,
ae .cd
x
V3
= 0, bd. ce
(3 + V5) z = 0, be . de
(3 - V5) 2 = 0, be . cd.
126. Of the 15 circles, 3 are the lines x yV3 = 0, =0, a' + yV3=0, viz.
these are lines at inclinations 30, 90, 150 to the axis of x. The equations of the
remaining 12 circles are
n + (3 - V5) y = o,
n - (3 + \/5) y = o,
(3 + V3) fl - 2 (y - V3) = 0,
(3 - V-5) H + 2 (y - a; V3) = 0,
(3 + V5) - 2 (y + x V3) = 0,
(3 - V5) + 2 (y + V3) = 0,
272
212
ON THE SCHWARZIAN DERIVATIVE
[745
viz. these are pairs of circles having, for their common chords with SI = 0, the diameters
at inclinations 0, 60, 120 respectively. And, lastly, we have the circles
0, , 2fi + [(- 1 + V5) V3* + (3 + V5) y] = 0,
127. The first three of these have, for common chords with fl = 0, the diameters
whose equations are
viz. these equations are y = ( 2 + *J5) x V3, y = ^W#,
If, as in a
/O /K /O
foregoing table, 5 = 37 46', sin 6= --,, cos# = -^, and therefore tan 5= ' ; then the
2t \ \ Y O
inclinations of these diameters to the axis of x are respectively 60 6, 6 and
120 -0, or say 30 -(0-30), 30 + (0-30) and 90 -(0-30), where 0-30 = 7 46',
Le. 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 .6-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.
,
t
,
l
*
*2^/2
,.w
'~
J3(V5 + 1)
4
/5 + V5
V i
, 1.
1
* i
3 + N /5
1 + ^5
2^2
1 v/ 2
. 4
4
4
1
..i
1
/25 + 11 v/5
3+^5
"2^/2^/3
"v/2^3
"V 40
* 4^3
1
'
1
/S + v/5
8 V wT
V3
1
..1
1
S V 'W
1
cos- 1 $ = 70 28'
90
cos" 1 -J = 109 32'
cos- 1 4j = 54 46'
90
cos- 1 - * =125 44'
V 3
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 1
Incl. of adjacent faces
Incl. of edge to adjacent face
But we require further data in the cases of the dodecahedron and the icosahedron
respectively.
745]
AND THE POLYHEDRAL FUNCTIONS.
213
129. For the dodecahedron, taking the edge to be =s as before, then in the
pentagonal face
diagonal, g is = s . ^ (\/5 + 1),
altitude, k = s . % V(5 + 2 V5),
segments of do., e = s . J V(10 2 \/5),
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 . J (3 + \/5), meeting at right angles in ; then drawing the double ordinates SB,
each =s, through l and @ 3 respectively, and joining their extremities with 2 and
6 4 : the sides . 2 B and 4 B are then each =k, =s. i V(5 + 2 V5); and inserting
upon them the points A, <t> from the figure of the pentagon, we have several
geometrical relations ; viz. the line A A cuts the parallel sides .B<S) 2 , B t at right
angles, and when produced passes through the intersection of B, and B 4 : we have
OA, OB, = r, R, p respectively: the four points <1> form a square, the side of
which is g, =8.
214
ON THE 8CHVVAKZIAN DERIVATIVE
[745
131. We find also
8
OJf-a^.
MB =
40
2(5 + 2 V5)
It may be remarked that iu the figure J?6 2 , fi@ 4 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,
= 8.^(V-T + 1): the slope-breadths of the triangular faces are e, = s . $ V(10 - 2 \/5),
and those of the quadrangular faces are /, = s . I V(10 + 2 V5) ; the lines represented
by the other lines of the figure are in actual length each = s. We have thus a
745]
AND THE POLYHEDRAL FUNCTIONS.
215
section which is an isosceles triangle, base g, other sides each =/; and the square
of the altitude is thus =/- i^ 2 =i* 2 , o r 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 each
= p =s. ^(1 + \/5) meeting at right angles; and then the distances A.,, A 4 each
= $s; and complete the hexagon. This gives the sides AS lt A 3 each =s.|\/3, the
altitude of the triangular face, side =s; and then, taking QB one-third of this,
we
at
Moreover, joining
a point M: we find
an gl es to A& lt and OA, OB, 0=R, r, p respectively.
and OA.,, we have these lines cutting at right angles in
, . /5 + 2 V5
*V~20~~'
15 + V5
V -HT 1
tit-'ffip -****.
5-V5
10 -
134. It may be remarked that -4,<5> 3 , 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 , A t respectively, we have a pentagonal pyramid, summit
A it standing on the first pentagon, and an opposite pyramid, summit A t , 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.
HIGHER PLANE CURVES.
[From Salmon's 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. 383385.]
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 soy = 0; and reciprocally a point-pair can be represented in line-
coordinates by an equation fj; = 0, but we have to consider how the point-pair can be
represented in point-coordinates : an equation a? = 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 = 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
a 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 i/ = 0, z0 for the given lines, x=0 for the line joining the given
points, and (x=0, y oz = 0), ( = 0, y #2 = 0) for the given points; the equation of
a conic satisfying the required conditions and containing one arbitrary parameter 6, is
a? + 26xy + 20 V() xz + s (y - oz) (y - /3z) = ;
747] NOTE ON THE DEGENERATE FORMS OF CURVES.
or, what is the same thing,
219
and this equation, considering therein 6 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, hQx, y, z) 2 =0,
we have
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 \/(6) : V(c) : V(/) ' 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 determinate equation,
viz. considering the condition imposed upon 0(0= infinitesimal) as a determination of
0, 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 P'Q^ . . . = ; 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 Q> 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 n-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? . . . = ; 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 3 . . . = 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"- = has been considered by Cayley,
Comptes Rendus, t. LXXIV. p. 708 (March, 1872), [515], who states his conclusion as follows:
282
220 NOTE ON THE DEGENERATE FORMS OF CURVES. [747
" there exists a quartic curve the penultimate of a?y* = 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 x = 0, y = 0), six situate at pleasure on the other line x = ; 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 QUAETIC.
[From Salmon's Higher Plane Curves, (3rd ed., 1879), pp. 387389.]
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, 1876, pp. 456 472 ; and see also Weber's Theorie
der Abel'schen Functionen wm 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%
where each of the letters is = or 1 ; there are thus in all 64 such symbols, but they
are considered as odd or even according as the sum aroc' + /3/S' + 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, /S, 7, a', /3', 7' constants chosen at pleasure,
and then a", /3", 7" determinate constants, such that the equations
z+
=0,
a'V; + ff'y + 7"* + + |> + 4 = 0,
a p 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 , rj, f,
each of them as a linear function of (x, y, z) ; and the equations of the bitangents of
the curve V(a ) + V(2^) + V(*?") = ( see Weber, p. 100) are
18
111
111
, = 0,
28
001
Oil
jr-0,
38
Oil
001
,=0,
23
010
010
= o,
13
100
110
77=0,
12
110
100
r-o,
48
101
100
x + y + z = 0,
14
010
Oil
+ y + * = 0,
58
100
101
ax + fiy + yz = 0,
15
on
010
l+^+y-0,
68
110
010
a'x + p^w + y'z = 0,
16
001
101
, + yS' y + y' z = '
78
010
110
f*+fr, +<-.-<
17
101
001
1,^'y^"^
24
100
111
++,-*,
34
110
101
* +y+ r=o,
25
101
110
ax + \ + yz = 0,
p
35
111
100
i
748]
ON THE BITANGENTS OF A QUABTIC.
223
26
111
/ "n ' n
001
ft' 7
36
101
Oil
a' x + ft'y + -, =0,
fl7
Oil
a"*+4> + 7 "*=0
101
37
001
f
a"x + @"y + -77 = 0,
111
7
67
100
100
X I, Z
-o,
1 - #y 1 - 7 a 1 - a/9
57
110
Oil
x y z
= 0,
1 S*~* 1 'a' ~^~ 1 'ff
56
010
111
x y z
= 0,
l-/3" 7 " l- 7 "a" 1-a"^"
45
001
001
, 7
f
a(l-/8y) * /9(l- 7 o) ' .
y (1 - a/3)
46
Oil
110
^
?
^'(l-aW
47
111
010
h ^ +
f
"/I *." iO"\
( X ot o j
= 0,
= 0.
The whole number of ways in which the equation of the curve can be expressed
in a form such as V(#) + ^(yi) + V(f ) = 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 13
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 1 1 , viz. (see p. 233) the four bitangents are such
that their points of contact are situate on a conic.
224
749.
SOLID GEOMETRY.
[From Salvions Treatise on tfie 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 THEORY OF RECIPROCAL SURFACES.
[From Salmon's Treatise on the analytic geometry of three dimetisions, (3rd ed., 1874),
pp. 539550.]
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.
B, number of nodal edges of the cone.
K, number of its cuspidal edges.
p, class of nodal torse.
a-, class of cuspidal torse.
b, order of nodal curve.
k, number of its apparent double points.
f, number of its actual double points.
t, number of its triple points.
j, number of its pinch-points.
q, its class.
c, order of cuspidal curve.
h, number of its apparent double points.
Q, number of its points of an unexplained singularity.
%, number of its close-points.
C. XI. 29
226 ON THE THEORY OF RECIPROCAL SURFACES. [750
<u, number of its off-points.
r, its class.
, number of intersections of nodal and cuspidal curves, stationary points on
cuspidal curve.
7, number of intersections, stationary points on nodal curve,
t, number of intersections, not stationary points on either curve.
C, number of cnicnodes of surface.
B, number of binodea.
And corresponding reciprocally to these:
n', class of surface.
a', class of section by arbitrary plane.
8', number of double tangents of section.
K, number of its inflexions.
p, order of node-couple curve.
a ', order of spinode curve.
b', class of node-couple torse,
if, number of its apparent double planes.
f, number of its actual double planes.
t', number of its triple planes.
j', number of its pinch-planes.
tf, its order.
c', class of spinode torse.
h', number of its apparent double planes.
ff, number of its planes of a certain unexplained singularity.
X', number of its close-planes.
to', number of its off-planes.
;', its order.
ff, 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.
i, number of common planes, not stationary planes of either torse.
(7, number of cnictropes of surface.
B 1 , 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 a- 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 tr : 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 <r.
G02. In regard to the nodal curve b, we consider k the number of its apparent
double points (excluding actual double points) ; f 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 = V b 2k 2/ 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, %, a) 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 = c 2 - c - 2& - 3/9.
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
292
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 t 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 formulae for a (n - 2), b (n 2), c (71 2), we have to write
K B instead of *, and the formulae are further modified by reason of the singularities
6 and to. So, in the original formulae, for a(n 2)( 3), b (n 2) (n 3), c (n 2) (n 3),
we have instead of 8 to write B G 3<a, and to substitute new expressions for
[06], [oc], [be]; viz. these are
[ab] = ab 2p j,
[ac] = ac 3<r x <a,
[be] = be - 3/3 - 2 7 - i.
The whole series of equations thus is
(1) a' = a.
(2) /'=/
(3) i'=i.
(4) a = n(n-l)-2b-3c.
(5) *' = 3n(w-2)-66-8c.
(6) 6-' = iw(-2)(n 2 -9)-(n 3 -?i-6)(2& + 3c) + 26(&- l) + 66c + fc(c- 1).
(7) a (n - 2) = K - B + p + 2<r + 3ta.
(8) 6(w-2)= p + 2/3 + 87 + St.
(9) c(?i-2)= 2<r + 4/3 + 7 + <9 + a>.
(10) a(n-2)(n-3) = 2(-(7-3a) + 3(ac-30--x-3a>) + 2(a6-2/> -j).
(11) 6(n-2)(n-3) = 4& + ( a b-2p-j ) + 3(6c -3/3- 2 7 -i).
(12) c(n-2)(n-3) = 6A + (ac- 3<7- x -3w) + 2(6c-3/9- 2 7 -i).
(13) q = b>-b-2k-2f-3y-6t.
(14) r = c a -c-2A-3 / 8.
750]
ON THE THEORY OF RECIPROCAL SURFACES.
229
Also, reciprocal to these,
(15) o'= /i'('-l)-26'-3c'.
(16) * =3ft'(n'-2)-66'-8c'.
(17) 8 = $' (' - 2) (n'- - 9) - (w' 2 - n' - 6) (26' + 3c') + 26' (6' - 1) + 66V + f c' (c ; - 1).
(18) a (' - 2) = /c' - B' + p + 2<r' + 3m'.
(19) 6'(n'-2) = p' + 2# + 87' + 3f.
(20) c' (' - 2) = 2o-' + 4/3' + 7' + 6C + '.
(21) a (n' - 2) (' - 3) = 2 (8' - C' - 3o>') + 3 (a'c' - 3<r' - x ' - 3o>') + 2 (a'6' - 2p' - f ).
(22) 6' (B' - 2) (n' - 3) = 4' + (a'b'-2p'-f ) + 3(6'c'-3/8'-2 7 '-0.
(23) c' (n' - 2) (n' - 3) = 6A' + (a'c'-. So-' - x '- 3a)') + 2(6'c'-3/3'-2 7 '-0.
(24) ? ' = 6' 2 - 6' - 2i' - 2/' - 3 7 ' - 6f.
(25) r' = c' 3 - c' - 2A' - 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(n-
(27)
iu each of which two equations S is used to denote the same function of the accented
letters that the left-hand side is of the unaccented letters.
(28)
71 - 2) (1 In -24)
+ (- 93w + 252) c
+ 22(2/8+37+30
+ 27(4/3+ 7 + 0)
- 24(7 - 285 - 27j - 38 X - 73w
+ 4C" + lOtf + 7 + 8' - 4o>'.
Or, reciprocally,
(29)
2'(n'-2)(lln'-24)
+ (-66' + 184)6'
+ (- 93n' + 252) c'
+ 27(4/3'+ y'+ff)
- 246" - 285' - 27?" - 38 X ' - 73w'
+ 4(7
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) (n - 2) (n - 3) - (n - 3) (b + c) + $(q + r) + 2t +/3+f i+\-tf,
609. The equation (28) (due to Prof. Cayley) is the correct form of an expression
for &, 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 a as
standing for its value n(n- 1)- 26 - 3c, we have from the first 25 equations
6
a
2,
+ 2
3 - c
-
= 2,
- 2
o(-
2)- + fl-p-
2o--3
= 2,
- 4
6(n-
2)-p-2-3 7
-3t
= 2,
-6
c (n
2)-2r-4/8-7
-6-
<u =2,
+ 2
n + K
_o-_2C-4fi-
2J-3
x -3 = 2,
-3
2g-S
o + B + i
= 2,
- 2
3r+c
_5 <7 _ / g_4^ +
X-"
= 2;
multiplying these equations by the numbers set opposite to them respectively, and
adding, we find
- 2w s + 12w 2 + 4-n + b (I2n - 36) -f c (12w - 48)
_ Qq - Q r - 4(7- IOB - 41/9 - 30 7 - 24< - 7j - 8 X + 20 - 4w = 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 3n 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
'= o(o-l)-28-3,
c' = 3a(a-2)-6S-8,
b' = ia(a- 2)(a - 9) - (a 2 - a- 6) (28 + 3) + 28(8 - 1) + 68* + *(- 1).
From (7), (8), (9), we have
(a- b- c)(n-2) = K -B-6/3-4,y-3t-0 + 2(0,
750] ON THE THEORY OF RECIPROCAL SURFACES. 231
substituting these values for K and B, and for a its value = n(n 1) 26 3e, we
obtain the values of ', c, b' ; viz. the value of ' is
n' = n (n - 1)- - n (76 + 12c) + 46 a + 86 + 9c- + 15c
- 8k - I8h + 18/3 + 12y + I2i - 9t
- 2C - 35 - 36>.
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) = n 2 - n -f (- 4w + 6) = a + 26 + 3c + (- 4n + 6) ;
making this substitution in the equations (10), (11), (12), which contain (n 2)(?i 3),
these become
a (- 4 + 6) = 2 (8 - C) - a- - 4/> - 9<r - 2j - 3 X - low,
6 (- 4tt + 6) = 4fc - 26 J - 9/3 - 67 - 3i - 2/a - j,
c (- 4ra + 6) = 6A - 3c 2 - 6/3 - 4? - 2i - 3<r - x - 3,
which are the foregoing equations (0); adding to each equation four times the corre-
sponding equation with the factor (n 2), these become
a 2 - 2a = 2 (S - G) + 4 (K - B) - a - 2j - 3x - 3w,
26 s - 26 = 4 - + 67 + 12 - 3i + 2p - j,
3c J - 2c = 6A + 10/9 + 40 - 2i + 5a - x + co.
Writing in the first of these a" 2a = n' + 28 + 3 , and reducing the other two by
means of the values of q, r, the equations become
n' - a = - 2(7- 45 + K - a - 2j - 3^ - 3w,
40 + to,
which give at once the last three of the 8 2-equations.
The reciprocal of the first of these is
<r' = a-n + K- 2/ - 3 X ' - W -
viz. writing herein
a=tt(w-l)-26-3c and /c' = 3n(n -2) - 66 -8c,
this is
o-' = 4n (n - 2) - 86 - 1 Ic - 2j' - 3 X ' - 2C" - 4B' - 3w',
giving the order of the spinode curve ; viz. for a surface of the order n without
singularities, this is = 4n (n 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 b ( 4n + 6) add twice the value of b (n - 2) ; and to twice the value of
c ( 4n + 6) add three times the value of c (re - 2), thus obtaining equations free from
p and <r respectively; these equations are
b (- 2n + 2) = 4& - 26 - 5/3 - Si + Qt -j,
c (- 5n + 6) = 12/t - Gc 3 - 5 7 - 4t - 2 X + 36 - 3,
equations which, introducing therein the values of q and r, may also be written
6(2n- 4) =2q+ 5/9 + 67 + 6< + 3i +j + 4/,
c (5n - 12) + 30 = 6r + 18/3 + 5-y + 4t + 2 X + 3o>.
Considering as given, n 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 /8, 7, 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
6, , o>, 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, CQP, Q) 2 = 0, and the first equation is
b (- 2n + 2) = 4/fc - 26" + 6 -j ;
or, assuming t=0, say ;'= 2 (n 1)6 26 2 +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, C%P, Q) 2 = 0, where AC-B l = MP + NQ; and the
second equation is
c(- on + 6)= 12A- Gc 2 - 2 X + 30- 3a>,
or, say 2^ + 3w = (5n-6)c-6c 2 + 12/t + 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, 6, x : A y, i : C, B,
then of the 26 relations, 17 determine the 17 quantities
a, S, K, p, <r :j, q : r, to
n':a',V, 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', j', q : h', ff, X ', a,', r' : ft', y' : C', H.
750]
ON THE THEORY OF RECIPROCAL SURFACES.
233
Taking then further as given the 5 quantities j', %', a>', C', B',
equations (18) and (21) give />', a-',
equation (19) gives 2/3' + 87' + 3',
(20) 4/3'+ y'+ff,
(28) P + W,
so that, taking also t' as given, these last three equations determine /3', 7', 0' ; and
finally
equation (22) gives k',
(23) /,',
(24) ? ',
(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 l),
n
a
ff
K
V
k'
= n (n - 1) (n - 2),
= n (n - I)",
= n(n 1),
= $n (n - 2) (n 2 - 9),
= 3n (n - 2),
= \ n (n - 1) (n - 2) (n 3 - n 2 + n- 12),
= n (n - 2) (n 10 - 6n + 16n" - 54n 7 + 164n" - 288n 5
+ 547n 4 - 1058n s + 1068n 2 - 1214?* + 1464),
=^n(n- 2) (n 7 - 4n + 7n 5 - 45 4 + 114?i 3 - lib; 2 + 548n - 960),
= n (n- 2) (n - 3) (w 2 + 2w - 4),
c' = 4 ( - 1) (?i - 2),
/t' = n (n - 2) (16n 4 - 64n 3 + 80n 2 - 108n + 156),
' = 2 (n - 2) (3i - 4),
o-' = 4t (n - 2),
/9'=2n(7 l -2)(ll-24),
7' = 4n (n - 2) (n - 3) (n 3 - 3n + 16),
the remaining quantities vanishing.
C. XI.
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's 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 fonn
a cone of the order p, and class v, with y + 17 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 p = v = 2, y = i) = z = %=u = v = Q, and the binodes /& = 2, 77 = 1,
v = y = &c. = 0.)
And, further, a number of singular planes, viz. planes any one of which touches
along a curve of the class /*' and order v, with y' + V double tangents, of which y'
are generating lines of the node-couple torse, z' + " 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'=r}' = z'=? = u ' = v' = 0, and the bitropes /t'=2, rj' = l, v = y' = &c. = 0.)
616. This being so, and writing
a- = v + 2
the equations (7), (8), (9), (10), (11), (12), contain, in respect of the new singularities
additional terms, viz. these are
6(7i-2) = . ..+2[yO*-2)].
c (-2) = ... + 2 00* -2)],
o (n - 2) (n - 3) = . . . + 2 [x (- 4p + 7) + 2r, + 4fl,
b (n - 2) ( - 3) = ... + 2 [y (- 4 M + 8)] - 2' (4t' + 3t/),
c (n - 2) (n - 3) - . . . + 2 [*(- 4p + 9)] - 2' (2t/),
and there are of course the reciprocal terms in the reciprocal equations (18), (19),
(20), (21), (22), (23). These formulas 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
AUFEASSUNG 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 z x+h , a function of x + h, expanded in a doubly infinite, necessarily
divergent, series of integer or fractional powers of h, according to the law
K=+OO
2
(2)
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 oo to + <x> , 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. 331344.
302
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 seriea" 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 C n = Y (n), and hence that assuming that this is so in
general, or writing
F (n) . e* = 2 r [n - l] r a? 1 - 1 -*,
we have this equation as a definition of F (n). 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 z x+h in powers of A.
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. 260263; 439, 440.]
IN the paper " Ueber endliche Gruppen linearer Transformationen einer Verander-
lichen," Math. Ann. t. XII. (1877), pp. 23 46, Prof. Gordan gave in a very elegant form
the groups of 12, 24 and 60 homographic transformations - -,. The groups of 12
Off* T Ct
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
abcde.
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
1 , ... (where in the case in question ^ is a functional symbol of
are 1,
the like homographic form, SY = ^ w). But instead of obtaining the new form in
this manner, I found it easier to use the values of the rotation-symbol
cos - + sin - (i cos X +j cos Y + k cos Z)
for the axes of the icosahedron or dodecahedron, given in my paper "Notes on
polyhedra," Quart. Math. Jour. t. vil. (1866), pp. 304316, [375]; viz. if for any axes,
X, fj., v denote the parameters of rotation tan - cos X, tan - cos Y, tan - cos Z, then,
288
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," Matii. Annalen, t. xv. (1879),
pp. 23X 240, [660], the corresponding homographic function of x is
(- V - \) X + \ + ifi,
(X ip.) x + v i
where i denotes V 1 as usual.
The new formulae for the group of 60, or icosahedron group, of homographic
functions - ? are contained in the following table, where the four columns show
the values of the coefficients a, /9, 7, & respectively: and where in the outside column,
the substitution is represented as a permutatiou-symbol on the five letters abode:
moreover for shortness 6 is written to denote \/5.
THE GROUP OF 60.
y
1
1
1
1
2
-1
1
lib . cd
3
1
1
uc . bd
4
-1
1
ad. be
5
2
-3+e+( i-e)
-3+e+i(-i+e)
-2
be .de
6
2
-3+e+i(-i+e)
-3 + 6 + i( 1-6)
-2
ae . Ic
7
2
3^-e+i(-i+e)
3-6 + i( 1-6)
-2
ad. ce
8
2
3-e+i( i-e)
3-6 + i(-l + 6)
-2
ad . be
9
2
-i-e+i( i-e)
-i-e+t(-i+e)
-2
ae . cd
10
2
-l-e + i(-l + 6)
-i-e+( i-e)
-2
ab . de
11
2
i+e+ t -(-i+e)
l + 6 + i( 1-6)
-2
be .cd
12
2
i+e+i( i-e)
i + e+i(-i+e)
-2
ab . ce
13
2
-i-e+i (-3-e)
-i-e+i( 3+e)
-2
ac . be
14
2
-l-e + i( 3 + 6)
-i-e+i(-3-e)
-2
bd.ce
15
2
i+e+i( 3+e)
l + 6 + i(-3-6)
-2
ae . bd
M
2
l + 6+i(-3-6)
l + 6 + i( 3 + 6)
-2
ac. de
17
-i
i
1
1
abc
18
-1
i
1
t
acb
19
1
-i
1
t
adc
20
-'
i
1
-1
acd
21
i
1
1
-1
adb
22
1
i
1
-i
abd
23
-1
-t
1
-i
bed
24
i
-<
1
1
bdc
752]
ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE.
239
25
-l-G + i( 3 + 6)
2
-2
-i-e+i(-3-e)
aec
26
i+e+i( 3+e)
2
-2
i+e + ((-3-e)
ace
27
l + e + i(-3-6)
2
-2
i+e+i( 3+e)
bed
28
-i-e+t(-3-6)
2
-2
-i-e + (< 3 + e)
bde
29
-3+e+j( i-e)
2
2
3-e+( i-e)
bee
30
-3 + 6 + i(-l + 6)
2
2
3-e + t(-i+e)
bee
31
3-e+i(-i + e)
2
2
-3+e+i(-i+e)
aed
32
3-e+i( i-e)
2
2
-3+e+t( i-e)
ade
33
2
-i-e+t(-i+e)
l + 6 + i(-l + 6)
cde
34
2
l + 6 + i( 1-6)
-l-6 + i( 1-6)
ced
35
2
-i-e+i( i-e)
l+6 + i( 1-6)
aeb
36
2
l + 6 + i(-l + 6)
-i-e+i(-i+e)
abe
37
-i-e + i(-3-6)
2
2
i+e+j(-3-6)
abcde
H
-i-e+i( i-e)
2
2
i+e + i( i-e)
acebd
39
-l-6 + i(-l + 6)
2
2
l + 6 + i(-l + 6)
adbec
40
-i-e + i( 3+e)
2
2
i + e+i( 3+6)
aedcb
41
l + 6 + i( 3 + 6)
2
2
-l-6 + i( 3 + 6)
adceb
42
i+e+t(-i+e)
2
9
-i-e+i(-i+e)
acbde
43
i + e+i( i-e)
2
2
-l-6 + i( 1-6)
aedbc
44
l + 6 + t(-3-6)
2
2
-i-e+i(-3-e)
abecd
45
-l-6 + i(-l+6)
2
-2
-i-e+j( i-e)
acbed
46
-3+e+t(-i+e)
2
-2
-3+e + t( i-e)
abdce
47
3-e+f (-1+6)
2
-2
3-e + j( i-e)
aecdb
48
i+e+i(-i+e)
2
-2
l + 6 + i( 1-6)
adebe
4U
l + 6 + i( 1-6)
2
-2
i + e+t(-i+e)
aecbd
50
3-e+t( i-e)
2
-2
3-6 + i(-l + 6)
acdeb
51
-3 + 6 + i( 1-6)
2
-2
-3+e+t(-i+6)
abedc
52
-i-e + i( i-e)
2
-2
-i-e+t(-i+e)
adbce
53
2
-3 + 6 + t(-l + 6)
3-e + i(-i+6)
2
aebdc
54
2
-i-e+i( 3+6)
i+e+t( 3+e)
2
abced
55
2
i+e+i(-3-e)
-i-e+j(-3-e)
2
adecb
56
2
3-6 + i( 1-6)
_3 + e+i( i_e)
2
acdbe
57
2
-3+e+i( i-e)
3-e+t( i-e)
2
abdec
58
2
-i-e+f(-3-e)
l + 6 + i(-3-6)
2
adcbe
59
2
l + 6 + i( 3 + 6)
-l-6 + i( 3 + 6)
2
aebcd
80
2
3-e+(-i+e)
-3+e+j(-i+e)
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
GKOUPS OF 12 AND 24.
o ft y S
1
1
1
1
2
-1
1
'/' . i-il
3
1
1
ac . lul
4
-1
1
ad. be
5
-i
i
1
1
abc
6
-1
i
1
i
acb
7
1
-j
1
t
adc
8
-t
i
1
-1
tied
9
t
f
1
-1
mil,
10
1
t
1
- f
abd
11
-1
-i
1
- i
bed
12
i
-i
1
1
Me
13
f
1
adbc
14
-{
1
acbd
15
t
1
cd
16
*
t
-1
ab
17
1
-1
1
1
acdb
18
-t
-1
1
i
bd
19
t
1
1
*
abed
20
1
1
1
-i
be
21
-1
-1
1
-i
abdc
22
t
-1
1
-i
ac
23
f
1
1
-t
adcb
24
-1
1
1
1
ad
The groups of 60 and 24 thus each of them contain the group of 12,
x >
\-x
.1 4-a;
l T^x'
X + 1
X -I
It may be remarked that, to verify the periodicities of the forms contained in the
group of 60, we have as the conditions that
K may be periodic of the order 2, ' / =0, that is, a + 8=0,
yx + o ao py
3,
5,
= 1,
752] ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE.
For instance, in the form
we have
and therefore
as it should be.
241
[_ i _ Q + j (_ 3 _ 0)] x + 2
+ i(-3-0)] '
) 2 , =-20-80,
+ 8)*__4(3 + @) 2 _8 + 6 r
-^ --8(3 + 0)' 2 ~* (e *
Cambridge, 11 AW 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. XI.
31
242 [ 753
753.
ON A THEOREM RELATING TO THE MULTIPLE
THETA-FUNCTIONS.
[From the Mathemutische Annalen, t. XVII. (1880), pp. 115122.]
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 Theoiie der Abel'schen Functionen von drei Variabeln,
(Leipzig, 1880), which theorem is there presented in the form :
e-'<" ..... : "' "' ) (, + 25,', ... ;r,v) = e'*" ^'- O (u,, ... ; p + /, v + v), (3')
but which I write in the slightly different form
exp. [- H (u ; /*', v')] . B (u + 2*r' ; p., v) = exp. [- 2?ri/w/] . (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
G(w,, ..., u f ; n,, ...,n p )
of 2p variables, p being the number of the arguments , ..... u f (in fact, the periods
are not reduced to the normal form, but are arbitrary); and the characters i/,,...,j/ p ;
fo ..... fjL p , instead of having each of them the value 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 j, ...,,,, and so in other cases), is that the function
6 (u ; M + /*'. " + "')
753]
ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS.
243
with the new characters fj. + p and v + v is, save as to an exponential factor, equal
to the function (u + 2t*' ', p, v) with the original characters p., v, but with the new
arguments u + 2ta'.
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 = (u lt ..., u p ):
or if, to fix the ideas p = 3, then u = (u ls u. 2 , u t ), and so in other cases.
But I use certain other letters a, etc. to denote squares or matrices each of p*
letters ; thus, if p = 3 as before,
a n , a ls , a ]3
and in any such case the transposed matrix is denoted by the same letter enclosed
in parentheses
Oil, Oin si
OH. 2, 32
i3, etas, ay,
The sum u + v of the row-letters u, = (v^, u t , u,) and v, = (v lt v 3 , v 3 ) denotes the
row (,+!'!, Wj + Va, iit + v,): and in like manner the sum a+b of the two matrices,
or square-letters a and b, denotes the matrix
bi,, Oj 3 + 6 13
b,,, On + bn
and similarly for a sum of three or more terms.
The product uv, =(MI, w 2 , u 3 )(vj, v. it v,), of the two row-letters u, v denotes the
single term u l v 1 + u,v, + u 3 v,. We have uv = vu.
The product
aw,
,, M 2 , M 3 ),
of a preceding square-letter a and a succeeding row-letter u, denotes the set or row
(ji, a, a , OU)(MI, M 2 , u,), (an, a-a, 0,^(11^, u. 2 , u 3 ), (a si , a^, )(!, u 2 , u,);
the notation ua is not employed.
312
244 ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS.
The product
[753
auv
On,
(MI, MI, M,)(,, V 3 , V,),
of a preceding square-letter a followed by the two row-letters u and v, denotes the
single term
(Oil. On, O,,)(MI, M,, W,)w 1 +(a n , On, OB)(MI, MS, Mj) t), + (dj, , Ojj, ajs)(,, tt,, M s ) V,.
Observe that auv is not in general = ami ; but it is easy to verify that auv = (d)vu ;
and hence if (a)=o, that is, if the matrix a be symmetrical, then auv=avu.
A product of two matrices
denotes a matrix
(flu,
ab, =
On, u ia> Ojj
in, in, iw
|
OJL O^, OB
in, ia, ias
031, 0,,, 03,
i, in, i
(in, in, ijn), (iia, iaa, is2)> (iis, i;o, iss)
, 0,3)
l
, 0.,,)
>
viz. the top-line of the compound matrix is
(a,,, a,,, Oi,)(i u , 6 a , i 3 ,), (OIL ais, o ]3 )(6 IS ,
u , a, 2 ,
, by,),
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 (06) 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 6 are each symmetrical, we do not in general have
ab = ba, but only (a&) = 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, M, ^, v, /*', v, n, or', f are row-letters; a, 6, h, <a, <a', rj, 17'
are square-letters : a and b are symmetrical, viz. a = (a), b = (b).
753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS.
And I write
()(u, v}*, =(a, h, b)(u, v)-
2huv + bv*
(11 -01 11 v
(Ul> 2, U S)
245
1 ) 22
lai, 32,
+ 2 /*, ^ 2 ,
l, " 3 , I> 3 ) 3
to denote the general quadric function of the 2p letters u, v, with
coefficients. It is assumed that the determinant formed with the ^p(p + l) coefficients
b is negative: this is the necessary and sufficient condition for the convergence of
the series.
Definition of (u ; p, v).
( ; /i, v), the general theta-function with p arguments u, and 2/s characters /*, i>,
is the sum of a p-tuple series of exponentials
(u ; /t, v) = 2 exp. [() (M,
Imp (n + v)],
where each of the letters n, =(n,, ..., n p ), has all integer values (zero included) from
oo to +00.
The general theorem in regard to (u; ft, v).
This is
exp. [- H (u ; /*', v')] . (u +
v) = exp. [- 2m/j.v] . (it ; p. + pf, v + v),
establishing a relation between the function @ (w ; /* + /u,', i/ + 1/), with arbitrary character-
increments //, v', and the function @ (u + 2or' ; ^4, v) with the original characters, but
with new arguments w+2w'. Also H(u; p, v') denotes a function, linear as regards the
arguments u, but quadric as regards /*' and v' ; liripv is a single term depending
only on /* and v ; 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-FUNCTION8. [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 + 2*', n + v),
to denote the quadric function ()(M + 2r', n + j/)*; we ought to have
-H(u\ /*', v')+G(u + 2v', n + v) + 2mp(n + v)
= - 2mfi.v' + G (u, n + v + v) + 2m (ft, + ft.') (n + v + v),
or say
H(u ; ft.', v')= G(u + 2vr', n + v)-G(u, n + v + v')- 2m (n+v + v')p.'.
In this equation, if true at all, the terms containing n must destroy each other;
assuming that they do so, the equation becomes
H(n ; ft.', i/') = G (u 4 2w', v) - G (u, v + v) - 2m (v + v) /.
Consider first the terms in n : the right-hand side is
= a (u + 2vf'y + 2A (u + 2vr') (n + v) + b(n + i/) 2
- * 2hu (n + v + v) b(n + v + v')-
and the terms herein which contain n thus are
2h (u+ 2w') n + bn* + 2bnv
2kun - bn- - 2bn (v + v') - 2-rrinfj.',
which, b being symmetrical, may be written
= 2 (2/4*7' - bv - mfi) n,
and these terms will vanish if, and only if
2Aor' - bv' - m/j.' = 0,
a system of p equations connecting -as' ', p, v.
Assuming them to be satisfied, the remaining relation,
H ( ; /, i/') = G (u + Zw', v)-0 (u, v+v')- 2m (v + v) ft',
becomes
H (u ; ft.', v') = a (u + 2O" + 2h (u + 2w') n + bv-
- au l - 2hu (v + v ')-b(v + vj - 2m (v + v) p.
Here, a and b being symmetrical, we have
a ( + 2w') 2 = aw" + 4ar'w + 4aw' a , b (i> + v') 3 = bi>* + 2bv'v + bv'-,
753] OX A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 247
and the value therefore is
= 4a (ar'u + cr'") + 2h (Zm'v - uv) - b (2i/i/ + i/ 2 ) - Ztri (v + v) p.
On the right-hand side, putting the term in h under the form
-2h(u+*r')v+ ZhTsr' (2i + v), = - 2 (h) v (u + w') + 2Ar' (2v + v),
and the last term under the form
irift (Zv + v) - iriftv,
the equation becomes
H (u ; ft', v') = (4aw' - 2 (h) v') (u + w') - nifty
+ (2hvr' - bv - Trip) (2v + v'),
where the second line vanishes in virtue of the foregoing equation
' - bv - Trip! = ;
the equation thus is
H (a ; ft', v') = (4ow' - 2 (h) v) (u + ') - wt/tV,
which equation, regarding therein w' as a linear function of /*' and v, shows that
H (u ; ft, v) is a function linear as regards u (and containing this only through u + is'),
but quadric as regards /*', v.
Introducing the new row-letter f, we may write
H(u; ^, i')=
viz. the expression on the right-hand side is here assumed as the value of the
function
H(u; ft', v), =G(M + 2w', v)-G(u, v+v')-2m(v+v") ft ;
and the theorem then is
exp. [- H (u ; ft, v')] . (it + 2w' ; ft, v) = exp. [- 2iriftv r ] . 6 (u ; /* + /*'. " + v),
where, by what precedes,
Zhia' bv' Trift = 0,
2aw'-(A)i'-f =0,
2/j equations for determining the 2p functions w', f ' as linear functions of ft, v :
which equations depend on the p (2p + 1) constants a, b, h.
Suppose that the resulting values of or' and f are
as' = (Oft' + ta'v',
where <a, &>', 77, 17' are square-letters ; then, regarding a, b, h as arbitrary, the 4/a 2
new constants <a, o>', 77, 77' cannot be all of them arbitrary, but must be connected
by 4p 2 p (2p + 1), =p(2p 1) equations.
248
ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. [753
We may regard o>, <a', rj, if as satisfying these p(2p l) equations, but as being
otherwise arbitrary ; the foregoing equations then are
2/mr'- bv iri/i' = 0,
9aur'-(h)i>'-? =0,
or' = OJ/A' -h ft>V,
rt , / /
= tin + t)t>,
which lead to the equations connecting a, b, h with u>, a>', 17, 77'.
The first and second equations, substituting for vr' and " their values, become
7ri)/ + (2Ao/-&)i/ =0,
(2oa> - 77 ) /*' + (2a' - (A) - V) "' = 0,
or ft,', v being arbitrary, we thus obtain the 4p" equations
2aa> -77 =0,
2Aa> in =0,
2aa>' - 77' - (/t) = 0,
2Aa>' - 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,
We can eliminate a, b, h from these equations and thus obtain the p(2p 1)
equations before referred to, which connect the 4p 2 constants <o, as', 77, 17'. I give, but
without a complete explanation, the steps of the elimination.
The equation 2ao> 77 = 0, may be written in the form
that is,
or since (a) = a, this is
from the original form, and the new form respectively, we find
2 (a>) ao> - (w) 7) = 0, 2 (a>) a (w) - (17) w = ;
and comparing these
() i] (ij) a> = 0, (first result).
The equation 2oo' - ?/' - (h) = 0, or say (h) = - rf + 2ao>', may be written in the form
753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 249
that is, since a = (a),
and we thence deduce
hco = (rf) o) + 2 (to') aa>.
But from the equation 2ao> t) = 0, we have 2 (a/) au> (&>') 77 = 0, and the equation
thus becomes hos = (77') w + (&>') ij ; which, in virtue of 2ha> m = 0, becomes
+ (o)')r), (second result).
From the equation above obtained, h = (if) + 2 (to') a, we have
hot = - (rf) <a' + 2 (&>') aw' ;
in virtue of 2ha>' 6=0, this becomes 2 (77') &>' 4- 4 (<') aw' = 6 ; an equation which
may also be written 2 ((T/) o>') + 4 ((&>') aw') = (6), or, what is the same thing,
2 (<o r ) i{ + 4 (a/) (a) a>' = (6) ; or since (a) = a and (6) = 6, this is
- 2(>^+ 4 (0 *'-&:
and comparing with the original equation
we obtain
(')'?'-('7')< u '=0. (third result).
We have thus the three systems
(o>) i) (TJ) a> = , \p (p 1) equations,
(w') 17 - (V) w = ^T", p 2
in all p(2/j 1) equations. As to these systems, observe that (o>)r), (tj) a>, etc., are
all of them matrices of p s terms; each of the three systems denotes therefore in the
first instance p 2 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; 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 p 1 to
^p (p 1) ; and similarly in the third system the number of equations is = Jp (p 1) :
but for the second system the number of equations is really =p 2 . It is hardly
necessary to remark that in this second system ^-jri 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. XI.
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
that is, "
sn acnadn asn 2 du _ &a ,. <&(u a)
U ~ + * g
leads not only to the relation
2&' ' K f ~F\ f f
log u = % log + ( 1 - =) w 2 - A 2 I du du sn- u,
i" V M./ 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, en a,
dn a then become a, 1, 1, respectively, the equation is
a@"0 u-a&u
k?a I sn 2 M du = u -^r + \ log
=>M + a&u '
T9_ Jf
that is,
&u "0 r
J
or, integrating from u = 0, this is
!/ 2 . fc 2 I du I dwsn'w,
Vj() J n J 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 logQw.
Next, differentiating twice the equation for II (a, a), and once the equation obtained
, &u
for -fr- , we have
BM
d I sn 2 u \ . "6 @' 2 , "@ 0' 2 ,
Ar'snacnadnaT- , IT- - } = *- (it a) * ;=-- (it+a),
du \l k 2 sn 3 a sn 2 u] @ 2 2
and
0"0 _ @'2 "0
ji _ i2 CT|2 it
-0T- - 0o u '
where, for shortness, - M is written to denote - nii , and the like in
\J \J U
the first equation ; the right-hand side of the first equation therefore is
- k 1 {sn 2 (u - a) - sn 2 (u + a)},
or the equation becomes
d sn 2 u
2 sn a en a dn a T- = sn 2 (u + a) sn 2 (u a).
du 1 - & 2 sn 2 itsn 2 a
that is,
4 sn u sn' M sn a en a dn a
7= r - - = sn- (it + a) - sn 2 (u a).
(1 - A; 2 sn 2 u sn 2 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'w were unknown) it would appear that sn'w 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
= sn 2 (M + a) sn 2 (u a)
{sn (u + a) + sn (u a)} {sn (u + a) sn (w a)}.
The factor {sn (u + a) + sn (u a)} becomes = 2 sn u for a = 0, and this suggests that
the factor sn u 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 = ; 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(w a)}, and by symmetry en a, dn a belong to the
first-mentioned factor. And we are thus led to assume
sn (M + a) + sn (u a) = 2M sn M en a dn a,
sn (u + a) sn (u a) = 2M' sn a en u dn M,
where
denom. = 1 k 1 sn 2 a sn 2 u,
and MM' = 1. Some further investigation is wanting to show that M and M ' are
constants, but assuming that they are so and each = 1, the formulae give at once the
ordinary expression for sn (u + a) ; that is, we have the addition-equation for the
function sn.
322
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
powers and roots of the function 5 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. CXLVIII. (1858), pp. 1737, [152]. I had recently
occasion to reconsider the question, and have obtained for the nth function <f> n x, where
<fxe = j , a form which, although substantially identical with Babbage's, is a more
ex + a
compact and convenient one; viz. taking \ to be determined by the quadric equation
the form is
ex a)'
The question is, in effect, that of the determination of the th power of the
matrix ( a, b ); viz. in the notation of matrices
' c, d |
> 6 ) (x, y),
c, d
ad be'
755]
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH.
253
c n , d n
, =( a
means the two equations x l = ax + by, y^ = cx + dy ; and then if # 2 , i/ 2 are derived in
like manner from x lt y lt that is, if X 3 =ax 1 + by 1 , y^=cx 1 + dy l , and so on, x n , y n will
be linear functions of x, y; say we have x n a n x + b n y, y n = c n x + d n y : and the nth
power of ( a, b ) is, in fact, the matrix ( a n , b n ).
I c, d\
In particular, we have
( a, b ) 3 , = ( a.,,
c, d c a ,
and hence the identity
( a, b )*-(a+d)( a, b ) + (ad-bc)( I, ) = 0;
c, b(a
c, d
0, 1
c, d
viz. this means that the matrix
( 02 - (a + d) a + ad - be, b. 2 -(a + d)b ) = ( 0, ),
c 2 (a + d) c , d 2 (a + d)d + ad bc 0,0
or, what is the same thing, that each term of the left-hand matrix is = ; which is
at once verified by substituting for a 2 , &.,, c 2 , d. 2 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 like 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
M* - (a + d) M + (ad - be) = 0,
the matrix ( 1, ) being in the theory regarded as =1; viz. M is determined by
0, 1
a quadric equation ; and we have consequently M n = a linear function of M. Writing
this in the form
the unknown coefficients A, B can be at once obtained in terms of a, /9, the roots
of the equation
v? (a + d) u
viz. we have
a" -A
or more simply from these equations, and the equation for M n , eliminating a, j3, we
have
M n , M, 1
= 0;
254
that is,
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH.
(a - /3) -
[755
But instead of o, , it is convenient to introduce the ratio X of the two roots,
say we have o = X/9; we thence find
riving
for the determination of X, and then
= ad bc,
ft-
ad be
a + d
: x+r
(a + d) X
X+l '
The equation thus becomes
or we have
-(\ n -\)^ t+1 = 0,
{(X-l)Jf-(X-X)/S}.
It is convenient to multiply the numerator and denominator by X + l, viz. we
thus have
The exterior factor is here
1 la + dX""
x s - 1 Vx + 1
moreover (X+l)/3 is =a + d: hence
M=( a, b ),
c, d|
and
the formula thus is
= ( a, b )-( a + d, ),=(-rf, i );
c, d
X s -
, a
a,
c , a
'-X)( -d, b
viz. we have thus the values of the several terms of the rath matrix
M n = ( a n , b n );
c n , d n \
755]
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH.
255
and, if instead of these we consider the combinations a n a; + b n and c n x + d n , we then
obtain
a n x + b n =
- X) (-
6)),
ex -a)};
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
c<p + d = 1 ; and if n = 1, they become a^x + b 1 = ax + b, c 1 x + d 1 = cx + d.
= x,
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 { j are
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 (\-l)(a + d) we write
' , the formulae become for n = m
fa + d\ m
viz. we have here
_
' d '"~
+ b m
c m x + d m
a + d
= x,
STT
or the function is periodic of the mth order. Writing for shortness ^ = , s being
any integer not = 0, and prime to n, we have X = cos 2^- + i sin 2^, hence
1 + X = 2 cos ^ (cos S- + i sin S-),
or - - =4cos J ^; consequently, in order to the function being periodic of the nth
A.
order, the relation between the coefficients is
TT (a + dy
4 cos 2 = - -, r- .
n ad be
The formula extends to the case m = 2, viz. cos J (STT) = 0, or the condition is
a + d = 0. But here X + 1 = 0, and the case requires to be separately verified. Recurring
to the original expression for M ', we see that, for a + d = 0, this becomes
that is,
or the result is thus verified.
a?+bc,
, d' + bc
1,
0, 1
CyK
= x,
256 ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. [755
But the case m = 1 is a very remarkable one ; we have here X = 1, and the
relation between the coefficients is thus (a + dy = 4 (ad be), or what is the same thing
(a dy + 46c = 0. And then determining the values for X = 1 of the vanishing fractions
which enter into the formulae, we find
On* + & = ^ ( + <*)"-' {( + 1) (o + 6) + (n - 1) (- da + b)},
c n x + d n = ^(a + d) n ~ l {(n + 1) (ex + d) + (n - 1) ( ex- a)},
or as these may also be written
a,,x + b n = , (a + d)"- 1 {x [n (a-d) + (a + d)] + 2nb],
c n x + d n =^(a + d)' 1 ' 1 {x . 2nc + [- n (a - d) + a + d]},
which for =0, become as they should do a x + b a = x, c x+d a = l, and for n=l they
ft CT I /)
become a,x + 6, = ax + b, dx + d l = ex + d. We thus do not have - 1 = x, and the
function is iwt 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)- + 46c = 0, we write 26 = k (a d), 2c = r (a - d).
K
then the function of a? is
ax + %k (a d)
and the formulae for the nth function are
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 \ is not = 1, we have X" indefinitely large or
indefinitely small. In the former case, we obtain
\(cx +d) + ( ex a)' ' c(\ +
which, observing that the equation in X may be written
\a-d _6(X-Mj
~
Xa d _ I v ._ , _ 7
c(X+ 1)~ \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
cw 2 - (a - d) u - b = 0,
that is,
au + b
M = = ,
CM + d
as is, in fact, obvious from the consideration that n being indefinitely large the nth and
(w + l)th functions must be equal to each other. In the latter case, as X is indefinitely
small, we have the like formulae, 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 = &-T-C; u is therefore the other root of the quadric equation. Hence, as n
increases, the function <j> n x continually approximates to one or the other of the roots
of this quadric equation. The equation has equal roots if (a d) 2 + 46c = 0, which is
the relation existing in the above-mentioned special case ; and here u = =- (a d), = -, ,
Ac ct ~ d
which result is also given by the formulae of the special case on writing therein n = oo .
C. XI.
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
1 1 J_
X=A+X-B + X-C~
The equation is
(X
that is,
and we have therefore
3Z - (A + B + C) = VP + B + <?)' - 3 (BC + CA + AB)},
or as this may be written
X = $(A+B+C)J{$(A+Bo> + Cafi).$(A + Ba>* + Co,)},
where w is an imaginary cube root of unity,
= cos 120 + t sin 120 suppose.
Taking an arbitrary point as the origin, let the imaginary quantity A, =
suppose, be represented by the point A, coordinates a and at'; and in like manner
the imaginary quantities B and C by the points B and C respectively.
Then Bo>, Bta 1 are represented by points B lt B,, obtained by rotating the point
B about the origin through angles of 120 and 240 respectively; Ca>*, Cot are repre-
756] A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY QUANTITIES. 259
sented by points C lt C a obtained by rotating the point C about the origin through
angles of 240 and 480 (= 120) respectively : and
are represented by the points 0, G lt (r 2 which are the C.G.'s of the triangles ABC,
ABiC lt AB^C 2 respectively. The formula therefore is
where, if a, a' are the coordinates of 0, then OQ is written to denote the imaginary
quantity a + a'i; and the like as regards OG lt 0(? 2 . Taking >j'(OG 1 .OG. 1 ) = OH, we then
have' H a point such, that the distance OH from the origin is = geometric mean of
the distances OG lt 0(? 2 , and that the radial direction* of the distance OH bisects
the radial directions of the distances OG lt OG 2 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 = ^(GG^ . GG^), 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, q, r real) is somewhat similar, but the construction is less simple ; we have
Writing herein q + r, r +p, p+q = l, m, n, the equation becomes
(I + m + n ) X* - 2 (IA + mB + nO) X + (- 1 + m + n) BC + (I - m + ri) CA + (I + m - ri) AB = 0,
that is,
{(I + m + n) X - IA - mB - nC}*
= (IA +mB + nCy + [V - (m + n)*\ BC + {m? -(n + Z) 2 ) CA + {n 2 - (I + m) 2 } AB.
Here the right-hand side is
= VA* + m'B 1 + n'C 1 + (I* - m? - n") BC + (- I 2 + m? - 2 ) CA + (- I 1 - m? + n") AB,
which is
= -P(C-A)(A-B)-m*(A-B)(B-C)-n' i (C-A)(A-B'),
and consequently is a product of two linear factors ; these, in fact, are
} {PA + ( - 1* - m 2 + n 2 A/A) B + % (- 1- + m> - n* + V A ) C} ,
I
* 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.
332
260 A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY QUANTITIES. [756
where
A = I* + m 4 + n 4 - 2mV - 2n?P - 2ton s .
It is to be observed that A, = (I* m 1 n 1 ) 1 4wi'n J , is negative ; hence, calling
the factors fA+gB + hC, f'A+g'B + h'C respectively, the coefficients /, g, h, and
/', g 1 , h' are imaginary ; moreover /+ g + h = 0, /' + g' + h' = 0.
The values of X thus are
(I + m + n) X = IA + mB + nC </{(fA + gB + hC) (f'A + g'B + h'C)},
IA + mB + nC
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 + coB + afC, A+w"B + a>C,
might be found without difficulty.
757]
261
757.
ON A SMITH'S PEIZE 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. 675774, [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 = - - (p v ~ p a ), where p, p l , p 3 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 iu
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 + V[(a a?) 2 ]
is one and the same function of x whatever be the value of x : this is so, if
V[(a #)"] denotes indifferently or successively the two functions + (a x) : but if, a
and x being real, \/[(a a;) 2 ] is taken to mean the positive value, then the function
a+x + V[( )*] is = 2a or = 2# according as a x is positive or negative.
Fig. l.
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
in general an abrupt change of value in -=- . Taking V as a coordinate y at right
Ct3C
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, qud one-valued function of (x, z), V is not any root y at pleasure of an
algebraical equation $ (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 -=- 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 F 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) 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 F= 2?r {V(c 2 + a?} x} ; or writing V+2ir=y, the section is made up of the two
distinct hyperbolas y(y 2x) = c 2 , and y(y + 2#) = c 3 .
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 V= 2iry 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
(1) y =
X
-(* -a)},
( 2 ) y = l
(3) ,-*.
+ (*-)},
- (x + a)\,
(4) y-=
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
Fi. 2.
equal or opposite curves: the rationalised equation of (1) is in fact
2a'(y + o)
(y + a)'-a>'
and by writing a for a, and in each equation x 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 + 6 + c = and x + y + z = 0, then
4 (ax + by + czf
- 3 (aa; + by + cz) (a*+b* + c 2 ) (a? + f + z 2 )
-2(b-c)(c-a)(a-b)(y-z)(z-x)(x-y)
54sabcxyz = 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
.11, 1, 1
a , b , c
a 2 , b-, c 3 | a?, f,
3, a +6 +c , a 2
x + y + z , ax + by 4- cz , d*a
x' + y' + z-, ax? + by 1 + cz 2 , a'-
and therefore when a + b + c = and x + y + z = 0, is
= 3 (ax + by + cz ) (a*a? + fry* + cV)
- 3 (a?x + b'y + c?z) (ax- + by- + cz 1 )
(ax +by + cz) (a 2 + 6 2 + c 2 ) (x 1 +
C. XI.
+ <?z
34
266 SOLUTION OF A SENATE-HOUSE PROBLEM. [758
or, as this may be written,
= 6 (ax + by + cz ) (aV +
- (ax +by + cz)
- 3 (ax +by + cz) (a"
- 3 (a'x + % + c*z) (ax* + by* + cz*).
Here the third and fourth lines, omitting the factor 3, are
2 (aW + &y + cV) + (ai 5 + a'6) (#y s + x'y) + (ac 2 + a'c) (a* J + a 8 *) + (fcc 2 + 6 2 c) (y^ 3 + y-z),
where, in virtue of the two relations, each of the last three product-terms is = abcxyz,
and the whole is thus
= 2 (a
+ Sabcxyz.
The product of the two determinants is thus
= 6 (oar + by + cz) (a'a? + fry
- (ax + by + cz) (a 1 + b* + c") (a? + y 2 +
9 abcxyz ;
and this being so the identity to be verified is
4 (ax + by + czf
- 1 2 (cue + by + cz) (
+ (18 - 54 =) - 36abcxyz = 0.
We have here the terms
1 2 (a'x* + 6y + c 3 z 3 - Sabcxyz),
= 12 (ax + by + cz) (a'x 3 + b*y* + c*z* 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 + czy
+ 12 (a'x 1 + 6y + c V - bcyz - cazx - abxy) = ;
758] SOLUTION OF A SENATE-HOUSE PROBLEM.
or, as this may be written,
4 (oftc 2 + 6y + c 2 * 2 - bcyz - cazx - abxy)
267
which under the assumed relations a + b + c = 0, x + y + z may be verified without
difficulty. It may be remarked that we have identically
8 (aV + by + c-z z - bcyz - cazx - abxy)
x( 3a 2 - b-- c 2 + 2bc - 2ca - 2a6)
- a 2 + 3b- - c 2 - 2bc + 2ca -
z (- a 2 - 6 2 + 3c 2 - 2bc - 2ca
a( 3af- y 1 - z" + 2yz-'2zx-2xy)\
+ (a + b + c) + b (- x* + 3i/ 2 - z* - 2yz + 2zx - 2xy) I ,
[ + c (- z 2 - y 2 + 3* 2 - 2^ - 2zx + 2xy) I
which is a more complete form of the last-mentioned theorem.
342
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 s # + #"7 + -fa = a given value k, then a, /3, 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 2 6 + b>c + c*a = k, then clearly the relation in question
a 2 y3 + fPy + <fa. = k, will be ^ satisfied if a, /3, 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 =6, 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
/3 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? 1x + 6 = having, roots (1, 2, 3), and that the given
relation is or/3 + [3fy + <fo. = \, then the cubic equation gives
and we have, besides, the relation in question
eliminating 7 we have
- /3 3 + 1 = 0;
or, as it is convenient to write these equations,
/S 2 + ct/8 + a 2 - 7 = 0,
*e- =o,
/S 8 - 3a 2 /3 - a 3 - 1 = 0.
759] ILLUSTRATION OF A THEOREM IN THE THEORY OF EQUATIONS. 269
If from these equations we eliminate /3, 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 /3 must contain as a factor the cubic function a 3 - 7a + 6, and be thus of the form
M (a 3 7oc + G) = 0, where M is a function of a; one result of the elimination is
a 3 7a + 6 = 0, and every other result is of the form just referred to, M (a 3 7a + 6) = 0;
hence we have definitely a 3 7a+6 = 0, viz. the roots of the equation M = do not
apply to the question.
/>
In verification, observe that the first and second equations give a.- 7 = - , that
is, a 2 6a + 7 = 0. To eliminate /3 from the first and third equations we first find
a/8 2 + (4a 2 7) /8 + a 3 + 1 = 0,
or say
and combining herewith the first equation
we obtain
that is,
7a+:
-3a 2 +7'
substituting in the first equation,
(7a + 1) 3
+ a(7a+l)(-3a 2 + 7)
+ (a 2 -7)(-3a 2 + 7) 2 = 0,
that is,
49 14 1
21-3+49 +7
9 0-105 +343 -343
9 - 126 - 3 + 441 + 21 - 342,
or, dividing by 3,
3a" - 42a 4 - a 3 + 147a 2 + 7a - 1 14 = 0,
which, in fact, is
(a 3 - 7a + 6) (3a 3 - 21a - 19) = 0,
of the form in question M (a 3 - 7a + 6) = 0. Thus a has any one at pleasure of the
three values 1, 2, -3, but a being known we have ff = _ = , and thence
-7a-l 3a 3 -14a-l
7^ flt -I 1_ == ~ *
^ O~2 -1-7 ^*7^ L 7
in particular, as = 1, then /3 = 2 and 7 = 3.
270 [760
760.
REDUCTION OF 7-^, TO ELLIPTIC INTEGRALS.
[From the Messenger of Mathematics, vol. xi. (1882), pp. 142, 143.]
WRITING 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 + Qscd
I + ffscd '
To find from this the value of y, = j/(l of), putting for shortness X = dscd, the
e thence have
_
formula is x = -- , and we thence have
where
l+3Z 2 = l-f
= 1 + 30=s 2 - 30 s (1 + fc 2 ) s 4 + 30*"s 6 ,
may be put equal to (1 + tf's 2 ) 3 , that is,
= 1 + 30 2 s 2 + 30V + 0V ;
viz. this will be the case if
30 4 = -30 2 (l+ J ), s = 30*;
that is,
these give
'
that is, &* = &>, if a> = ^ + i V3, an imaginary cube root of unity ; and then
v o \
760]
that is,
REDUCTION
f dx
N OF 7
; (i-x
TO ELLIPTIC INTEGRALS.
0=
as may be verified by squaring.
2V2
Hence finally, d and k denoting the values just obtained,
- 1 + Oscd
x =
1 + 6scd '
or, -\vriting as before, X = Oscd, we have
whence
and then
dl
that is,
or say
the required formula.
da;
du, =0(1
dx
/_* .
./ 1-^3
271
272 [761
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. 137147.]
GORDAN'S proof, the only one hitherto given, is based upon the theory of derivatives
(Uebereinanderschiebungen). It is shown that the irreducible covariants of the binary
quantic / are included in the series
(/. /)' (/ /) 4 > (/, A), (/, A) 2 , ...
of the derivatives of the quantic upon itself or upon some other covariant, and that
the number of the irreducible covariants thus obtained is finite. And not only so,
but for the quintic and the sextic the complete systems were formed, and the numbers
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, dQx, y) 8 , these are
the cubic itself (a, ...Ja;, yf, the Hessian (ac-6 2 , ...$>, y) 2 , and the cubico variant
(a*d-3abc + 26 s , ...$#, y) 8 ; and so in general for the quantic (a, ...$#, */)", we have a
series of fundamental covariants the leading coefficients whereof are the seminvariants
a, ac-b\ a'd-3aic + 2& 8 , a*e - 4a*bd + 6a6*c - 36 4 , &c.
It is known that every covariant can be expressed as a rational function of these, or
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 fundamental
covariants, and we may for the covariants substitute their leading coefficients, or say
the seminvariants; hence, every seminvariant is a rational function of the fundamental
seminvariants, and more precisely, every seminvariant multiplied by a positive integral
761]
ON THE THEOEEM 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,
obtained from
by the formula
= a?d? - Gabcd + 4OC 3 + 4& 3 d -
a, ac b-, a-d Sabc + 26 3 ,
a 2 V = (a 2 d - Sale + 26 3 ) 2 + 4 (ac - & 2 ) 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 b' 2 , a-d 3a6c + 26 s ,
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 /3y) . . . , or say (x a) (x @) ... of the quantic ; we have
here what may be called the monomial form of covariant, viz. the general monomial
form is
where in all the factors (whether a /3 or x a) which contain a, in all the factors
which contain /9, . . . , and so for each root in succession, the sum of the indices has
one and the same value, = suppose. Thus, for the cubic
we have the monomial covariants
(-<* -/9)(*y);
and so for the quartic
we have the monomial invariants
(a /3) (7 B), (a 7) (/3 B), (a <
Observe that the monomial form is considered as essential ; a syzygetic function of
C. XI.
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
notwithstanding these syzygies respectively,
are regarded as independent covariants of the cubic, and
(a -8)(0 -7), (/8-S)(a-7), and ( 7 -S)(a-/9),
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
-8) and ( a -y)(ft-B).
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
G(&, 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 (?(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
c+ ..., ab-fac + bc + ..., abc+..., &c.";
761]
OF THE COVARIANTS OF A BINARY QUANTIC.
275
and so, if (?(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 (.4), 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
(a-/3) m (a-7)''(/3-7) p --,
where in all the factors which contain a, in all the factors which contain /?, and so
for each root in succession, the sum of the indices has one and the same value = Q.
Writing 12 for the index of a /3, 13 for that of a y, and so in other cases, then
assuming always 12 = 21, 13 = 31, &c., the indices, taken each twice, form the square
12
13
21
23
31
32
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 R e , represents an invariant.
Thus, for the quartic (so o) (x /8) (as 7) (x B), the square Rg is a square of
four rows (or columns) representing the invariant
in which
03 -7)* 09 -S)
(7-S)
12 + 13 + 14 = 61,
21 + 23 + 24 = 6,
31+32 + 34 = 6,
41 + 42 + 43 = 6.
352
276 ON THE THEOREM OF THE FINITE NUMBER
There are three squares J?,, viz. these are the squares
[761
1
representing the before-mentioned invariants (a-/3)(7-S), (<*-7)(/9-S), (a-
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 = a^b^c", where 12, 13, 14 are each
of them a positive integer number (which may be =0). Or, what is the same thing,
the square R t (0=12 + 13 + 14) is a sum
= U.R 1 +IB.R 1 ' + U.R 1 ",
with positive integer coefficients 12, 13, 14, say for shortness it is a sum of squares
RI. And so any like expression with a negative coefficient or coefficients may, for
shortness, be called a difference of squares JR,.
Observe that, in general, two squares R t , R^, are added together by adding their
corresponding terms, the result being a square Re+j,; similarly, if each term of R+ be
less than or at most equal" to the corresponding term of R e , then (but not otherwise)
the square R$ may be subtracted from R t , giving a square R e -^.
In the case of the sextic
there are fifteen squares
#3
y
*
!, which may be represented as follows:
12.34.56
12.35.46
12.36.45
13.24.56
13.25.46
13.26.45
14.23.56
14.25.36
14.26.35
15.23.46
15.24.36
15.26.34
16.23.45
16.24.35
16.25.34 *.;
y*
761]
OF THE COVARIANTS OF A BINARY QUANTIC.
277
viz. 12.34.56 here represents the square R lt 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 f) M . But
it is not true that every square R 9 is a sum of squares R l ; this is not the case,
for the square R,
= 12.13.23.45.46.56,
representing the invariant
is not a sura of squares R l .
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 R^
It is required to show, first, that every square R e is a difference of squares R t ;
and thence, secondly, that it is a sum of a finite number of squares R^ (being, in
fact, squares .R, and R^).
For the first theorem we equate the general expression of R e with the assumed
value
B! . 12 . 34 . 56 + y 1 . 12 . 35 . 46 + z 1 . 12 . 36 . 45 + ... + z s . 16 . 25 . 34.
We thus obtain
fifteen equations satisfied by
12-y l + x l + t 1
#! = 34 _ 26 + r + s - t,
I3 = x i + y, + z t
# a =13 25 +p r +t,
14 = ar, + y 3 + z 3
#,= 14 -p -s
I5=x t + y 4 + z,
x t = 15 - 26 - 36 +p + q + r + s
16 = #5 + 2/5 + z f
# 5 = 45 q r ,
23 =X 3 + !E t +X i
y, = 12 - 34 + 26 -q-r-s + t,
24 = x, + ?/ 4 + y,
</ 2 =25 -p
25 = 2/ 2 + y 3 + z,
2/3= P
26 = 2 2 + Z 3 + Z t
2A = 36 -p-q
34 = x, + z t + z,
y, = 16 - 45 + ? + r -t,
35 = y, + y s + z.
*,= >
36 = 2/3 + ^ + *,
* 2 = r
45 = x, + z t + z.
5- 3 = ,
46 = x, + jh + y.
* 4 = 26 -r-s ,
56 = #1 + #2 + #3
*.- *,
278 OX THE THEOREM OF THE FINITE NUMBER [761
connecting *,, y,, *,,...,*, with the terms 12, 13, etc. of R, (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 x lt y lt z lt ...,*, 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 from the equations
12 + 13 + 14 + 15 + 16 = 0,
12 + 23 + 24 + 25 + 26 = 6,
16 + 26 + 36 + 46 + 56 = 0.
The equations are thus insufficient to determine the values of x lt y lt z l , ..., z t , 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 y a , z lt z,, z 3 , z, 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 x lt y,, Zi z t , 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 R e ,
= x l . 12 . 34 . 56 + y, . 12 . 35 . 46 + *, . 12 . 36 . 45 + . . . + z, . 16 . 25 . 34,
equal in general to a difference of squares R l .
Suppose in such difference of squares R l we have any term, say 12 . 34 . 56,
occurring with the coefficient 1. Since the expression represents a square R e , 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 - 1 2 . 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 R t (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 .R, ; and similarly each of the other
combinations is either a square R, or a sum of squares R lf 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 COVARIANTS OF A BINARY QTJANTIC. 279
or, whatever the negative terms may be, we get rid one by one of each negative
term ; and thus ultimately express R t as a sum of squares R t and R z . Or, what
is the same thing, the invariant R a originally expressed as a rational function of
invariants R^, is finally expressed as a rational and integral function of invariants
R, and R,.
Similarly for a root-quantic of any even order n, we have the general square
R e expressed, first as a difference of squares R lt and then as a sum of squares
R lt R,., or it may be higher squares R 3 , &c., but certainly as a sum of a finite
number of squares R t . For a root-quantic of any odd order n, the investigation
would be of a somewhat different form, since here there are no squares R lt but the
lowest squares are squares R 2 of a form such as 12.23.34.45.15; but the general
conclusion would still follow that every square R 6 is a sum of a finite number of
squares Rj,. And a like reasoning would apply to covariants 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, ...$, y) n . Consider for a moment the before-
mentioned case of the invariants of the root-quartic
(x - ay) (x - fty) (x - 7 y) (x - By),
now put
= -(a, b, c, d, e$x, y) 4 ;
tv
and to make the reasoning clearer, take a, b, c, f, g, h = (a - B) (ft 7), (ft - 6) (7 - a),
( y -B)(a-ft), (a-S)(7-/S), (ft-B)(a-y), (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, ...$#, y) 4 , qua rational and integral function of the coefficients, is,
when expressed in terms of the roots, a rational and integral function of the roots,
and then qua, invariant is a sum of monomial invariants, and as such a rational and
integral function of a, b, c, f, g, h. But every such rational and integral function
of a, b, c, f, g, h is not a symmetrical function of a, 0, 7, B, and consequently not
in the present theory an invariant of (a, ...$#, 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
(r(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 bc + ca + ab and (b-c)(c-a)(a-b); the conclusion being,
that the invariants of the quartic (a, 6, c, d, e~$x, y? are all of them rational and
integral functions of the last-mentioned two functions, that is, of
/, = ae - Ibd + Sc 1 , and J, = ace - ad* - \fe + Zbcd - c 3 .
As regards the group G(&, b, c, f, g, h) of 6 substitutions upon a, b, c, f, g, h,
observe that the 24 substitutions of (a, /9, 7, B) operating upon a, b, c, f, g, h give 6
substitutions taken each four times; for instance, the substitutions 1, a/9 .78, 07 . @S,
a&.fty leave each of them a, b, c, f, g, h unaltered, that is, they each give the
substitution 1. And we thus find for the group (?(a, b, c, f, g, h) the 6 substitutions
1,
abc . fgh,
acb . fhg,
af . bh . eg,
ah . bg . cf,
ag . bf . ch.
For the functions of a, b, c, f, g, h, which remain unaltered by the substitution of
this group, observe that we have f, g, h = a, b, c ; so that any function of
the six letters may be represented as a function of a, b, c. An odd symmetrical
function, for instance abc, does not remain unaltered, for it is by any one of the last
three substitutions changed into fgh, that is, into abc; on the other hand, the
two- valued function (b c) (c a) (a b) does remain unaltered : the functions which
remain unaltered are therefore the even symmetrical functions of a, b, c (that is, the
symmetric functions a 2 + b 2 + c 2 , or ab + ac + be, &c., which are of an even order in
a, b, c conjointly), and the same even functions multiplied by (b c) (c a) (a b) ;
and having regard to the relation a + b -I- c = 0, all these can be expressed as already
mentioned as rational and integral functions of be -I- ca + ab and (b c) (c a) (a - b).
The proof applies to the general case of the theorem (C), viz. taking the theorem
(A) to be proved, and putting the root-quantic
(x -ay)(x-py)... = - (a, ...$#, y),
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. 244258.]
I WISH to reproduce in part 33, "Coincidenz von Schnittpunkten einer Geraden
mit einer Flache " of Schubert's very interesting work Calcul der abzdhlende 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 Postulanduin 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 ; p^ the condition that the point 1 lies on a given
plane ; then gp l 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,
g t the condition that it lies in a given plane,
ff p the condition that it passes through a given point,
then ffe+ffp 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,
p g the condition that the point shall lie in a given line,
then p' = p ff .
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,
g t the condition that it lies in a given plane,
g p the condition that it passes through a given point,
then g'=g e +g p .
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 SUKFACE.
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 lie on a given plane,
a 2
^f >j > ))
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 ep+qg 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 8 lt , 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, plus 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, , m 2 for the orders of the surfaces
iS, , <S 2 respectively ; then it is to be shown that Xe, Xp, Xq, Xg are each = km^m^
(values which satisfy the equation). First Xe : the points 1 and 2 here coincide at
a point of the curve of the order TO,ra 2 , which is the intersection of S, and S?;
the regulus meets this curve in kn^m* points, and through each of these we have a
line of the regulus having upon it the two coincident points ; that is, Xe = km^m^.
Next Xp: the point 1 is here on the plane curve of the order m lt which is the
intersection of <S, with the corresponding given plane ; the regulus meets this plane
curve in km, points ; through each of these we have a line of the regulus intersecting
S., in TO, points, any one of which may be taken for the point 2 ; that is, the
number of subjects is Xp, = km, . m 2 . Then Xq : in precisely the same manner we
have Xq = km,.m l . Lastly Xg : the given line meets the regulus in k points, and
362
284 ON SCHUBERT'S METHOD FOR THE [762
through each of these there is a line of the regulus meeting <S>i in wi, points, any
one of which may be taken for the point 1, and meeting <S> 2 in m, points, any one
of which may be taken for the point 2; the number of the subjects Xg is thus
Xff, = k.m l .m t .
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 f,
bearing upon it the two points 1 and 2 ; we consider in conjunction therefore a given
line X, and through this draw the planes P,, P a 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 /9' 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 /9 positions
of the point 2 ; then, 1 lying in a given plane, the number of subjects is a/9, or
we have Xp = a/9 ; and, similarly, Xq = a'/9'. Take now for the point P! an arbitrary
plane through X; then, 1 lying on this plane, the number of the points 2 is =0/8,
or, since each of these determines with X a position of the plane P 3 , the number
of these planes is = a/9, that is, it is = Xp ; and, similarly, taking P 2 an arbitrary
plane through X, the number of the planes PI is a'/S 7 , that is, it is = Xq ; viz. the
two planes P,, P 2 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 f 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
e=p + q-g.
762]
CONTACTS OF A LINE WITH A SURFACE.
285
The like remarks would apply to the before-mentioned equation (subject a point)
p*=p y : either adding to it a onefold condition X, and so taking it in the form
Xff = Xp g , or understanding it in its original form p*=p g 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 p z =p g has, the equation e=p+qg 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 Jf + do. of M=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 + qg), then the equation in question, Me = Mp + Mq Mg,
is to be interpreted as meaning
KM e = KMp + KMq - KMg,
that is,
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
(A + B) e = (A + B)p + (A + B) q - (A + B) e.
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
we may write
(A +B- C) e = (A +B- C)p + (A + B - G)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) = ;
286 ON SCHUBERT'S METHOD FOR THE [762
that is,
c'-(p + q-9? = Q,
or
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 w, 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. e,g, = n(n-l),
2. e t b,g e = n,
3. f s g e = 3(n-2),
4. e,ff p = (-l)(n-2),
5. 3 6 3 2 = 2)i,
6. e~g e = ii(-2)(n-3)(n+3),
7. e*g p = in (-!)( -2) (re -3),
8. e^b,*
9. e-sAc, =
10. e 4 g =
11. eA = ?i (lire -24),
12. e^g = n(n-3)(7i-
13. e K b, =
14. e a b, = n(n-2)(n-4)(n'
15. e/; = ^re(n-3)(re-4)(ra-5)(re 11 -|-3n-2),
16. t^b. = in(w-2)(re-4)(rc-5)( s + 5tt+12),
17. 5 = 5ren-47w-12
18. e t , = 2n(n-4)(n-5)(n + 6)(3n-5),
19. 33 = iw(re-4)(n-5)(n-(-3n s +29n-60),
20. e,,. = in(-4)O-5)(n-6)(re 3 +9re 2 +20re-60) )
21. e^ = 1 J 5 n(n
22. 6^6, = ire(ft
23. e,6, s = (
24. e.b^di = 7i s (n - 4) (2n 2 - 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. e a denotes a
2-pointic intersection, 632 a 3-pointic and a 2-pointic intersection. The letters b, c, d refer
to the points of contact or intersection, thus 63563, b 3 is the point of 3-pointic intersection ;
^b s , &i is one of the points of simple intersection; 6 : is also the condition that the point
in question lies on a given plane ; g, g,, g e , g p have their ordinary signification explained
a little further on. Thus (15) e^g 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 formulae used in the investigation of the foregoing
results.
Subject a line having upon it a point,
Postul.
p the condition that point is in a given plane 1
Pg line 2
g line meets a given line 1
g e is in a given plane 2
g p passes through a given point 2
g, 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.
Pa = P*
2 (logical)
Pg = P~ + 9e
2
9* = 9e + 9p
2
9* = 99'
3 (logical)
9> =99p
3 (logical)
Pffp = p" + 9>
3 (demons, infra)
p< =
4
g<9j> =
4
9<? =G
4
2 />
4
P 3 9 =P'9e
4 (demons, infra)
P9> ~P 2 ffp
4
Pffi =P*ge + (;
r 4
P 3 g e = o
5
p 3 g p =pG
5
Jfy. =p
288
ON SCHUBERT'S METHOD FOR THE
we have =9e+P t -pg, = 9,+9p-g*, and thence
0=
-P9e~P9p + P9*
= P*9<- from pg=p*+g,, we have p*g=p t + p*g e = p' ! g e , since ja 4 = 0,
pg, = p*g e + G g, =gg e pg e = pgg e = (p* + g e )ge = pg, + 6,
[762
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 p l , p t , ... refer to these points respectively; thus, ^>, is the
condition that the point 1 may lie on a given plane ; and then, writing
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 e a , ee' by e 3 , ee" by e^, &c.
We thus have
-5 r ) (PI
(p, +p 3 -g) (p 3 +p t -
(Pi + p. - g) (p, + p 3 - g) (P! +p t -g) (p, + p e - g),
a -g)(p i +p i -g)(p<+p s -g\
-g)(p t +p i
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
pfpf, Pi'paps, Pip*p 3 p t , and G.
(Schubert says, as a linear function of these four symbols and Pip 2 g e ', but in fact p^g?
is =p-?p.?.)
Observe, first, that we may, p. 287, in all the general equations instead of p write
PI, PI, & c - > anc l, further, that any symbol containing for instance pf 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 p l s = Q.
We thus have p^p^ + ge, thence Pi 1 g=pi*+Piff f > that is, p 1 *g=p 1 g e ; and thence
further p l 3 g=p l t g e , that is, p^g e =0.
Again, from p. 2 g=p^ + g e , p 1 g = p l * + g e , we have
Pi (P* + 9') = PlPl (Pi + 9e\
which, in virtue of p^g e = and pfp* = 0, becomes
As a simple instance of the reductions, take
a<7> = (PI+PI - g)9s-
Here
Pi9>, -p-.9i, = p*9e + G, = G, since p l -g e = 0;
and
99 ~ 9*ff* = (9e + 9p) 9e = 9* + 9e9p = G > since 9* = > 9'9p = G S
whence the value is
As a more complicated example, take
e >, = (PI +p* - 9) (PI +p> - 9) (PI +Pt-g) (PI +p* - g)-
Observe that, after the multiplication is effected we may, in any way we please,
interchange the suffixes, p*psp4=p*pzp 3 , p^pf=p^pl, &c. ; the suffixes serve only to
distinguish from each other symbols in the same product (thus pf is different from
PipiPsP*), but there is nothing to distinguish one point of intersection from another.
Thus the foregoing expression containing the terms (p? + p 3 + p 4 + p s ) (PI gj*, these may
be combined into the single term 4p a (p l g) 3 ; expanding in powers of p\ g and
reducing in this manner, the value of e, is, in fact, found to be
= (Pi~i
Developing this in powers of g, omitting the terms containing pf which vanish, and
further reducing, the value is
6p*p,p, + Spipip 3 p* + g(- 12^> ~ 16^ip 2 p 3 ) +g* (6/V + ISja^) - Sp,^ + g 4 .
c. XI. 37
ON SCHUBERT'S METHOD FOR THE
[762
We have
g* = 20, tw' =p l g t =pSg e + G, = G.
Next for the terms in g*, from Pig=p?+g t we have
Pi 9=
and thence
or, since p^g, = as before, the whole term is = ISpSptf + 24G. The terms in g thus
become =g(6p*pt IGpiptpj), and from the same equation Pi9 = pi' + ge we find
Pi'P*9=P*P* and f t ptpig-pfp&+Pi t pf'
The value is thus finally found to be
= - lOpfpf - lOpfaps + 5p 1 p 3 p,p t + 106.
The whole series of like results is
PiPiPs PiPiPsP* G
1. c 3 g.
+ 1
2. b^f e
+ 1
1
3. c 3 ff t
3
- 3
4- 9 P
+ 1
5. V
- 2
+ 1
+ 1
6. 2i g t
+ 4
3
7- 2,, fff
+ 1
8. V
- 3
+ 2
+ 1
9. t, AM
- 2
+ 1
+ 1
10. 4 y
- 2
+ 4
- 2
11. 6 4
6
+ 1
+ 4
12. * a g
3
+ 6
- 2
13. 6 S
7
1
+ 2
+ 4
14. 6 a
- 6
- 3
+ 3
+ 4
15. 6eja gr
4
+ 8
- 2
16. 2,, 6,
7
4
+ 4
+ 4
17. c,
- 10
- 10
+ 5
+ 10
18. ta
- 10
- 16
+ 8
+ 10
19. 2CJ,
- 9
- 18
+ 9
+ 10
20. 2 sffl
- 9
-24
+ 12
4- 10
21. 24^
- 8
-32
+ 16
+ 10
22. 6f~,, 6
6
- 12
+ 8
+ 4
23. n 3 6,
3
+ 3
+ 1
24. j 6,0,^,
1
1
+ 2
762]
CONTACTS OF A LINE WITH A SURFACE.
291
But in these formulae p?p.?, p*p*p 3 , Pip- 2 p 3 p t , 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
^2 ^3 ^22 ^4 ^32 ^222 ^5 ^42 ^38 ^S22 ^2222 ^222^1 ^3^1
i = 23445656678 7 4 5,
and the values of the symbols are
p*p* =n 2 (m-2)(n-3) ...(n-i+I),
PipipsP* = 2 (2n 2 - 6n + 3) (n - 4). . .(n - i + I),
G = n (w-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(n !)( 2)(n 3) subjects satisfying the prescribed condition (that the line of the
subject may coincide with the given line). Hence here G = n(n !)( 2)(w-3);
and so in general G = n(n !)( 2)...( i + l).
Next, for pi 2 pf. 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 l ; and, similarly, p is the condition
that 2 may lie in a given line Z 2 - We take any one of the n intersections of L %
with F for the point 1, and any one of the n intersections of L 2 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,
' = TI J ( 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 Z,;
we therefore take for 1 any one of the n intersections of L^ with F; p% is the condition
that 2 may lie in a given plane P t , it lies therefore in the curve of intersection
of P 2 with F; and, similarly, 3 lies in the curve of intersection of a plane P 3 with
F; the two planes intersect in a line meeting F in n points a-, and the two cones,
vertex 1, which stand upon the plane curves respectively, intersect in the n lines
joining 1 with the n points a, and in n* n 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 n(n 2 n), ="( 1) lines; the remaining points 4, 5, ..., i are
any i 3 of the remaining n 3 intersections of the line with F ; hence the formula
3 = n*(n - l)(n-3)(n-4,)...(n-i+ 1).
For pipip 3 p t . 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
372
292 ON si iirr.Kin-'s MKTHOU FOR TIII: [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 2n' 3n?; this scroll meets
the fourth section in n (2n* 3n l ), = 2n 4 3n* points ; or we have this number of lines
meeting each of the four sections. But among these are included 3n s (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 n 1 n other lines, and we have thus n (n' n),
= n ! (n 1) lines; and similarly from the intersections of 2 and 4, and from the inter-
sections of 3 and 4, n'(n 1) and n a ( 1) lines, in all 3n*(n 1) lines. Hence the
number of lines meeting the four sections is
2' - 3n' - 3n" + 3n 5 , = 2w 4 - On' + 3> ;
taking any one of these for the line of the subject, the remaining points 5, 6, ..., i are
any t 4 of the remaining n 4 intersections, or we have the required formula
Pipip>p4 = i? (2ra 3 - 6n + 3) (n - 4). . .(n - i + 1).
The four numbers p*p*, p\"p^, Pippp4, G for any line of the table being now
known, we can at once calculate the required values e t g t , &c., as the case may be ; for
instance,
t' = 5, e, = - IQpfpf = -lQn*(?i-2)(n-3)(n-4)
-10n J (n-l)(-3)(w-4)
+ 5n" (2w 2 - 6n + 3) (n - 4)
+ WG +10n (n-l)(-2)(n-3)(n-4)
= on (n-4)(7n-12).
In fact, throwing out n (n 4), the remaining terms give
-lOtt'+SOn"- 60w
- 10n 3 + 40 2 - 30n
+ 10n-30 2 + Ion
+ 10n 8 - 60n s + HOn - 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 F n there is a curve order r, then of the tangent planes of F n
along this curve there pass r(n-l) through an arbitrary point of space; aliter,
class of torse is =r(w 1).
In particular, for curve of 4-pointic contact, r = re(ll-24), class of torse is
= n(n-l)(lln-24).
No. of tangent planes through line, or class of surface, =n(n-l) 2 .
762] CONTACTS OF A LINE WITH A SURFACE. 293
26. e 3 b s g = 6363- + f 3 g e = '2n + 3 (n 2), = n (3ra - 4).
e 3 &3<7, =(3 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(?i 2)(lln 24).
27. Order of parabolic curve = 4n (n 2).
Order of regulus formed by parabolic tangents
= 2n(n-2)(3n-4).
The parabolic curve and curve of contacts of an e 4 tangent meet in
points, i.e., they touch in 2n(n 2) (lira 24) points.
28. Umbilici. No. is =2n(5n a -14n + ll).
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 t ,
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
= In (n - 2) (n - 4) (3n 2 + 5n - 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) (n 3 + 3n 2 + 13>i - 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)(27w - 13n 2 - 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) (4 6 - 4m 4 - 95n 3 + 99w 2 + 544n - 840).
34. The curve of contacts b 3 of an 63., 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 b 3 of an 32 tangent has, with the curve of contacts of
an e 4 tangent, 2-pointic intersections at the contacts of an e 5 tangent ; and
has also simple intersections with the same curve, 1 at the contacts b t of
an fu tangent, 2 at the points where the chief tangents are e 4 and 633.
294 [763
763.
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
[From the Quarterly Journal of Pure and Applied Matliematics, vol. xvil. (1881),
pp. 258276.]
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
(x? + a;,') (y* + y,") = (a^, + x^)* + (x^ - a^,)".
For the 4 squares (for which the nature of the theorem is better seen) it is
(a;, 8 + x? + a;, 1 + x?) (y, 2 + y, a + y," + y*) =
763] ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
or, as this may be written,
295
x?
y
= (12 + 34) 2
+ (13 - 24) 2
+ (14 + 23) 2 ;
where 12 is used to denote x^ sc^, &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 :
(*,'+... +* 8 >)(yr + ...+y 8 2 ).
= (12 + 34 + 56 + 78) 2
+ (13 + 24 + 57 + 68) 2
+ (15 + 26 + 37 + 48) 2
+ (16 + 25 + 38 + 47)"
+ (17 + 28 + 35 + 46) 2
+ (18 + 27 + 36 + 45) 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 e a , that of 34, e u ; and so in
other cases. It is however assumed that , e M , e M , e rs ; e,,, e u> e, 5 , e 16 , e 17 , e a are
each = + 1.
We have then on the right-hand side triads of terms such as, 2 into
e a e 12 . 34 + e u e M 13 . 24 + e, 4 e a 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
f\i f 3t = ~~ e lS e S4 = e 14 e Z3 j
or, using now and henceforward when occasion requires, 12, 34, &c. to denote e is , 634, &c.
respectively, we have
13 . 24 = - k,
14.23 = + ,
where k, =1, has to be determined (in the actual case we have 12 = + 1, 34 =+1,
13 = 1, 14=1; and therefore the first equation gives k=l, and the other two then give
24 = -1, 23 = + 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 -r 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
13
14
15
16
17
18
+ 1
+ 1
+ 1
+ 1
+ 1
+ 1
+ 1
23
+ 1
24
-1
25
+ 1
26
-1
27
+ 1
28
-1
34
+ 1
35
a
-6
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
6
78
+ 1
C. XI.
38
298 ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. [763
viz. the original assumptions 12 = + !, &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 6, c, d. The tetrad 2358 then
gives
23.58 + 1 c,
25.38 - 1-6,
28.35 +-1 a,
that is, -a = 6 = 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, b, c, d, the tetrad 3456 then gives
34.56 +11,
35.46 - a d,
36.45 + b c,
that is, 1 = - ad = be, hd similarly the last two tetrads 3478 and 5678 give
1 = ac = bd and 1 = ab = cd respectively ; substituting the values in terms of 9,
the several equations give only 0*=l, that is, #= + 1 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 iq 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.gk, 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
382
300
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
Four forms of second-half.
I. II.
[763
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
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]
OX 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.
[763
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
14 13 16 15
13 14 15 16
16 15 14 13
15 16 13 14
10 9 12 11
9 10 11 12
12 11 10 9
11 12 9 10
9 10 11 12
10 9 12 11
11 12 9 10
12 11 10 9
14 13 16 15
13 14 15 16
16 15 14 13
15 16 13 14
1234
2143
3412
4321
6587
5678
8765
7856
13 14 15 16
14 13 16 15
15 16 13 14
16 15 14 13
10 9 12 11
9 10 11 12
12 11 10 9
11 12 9 10
6587
5678
8765
7856
1234
2143
3412
4321
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 1C 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
ilown. 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.
MM
OJf THE THEOREM* OF THE 2, 4, 8, AJTD 16 8QUABBL
L IL IIL IV.
1
1*4
^^
mat*
5 <
7 8
9 10
11 12
13 14
15 16
1
357
6 8
9 11
10 12
13 15
14 16
1
458
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 !> 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
: I'i
1 8 9 15
1 8 'J 14
1 H 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
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
305
2
358
same
same
same
6 7
9 12
10 11
13 16
14 15
2
457
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 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
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
PJ 16
12 15
3
456
BttllHl
HHI1IK
same
7 8
9 10
11 12
i:i 14
15 16
C. XI.
89
306
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. IT. III. IV.
[763
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
S
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
5
678
same
same
same
9 10
11 12
13 14
15 16
763]
ON THE THEOEEMS 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
8
9 11
10 12
13 15
14 16
8 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
same
392
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
db + cd + ef+ &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 e, a , e M , &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 e, 3 , e u , &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
a = b = c = d = 6,
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, /9, 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, <r, T; 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, =\ 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 d, \, /*, v, p, er, 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 0, \, /t, v, p, <r, T
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
- 1
11
+ 1
12
_ i
13
+ 1
14
- 1
15
+ 1
16
- 1
3 4
+ 1
same
same
same
5
-e
6
6
7
e
8
-e
9
e - \
10
f x
11
- A.
12
-/ -x
13
<J -/*
14
h M
15
-9 f-
16
-h -u
310
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
[763
4 5
e
same
same
same
6
7
-e
8
-e
9
i
X
10
j
X
11
-3
-X
12
_ i
-X
13
ft
M
14
/
/*
15
-/
-M
16
-k
-/*
5 6
+ 1
+ 1
+ 1
1
7
-e
-6
-e
-6
8
6
e
9
m
V
in
V
m
V
m
V
10
n
V
n
V
n
V
n
V
11
o
- P
o
p
o
p
o
p
12
P
P
P
p
P
p
P
p
13
m
V
n
V
o
p
-P
p
14
n
V
m
V
-P
p
o
p
15
P
-P
p
m
V
n
V
16
-P
P
p
n
V
m
V
6 7
e
e
e
e
8
e
6
e
e
9
i
V
1
V
q
V
q
V
10
r
V
r
V
r
V
r
V
11
s
P
s
p
8
p
s
p
12
t
P
t
p
t
p
t
p
13
r
V
-?
V
-t
p
8
p
14
-9
V
r
V
s
p
- t
p
15
-t
- P
-8 -
p
If __
V
-9
V
16
s
-p
- t
p
-9 -
V
r
V
7 8
+ 1
+ 1
+ 1
+ I
9
u
<r
u
er
u
a
u
or
10
V
a- v
tr
V
0"
V
a-
11
w
- r w
T
w
T
w
T
12
X
r x
T
X
T
X
T
13
w
T
X
T
u
<r
V
tr
14
- X
T
w
T
V
<7
u
(7
15
u
<T
V
<r
w
T
3J
T
16
V
<r
u
<r
X
T
w
T
763]
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
I. II. III. IV.
311
8 9
y
CT
y
a
y
CT
y
<T
10
z
cr
%
V
z
cr
z
<T
11
a.
T
a
r
a
T
a
T
12
13
T
ft
r
ft
T
ft
T
13
-ft
T
a
T
z
cr
-y
cr
14
a
T
-ft
r
-y
<T
z
<7
15
z
CT
-y
cr
-ft
T
a
T
16
-y
(7
z
cr
a
T
-ft
T
9 10
+ i
+ 1
+ 1
+ 1
11
e
-\
e
-X
e
-X
e
-X
12
i
X i
X
i
X
i
X
13
m
- v q
V
u
cr
y
cr
14
1
V
m
V
y
<r
u
cr
15
u
(T
y
<T
m
v
1
V
16
y
cr
u
<r
q
v
m
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
z
cr
15
z
CT
V
<T
r
v
n
V
16
V
cr
z
cr
n
v
r
V
11 12
+ I
+ 1
1
+ 1
13
w
T
a
T
o
~ P
s
p
14
a
T
w
T
8
P
p
15
O
- p
s
p
W
T
a
T
16
8
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
T
X
T
16
p
P
t
~ P
X
T
ft
T
13 14
+ 1
+ 1
+ 1
+ 1
15
9
I*
g
- P-
g
~ P-
9
- P-
16
k
P-
k
P-
k
P-
k
P-
14 15
I
f-
I
P-
i
P-
I
P*
16
h
P-
h
P-
h
P-
h
P-
15 16
+ 1
+ 1
+ i
+ 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 5 915+-0.-0- that is,
3 9 515 X . p da- = Xp = /ti/,
3 15 5 9 + fi.-v
3 5 10 16 +-0. a-
3 10 5 16 - X.-p
3 16 5 10 +-/t. v
3 5 11 13 +-0. -r
3 11 5 13 - X. v
3 13 5 11 +-/*.- p
3 5 12 14 +-0. T
3 12 5 14 - -X.-v
3 14 5 12 + //. . p
Second arrangement.
3 5 9 16 +-0. a-
3 9 5 16 --X.-p
3 16 5 9 +-/. v
3 5 10 15 +-0. a-
3 10 5 15 - X.-p
3 15 5 10 + 11.. v
3 5 11 14 +-0. T
3 11 5 14 - \.- v
3 14 5 11 + /A. p
3 5 12 13 +-0. T
3 12 5 13 \.- v
3 13 5 12 +-/*. p
763] ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES.
Third arrangement.
3 5 9 13 + -0.-<7
3 9 5 13 X. p Of = Xp = fj,v,
3 13 5 9 +-u.-i;
313
0<r = Xp = fjiV,
' = \v = pp.
3 5 10 14 +-0. <r
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 + it. - p
3 5 12 16 +-0. T
3 12 516--X.-J/
3 16 5 12 +-/*. p
Fourth arrangement.
3 5 914+-0. a-
3 9 5 14 --X.-p
3 14 5 9 + p.. v
3 5 10 13 +-0. o-
3 10 5 13 - X.-p
3 13 5 10 +-/*. v
3 5 11 16 +-0. T
311 516 X. P 0r = \v = p,p,
3 16 5 11 +-/*. p
3 5 12 15 +-0. -T
3 12 5 15 --X.-j/ 0T = -Xz/ = - /tt p.
3 15 5 12 +-M- P
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.
Bcr = Xp = /UP,
- = Xp =
C. XI.
40
314 [764
764.
THE BINOMIAL EQUATION of -1=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 of 1 = 0, Trisection and Quartisection,"
Proceedings of the London Mathematical Society, vol. xi. (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 = i) 1 +..., Y=i)*+..., etc.,
with Reuschle's selected prime root g, can be (for the primes 5n + 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
X' = a b c d e
XY=f g h i j
XZ = k I m n o;
that is, we have
* = (o, b, c, d,e%X, Y,Z, W, T),
and thence, by cyclical permutations,
Y* = (e, a, b, c, d$ ), etc.;
viz. from the value of X' we have those of F 3 , Z 1 , W 2 , T*; 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 X p 1 = : QUINQUISECTION.
315
From the equation X+Y + Z+W+T=-1, multiplying by X and then substi-
tuting for X", XT, &c., their values, we obtain
-a=I+f+k +m+g,
-b = g + I + n + h,
c= h+ m+ o +i,
d= i +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=
k+l
a + b+ c
whence also
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 X*, 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 (f, 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 =^(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 + l, 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.
402
316
THE BINOMIAL EQUATION a?- 1=0: QUINQUISECTION.
[764
TABLE 1.
a
b
c
t/
p
/
y
h
t
j
k
i
m
n
11
- 2
i
- 2
- 2
2
1
1
1
1
31
_ ^
- 6
- 6
- 4
- 5
1
2
1
2
2
2
1
1
41
- 8
- 5
- 6
-6
- 8
3
2
1
2
2
2
2
1
1
61
- 10
- 9
- 12
-8
- 10
3
2
2
3
2
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
p
**'***!
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
765]
317
765.
ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE.
[From the Proceedings of the London Mathematical Society, vol. Xll. (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 Mecaniqw Analytiqiie, 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) five geometrical conditions, which
I represent by Ui = 0, U 2 = 0, . . . , U t = 0, the equation of equilibrium is
8 {(X'Sx + Y'Sy + Z'Sz) dm + T, 8 Z/i + T,S U, + T 3 & U 3 + T t & U t + T 6 B U,} = 0,
where T lt T t , ..., T s are the indeterminate multipliers, representing colligation-forces
which correspond to the five geometrical conditions respectively.
Taking (, ij, ) for the coordinates of a particular point P on the generating
line; p, q, r for the cos-inclinations of the line (whence Ui=p' + q* + r* 1 =0 is one
of the geometrical relations), and p for the distance of dm from P, we have
<e, y, z= f+ pp, r) + pq, ?+ pr,
Sx, By, Sz =
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, L, 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 0; retaining the same summatory symbol S, 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
(> *?> ?) ^d taking <r 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, , 17, f, p, q, r are given
functions of <r ; 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 2 + 5" + r 3 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'FP (=180 C -TT), and the inclination G'F 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 <7. Taking 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 = / - ttdo- ; also the inclination GP to G'F
is an infinitesimal, = @d<r : we have /, ft, <s> given functions of <r. It is to be
remarked that these conditions imply, inclination of G'F to tangent plane GPT at P
has a given value Ado-; in fact, if through P we draw a line Py parallel to P'G',
then, if P is regarded as the centre of a sphere which meets PG, Py, 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
/ ttdcr, I, and d<r, and of which the perpendicular g'm is = Ado- ; we have thus
an infinitesimal right-angled triangle, the base and altitude of which are fld<r, Ada;
g Qd<r
and the hypothenuse is do-; whence <e) 2 = fl a + A 2 . In the case of the developable
surface A = and 8 = fl. It may be remarked that, when the curve on the skew
surface is the line of striction, we have O = ; in fact, taking P to be on the line
of striction, the line
___.
qr' q'r rp' - r'p pq' p'q '
through (, 17, f) at right angles to the two generating lines, meets the consecutive
generating line X, Y, Z=' + pp', y' + pq', ' + pr' ; and the condition that this may
be so is easily found to be fl = 0.
Take, for a moment, p', q', r' for the cos-inclinations of the consecutive generating
line FG'; we have
Ip + mq + nr cos /,
lp' -f mq + nr' = cos (7 ld<r),
pp' + qq' +rr' = cos do- ;
and then writing p', q', r p + dp, q + dq, r + dr, and observing that the equation
p'* + q' 1 + r' 2 = 1 gives
pdp + qdq + rdr = -% (dp* + dq* + dr 2 ),
these equations and the before-mentioned two equations become
(Z7.)
I 2
lp + mq +nr cosl = 0,
( Ut) Idp + mdq + ndr fl sin Ida 0,
(U,) dp*
which equations, considering therein I, m, n as standing for their values -f , ~ , -'r- ,
acr d& der
are the geometrical relations which connect the six variables f, 17, p, q, r, considered
as functions of a. And in these equations /, fl, 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
9> 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 f, rj, f, 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 <r = 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, fl = 0,
then the five equations are
p*+ 2 s + r-- 1 =0,
P + m 2 + n 2 - 1 =0,
lp+ mq+ nr cos 7 =0,
dp* + dq* + dr* sin 2 7 d<r 3 = 0.
Idp + mdq + ndr = 0.
We assume first
C (P> ? r)=p i + q*-i jl tan s 7, =0;
and secondly
C(p, q, ') = ?-, =0.
Then, in the former case, we find the solution
P, q, r = - sin 7 sin <r, sin7cos<7, 0087;
, T), = cos <r, sin <r, ;
giving
x, y, z = cos(r psiny sin <7, sin <r + p sin 7 cos a, cos 7 ;
and consequently
the hyperboloid of revolution. And, in the latter case,
P, q, r = cos (<r sin 7), sin (o- sin 7), 0,
, *7, (T = cot 7 sin (<r sin 7), cot 7 cos (or sin 7), a sin 7,
that is,
, cotycosz + psinz,
,
whence
a; sin z y cos z = cot 7,
a skew helicoid generated by horizontal tangents of the cylinder x 3 + f = cot 2 7. This
is a known deformation of the hyperboloid.
765]
ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE.
Returning now to the mechanical problem, we have to consider the terms
S . T
+ T 3 S (lp + mq + nr cos /)
The first term gives, under the sign S,
T^pSp + q&g + r&r). (*)
The second term gives, in the first instance,
%*(ld&t+mdki + nd&Q;
do-
or, since in general
Sfldog = Q"Sf - fl'Sf + S(-dfl. 8|),
then, attending only to the terms under the sign S, these are
T 3 (ISp + mSq + nSr)
+ T,(p&l + qSm + r$n),
The third term gives
where the second line,
attending only to the terms under the sign S, gives
The fourth term gives
(*)
(*)
4
\da-
where the first line, written under the form
T t (dp , Sfc dq s
j- -f- dot + T* dot] + -j-
da \da- da dcr
and attending only to the terms under the sign S, gives
-(*)-(*)*-('*.
<*>
321
c. xi.
41
322
ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE.
[765
and the second line, attending in like manner only to the terms under the sign S,
gives
The fifth term, written under the form
and attending only to the terms under the sign S, gives
d , dp s d T dq K d -, dr
---
(*)
where in each case I have marked with an asterisk the lines which present them-
selves in the final result.
Hence, joining to the foregoing the force-terms
XBl; + YBi) + ZB!;+LBp + MBq + NSr, (*)
and equating to zero the coefficients of 8f, Srj, 8f, Bp, Bq, Br respectively, we have
= X - d Tl-Tv -T^
" j *2* / j - t -3r' j * 4 T
da da da da
0=F
- T 3 n - T 3 r -
dcr dcr d<r
da 'da'
d m dr
dp
dl;
d$
where it will be recollected that I, m, n stand for 3*. -^ , f, the variables being
da- dcr da-
f. '?. ?. P, q, r, and a. The elimination of T lt T,,, ..., T s from the six equations
should lead to a relation between f, rj, p, q, r, which, with the foregoing five
relations, would determine the six variables f, r), f, p, q, r in terms of cr.
In particular, the forces and moments X, T, Z, L, M, N may all of them
vanish; assuming that T,, T,,...,^ do not all of them vanish, we still have the
sixth relation, which (with the foregoing five relations) determines f, i\, f, 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 CURVATURE OF A CURVE ON A
SURFACE.
[From the Proceedings of the London Mathematical Society, vol. xn. (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 Geome"trie (Paris, 1850), see pp. 574 and 575, the following
formula,
ds
di
;r
ds
dG
du
C<
dE
cos sn
p. 2
which gives the radius of geodesic curvature of a curve upon a surface when the
position of a point on the surface is denned by the parameters u, v, belonging to
a system of orthotomic curves ; or, what is the same thing, such that
ds 1 = Edit? + Odif.
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 1 = Edp^ + 2Fdpdq + Gdq\
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' = PQ 2 . 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.
412
324 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766
The notation is that of the Memoir, " Disquisitiones generates circa superficies
curvas" (1827), Gauss, Werke, t. III.; see also my paper "On geodesic lines, in
particular those of a quadric surface," Proc. Lond. Math. Society, t. IV. (1872),
pp. 191211, [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 + J(ftc = adp + a'dq + % (a.dp> + 2a' dpdq + a" dq'),
dy + i<fy = bdp + b'dq + $ (dp + 2# dpdq + /3"cfy 2 ),
dz + \$z = cdp + c'dq + b(ydp* + Zy'dpdq + 7" dq") :
E, F, G = a t + b t + c\ aa' + bb' + cc', a'' + b"> + c''-' ; V* = EG-F*:
and therefore
ds* = Edp> + ZFdpdq + 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 6 ; 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 E Jf E 3 , etc., to denote the
j rr 7 rr
differential coefficients - , etc. In the first instance, 6 is taken to be an
ii/i aq
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 0.
The coordinates of a consecutive point for the one curve are then
x+da;+ Jt&x, y+dy + ^d*y, z + dz + $d?z,
where
dp = P 'd0 + WdP, dq = q'dff + i gfdffi ;
hence these coordinates are
x + (ap 1 + a'q 1 ) d0 + (a/-' + 2a>y -f a'V) dffi + (op" + a'q"
and for the other curve they are in like manner
x + (ap + a'q') d0 + $ (ap'' + Za'p'q' + a" ? ' a ) dffi + $ (aP" + a'Q") d^,
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, | {6 (p" - P") + b' (q" - Q")} dffi,
k{c(p"-P") + c(q"-Qr)\dfr,
and consequently the distance QQ' of the two consecutive points Q, Q' is
e, F,
The squared arc P(f is
= (E, F,
and hence, if as before 2p.QQ' = PQ*, that is, - = 2QQ' -H PQ 2 , then
1 = */(E, F, G%p" - P", q" - Q'J
p ~ (E, F, G%p', qj " '
the required formula for p.
Second formula for the radius of relative curvature.
We now take the variable 6 to be the length s of the curve measured from a
fixed point thereof, so that p, p", etc. denote ~ , -" , etc. We have therefore
cts c(/s~
l=(E,F,
and the formula becomes
1
But, differentiating the above equation as regards the curve, we find
o = 2 (E, F, G~$p', q'Jip", q") + (E, F, G$p, q') 1 ,
where E, F, G are used to denote the complete differential coefficients E-^p' + E^q', etc.
And similarly, differentiating in regard to the tangent geodesic, we obtain
and hence, taking the difference of the two equations,
/\ ^_ / TTf Jjt
Hence, in the equation for -, the function under the radical sign may be written
(E, F, G^p', qJ.(E, F, G%p"-P", q"-Q!J-{(E, F, G%p',
326 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766
which is identically
= (EG - F>) [p (q" - Q") - q' (p" - P")}'.
Hence, extracting the square root, and for VA'tr A 1 * writing V, we have
or say
= V{p'(q"-Q")-q'(p"-P>%
- p =V(p'q"-q'p")-V(p'Q"-q'P"),
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. vin. of this Collection, p. 160],
writing EG-F*= V\ and P", Q" in place of p", q", the differential equation of the
geodesic line is obtained in the form
- (Fp' + Gq') {#,/' + 2E,p i q' + (2F t - 0.) q'*}
or, denoting by ft 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 ft is a cubic function of p', q' : we have
ft = (2l, S, S, 3>&/. <?') 3 .
the values of the coefficients being
2l = 2#F,- EE,- FE lt
= 2EG, + 2FF r - 3FE 3 - GE, ,
6 = EG, + 3FG, - 2FF, - 2GE,,
D= FG 3 -2GF,+ GG t .
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" = ; therefore also Gq r *=l,
and, if R be the radius of geodesic curvature, then
Similarly for the curve q = const. Here <?' = (), <?" = 0; therefore Ep'* = I, and, if
be the radius of geodesic curvature, then
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 o
where the term in { } is
= 2lp' - ^ p' + 93p Y + Sp V 2 + $V 3 - ?'
The terms in 21 are
and those in 2) are
Hence the whole expression contains the factor p'cf, and is, in fact,
or substituting for 21, 53, S, 2) their values, this is
' (~ GE, + EG, + 2 y ' - 2^ - FE,
+ EG, -
FG l -
328 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766
say this is
. = p'
and the formula thus is
Taking <f>, to be the inclination of the curve to the curves q = const., p = const.,
respectively, and a> (= <f> + 0) the inclination of these two curves to each other, then
cos
Fp'+Gq' Ep' + Fq' F
<f> = -- - , cos 6 = - " , cos to = -TT=
'
Vff Vq' V
sin rf> = -*- , sin = -i , sin &) = --
hence - = p'-JE, . - = q'*JG, and the formula may also be written
sin o) sin co
1 sin 6 1 sin <f> 1
- TJ o =
p sm w ^ sin ia S
The Orthotomic Case F-Q, or ds* = Edp- + Gdq\
The formula becomes in this case much more simple. We have
1 = Ep'- + Gq", V=^EG, w = 90, sin^ = cos^;
and the term Lp' + Mq' Becomes = EG EG, if, as before, E, G denote the complete
differential coefficients E^p' + E^q' and G t p' + G,q'.. The formula then is
1 1 4-^"* _ 1 S*
where the values -^ and ^ are now =^-/^- and -rr-rJj,, respectively. But we have
moreover <f> = tan" 1 ~ -, , and thence
q vtr
= - V(p'q"-p"q) - Ip'q' (EG - EG) ;
or the formula finally is
1 cos <f> sin cb f
~ B o + <P = 0,
/-* / o
which is Liouville's formula referred to at the beginning of the present paper. It
will be recollected that d>' is the differential coefficient ^ with respect to the arc s
as
of the curve.
7f>6]
ON THE GEODESIC CURVATURE 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) + JY (Lp + Mq),
which belongs to the general case. I find that these terms are, in fact, = <j> + ^p' ;
or, what is the same thing (since o) = <f> + 6 and therefore t0ip' + <o. 2 q' = <f> + 0), are
= 6 ot-fi'. It will be recollected that <f> 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 ; w is the inclination to each other of the curves p = b, q = c,
which pass through the given point of the curve, <a^ the variation corresponding to
the passage along the curve q = c, viz., u> + ca^s is the inclination to each other of
the curves p = b + db, q = c; and the like as regards 6 and a> 2 .
For the demonstration, we have, as above,
where
V =
and moreover Ep'' 1 + 2Fp'q' + Gq'* = I. In virtue of this last equation,
Py + (Fp' + GqJ = G ;
and we have
where
or, snce
D = (Fp' + Gq')p'V- Vp' (Fp' + Gq') ;
= EG-F"; and thence IV V = GE - IFF + EG, we have
D = ' {(Fp' + Gq') (GE - 2FF +EG)-2 (EG - F*) (Fp' + Gq')}.
Substituting herein for E, F, G their values
term in { j becomes
= Ip'* + Jp'q' +
where
7 = FGE, -
J =
K =
+ EFG lt
(-EG + 2F") G, + FGE* - 2EGF, + EFG 2 ,
(-EG+ 2F*) G,.
But from the equation ca = tan"" 1 ^ , differentiating in regard to p, we obtain
the
c. xi.
42
330 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766
or, for a writing
, / , f . , ,.
p' (Ep' 1 + ZFp'q' + (?</'). = Ep Iff* + 2-gpq '
we have
+ - l p' = -V(p'q"-p'W +
The terms in p'* destroy each other, and the form thus is
4, _ Utp > = _ V(p'q" - p"q') - ^p'q 1 (Lp' + Mq'\
where
L = ~ G + GE '
and, upon substituting herein for I, J, K their values, we find
= -GE l + EG, + ~ 1 - 2FF, - FE 2 + 2EF, -
-~-
viz., these are the values denoted above by the same letters L, M. The final result
thus is
1 q'JG p'v'E
p-R s -++"J>
= o toq',
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," Annali
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 t 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
767]
331
767.
ON THE GAUSSIAN THEORY OF SURFACES.
[From the Proceedings of the London Mathematical Society, vol. XII. (1881), pp. 187 192.
Read June 9, 1881.]
IN the Memoir, Bour, "Theorie de la deformation des surfaces" (Jour, de VEc.
Polyt., Cab. 39 (1862), pp. 1148), the author, working with the form dtf = dv* + g*du*
as a special case of Gauss's formula ds 2 = Edp- + ZFdpdq + Gdq", obtains (p. 29) the
following equations which he calls fundamental :
[IV.]
-= .
g dv
dT d.Hg
+-=*-*-*
***!
du
= 0,
dg
where #, is written to denote ~, and where (see p. 26)
H is the curvature of the normal section containing the tangent to the curve
v = constant,
H l 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
(MC-
422
332
ON THE GAUSSIAN THEORY OF SURFACES.
[767
Writing for greater convenience K in place of the suffixed letter H lt also V
instead of g, so that the differential formula is ds* = di? + Vdu-, the equations become
1 d'V
V di
dT d.HV
t dv
T-HK,
dv
dV
" =0;
du
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
= 2" - HK,
or, what is the same thing,
iV=V(T*-HK),
I wish to show how these formulae connect themselves with formulae belonging
to the general form ds 3 = Edp* -f 2Fdpdq + Gdq". These involve not only Gauss's coefficients
E, F, 0, but also the coefficients E', F', 0' 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.
da;, dy, dz = adp + a'dq, bdp + b'dq, cdp + c'dq ;
d 3 x = adp 2 + Zafdpdq + a"dq*,
d?z = ydp* + Zy'dpdq + y"dq 3 ;
A, B, C = bc'-b'c, ca'-c'a, ab'-ab; V*
E' = Aa + p + C y , F' = Aa' + B/3' + Cy', G' = Aa" + BQ"
so that E', F', G' are, in fact, the determinants
a, b , c
9
a, b , c
,
a, b , c
a', b', c'
a', b', c'
a', b' , c'
OC B ty
OC /3 *y
*", $', 7"
The equation for the determination of the principal radii of curvature is
(E'p - EV) (G'p-GV)- (F'p - FVy = 0,
767]
ON THE GAUSSIAN THEORY OF SURFACES.
333
which, in the particular case F=0 (and therefore V* = EG), becomes
(Ep - EV) (G'p -GV)- F'*p* = 0,
or, as this may be written,
F'-
p Ev)(p GV) EGV"~'
an equation which corresponds with Bour's form
and becomes identical with it, if
E' = EVK, G' = GVH, F' =
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
= 0;
- dp~F' I 1 " pE')~ G' : ( pG'J '
1 V
or, we see that to dq = corresponds the value - = vr Tr , and to dp=0 the value
p J^j V
1 f" ~\
- = ^y . Hence the former of these values of '- corresponds to Bour's du = 0, that
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 J =EVK, G'=GVH.
The formula da 2 = Edp* + 2Fdpdq + Gdq- agrees with Bour's ds i = dtf+g ; >du?, if
p = u, q = v, E=l, F = Q, G = g\ With these values, F 2 = EG- F 2 = f, or say g- V,
and Bour's equation is, as it was before written, ds 2 = dv* + V-du\ And we have to
find the three equations which, putting therein p=u, q = v, E=l, F=0, G=V*,
E' = VK, F'=- VT, G = V 3 H, 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 F (E,G, - E 2 G, -
-2 (EG-
- 2Fu + G u ).
:5:!4
ON THE GAUSSIAN THEORY OF SURFACES.
[767
In fact, writing herein E=\, F=0, and therefore the differential coefficients of E
and F each =0, the equation becomes
which is
or finally it is
J- 2F'
V=V(T*-HK).
The other two of Bour's equations are derived from equations which give
respectively the values of E 3 ' FJ and F.,' GI ; viz. starting from the equations
E' = Aa. + Bft +Cy ,
F' = Aa' + Bft' + Cy,
G' = Aa" + Bft" + Cy",
we see at once that E t ' and FJ contain, E t ' the terms Aa + Bft..+ Cy,, and F^ the
terms Aaj + Eft + Cfy/, which are equal to each other (a, = a,' since a and a' are
the differential coefficients #,,, x^ of x, and so /?, = /9 1 ' and y, = y 1 '). Hence
J - F,' = As*
and similarly
C 3 y-A 1 a.'-B ) ft' -C.y';
2 f 1 lr* 1 / *
F,' - 6,' = A,af + B,$ +
Here, from the values of A, B, C, we have
A=bc'-cV; 4 1 =/8c'-7&' +by' -cff; A, = 0'c' - y'b' + by" - c/3" ;
B = ca' ac' ; B t = yaf ac' + ca' ay' ; B 3 = y'a' a'c + ca" ay" ;
C = ab' - ba f ; C, = ab' - 0a' + a/3' - ba' ; C, = a'b' - & 'a' + aft" - bet" ;
and, substituting, we find
.EY - Fi = 2a'aa' + aa"a ,
if, for shortness, a'aa' denotes the determinant
a', a, a'
V, ft, ft'
c'> 7. y
and so for the other like symbols. Observe that, with
a, a, a, a', a"
b, b', ft, ft', ft"
C, c', y, y, y"
767]
ON THE GAUSSIAN THEORY OF SURFACES.
335
we have in all 10 determinants, viz. these are aa'a, = E'; aa'a', = F'; aa'a", =G';
aa'a"; and the six determinants ana', aa'a", aa"a; a'aa.', a'a'a", a'a'a. The foregoing
expressions of E.,' Fi and FG-[ 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, aa' + bb' + cc = F,
a'a + b'b + c'c = F, a' a' + b'b' + cc' = G,
aa +/S6 +70 = %E lt aa' + /3V + yc' ^F.-^E.,,
a'a + pb + y'c = E t , a'a' + PV + 7 'c' = <?,,
a" a + pit + y" c = F,-^G t , a" a + /3"b' + y"c' = G t ;
and if from the first five equations, regarded as equations linear in (a, 6, c), we
eliminate these quantities, and from the second five equations, regarded as linear in
(a', V, c'), we eliminate these quantities, we obtain two sets each of five equations,
= 0.
a,
a',
a,
a',
a"
= 0, and
a,
a',
,
a',
a"
b,
b',
ft,
ft*
8"
b,
b',
ft,
ff,
B"
c,
c,
7.
7,
7"
c,
c',
7.
7'.
7"
E,
F,
4^.,
IE,,
F.-4G 1 ,
F,
.G,
Fi-bE.,,
4i,
4^2 i
These may be written,
Fa a' a" - ^E.a'a a" - J# 3 a'a"a - (F., - ,) a'aa' = 0,
- Ea a a" + ^E,a a a" + $E 3 aa"a + (F, -^GJ aaa' = 0,
Ea'a'a"- F aa'a"+ ^E,G'-(F,- ^G,) F' =0,
^a'a"a - Faa"a - ^E,G' + (F,-^G t ) E' =0,
Ea'a a' - F a a a + #,/" - EE'
and
Ea'a a' - F a a a + #,/" - \EE' = ;
Ga a' a" - (F, - \EJ) a'a' a" - 4 G.aV'a - 4 G,a'aa' = 0,
- Fa a' a" + (F, -$E,)a a' a" + G,aa"a + G,aaa' = 0,
Fa'z'a" - G a a a" + (?,<? -%G. 2 F' =0,
Fa'a"z - G aa'a -(F,-^E t ) G + \G.,E' =0,
Fa'aa - G act a' +(F l -^E t )F'-^G 1 E 1 =0.
Attending in each set only to the third, fourth, and fifth equations, and combining
these in pairs, we obtain
V'aa'a" + (
V-a'a' a" + (
V\i a" a + (-
,- FF,
l - GF 3
EF, - ^
V'a a a + (
V\i'a a' +
F' + (- ^FG, + 4 GE,)
.^ G' + (- ^FG t + FF, -
,) G' + (- GG, + GF, -
E' + ( ^FE, - EF, +
i - $ GE.,)
E = 0,
' = 0,
' = 0.
:;:it;
We thus obtain
ON THE GAUSSIAN THEORY OF SURFACES.
[767
E,' - F t ' = ,
F t ' - 0,' = {( ^FG l - FF, + $EG Z ) F'
J E' + (-<?, + FF l -
-- K- * GEl
^ G '+(-^GG,+ GF, - IFG,) E'}
or, finally,
E,' - F,' = - a {(- ^FG l + GE, - FF, + ^EG 3 ) E'
+ (- GE l + 2FF> - FF,) F' + (^FE t - EF l
'
+ (FG, - 2FF,
-EG l + ^ FE,) G'} ,
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
V t K+ VK 2 + V>T l
*\ VK}, = V,K,
that is,
or finally
which is Hour's third equation. And the second equation becomes
~
that is,
or finally
- 2 FT.ff,
-F s r ! -2FF !l 2 T - Ffr i -3F 2 F,fl"=- F J F 1 - -
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 following differential identity, viz., if
m 2 ga? '
) 2 '
where
then
(this is exactly Landen's form, except that he of course writes tk, 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=k' 1 , if as usual
/2 = l, the transformation is
t = i
-a?
C. XI.
43
338 NOTE ON LANDEN'S THEOREM. [768
leading to
The form in which the transformation is usually employed (see my Elliptic
Functions, pp. 177, 178) is
leading to
(1 + AQda; _ dy
where
X = lTA 7>
If, to identify the two forms, we write y = ^ r, and in the last equation
x ~~* fc
introduce t in place of y, ' the last equation becomes
dx dt
have
Vl - of. 1 - k>a? V{(1 - kj - P} {(I + Jfc') 2 - P] '
Comparing with Landen's form, in order that the two may be identical, we must
x V(l-yfc') 3 -< a
viz., this is
that is,
where the function under the radical sign is
(1 - k*)* -2(1 + P) t'+P(=T suppose) ;
and this must consequently be a form of the original integral equation
In fact, squaring and solving in regard to a? with the assumed sign of the radical,
we have
t--
x* = ^..
768] NOTE ON LANDEN'S THEOBEM.
corresponding to an equation given by Landen. And we thence have
339
which is the required expression for 1
The trigonometrical form sin (2$' </>) = c sin $ of the relation between y and so
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 Exertices, 1811, and also in the Traite
des Fonctions Elliptiqu.es, 1825, and by means of it he obtains an expression for the
arc of a hyperbola in terms of two elliptic functions, E(c, <f>), 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 necessaiy 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.).
432
340 [769
769.
ON A FORMULA RELATING TO ELLIPTIC INTEGRALS OF
THE THIRD KIND.
[From the Proceedings of the London Mathematical Society, vol. xill. (1882),
pp. 175, 176. Presented May 11, 1882.]
THE formula for the differentiation of the integral of the third kind
TT_ [ <ty
Jo(l + nsin 2 4>)A
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
sin 2 < = ar = sn 2 u, sin <f> cos </> A = y = sn u en u dn u,
and thus connecting the formula with the cubic curve
y == sc ( x ^ sc ) ( J. ~ K x),
The parameter must, of course, be put under a corresponding form, say n = ,
Or
where a = sn 2 0, b = sn en dn 0, 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
-------
7 /I ~ A/ \ I* ^^
aff a x du x a
We, in fact, have
(J X
-5- = 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,
Also
and hence
-=^ = en" u dn 2 u sn 2 u dn 2 u k* sn 2 M cn a u
du
_ _
%
_ _
dux -a (a- xf \" du du
Interchanging the letters, we have
- = 7 - -{-
dO a x (a x) 3(
and hence, subtracting,
{- a; - a + 2(1 + fc 2 ) a* + A-'ic 3
- - -- - . .,
do a x du x a (a a;) 2
(a - x}
= kf(a-
which is the required result.
If (a- a?
342 [770
770.
ON THE 34 CONCOMITANTS OF THE TERNAKY 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 ax' + by' + cz s + Glxyz, or, what is the same thing, the 34 covariants of
this cubic and the adjoint linear function %x + r/y + z : 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:
HESSK, Ueber die Elimination der Variabeln aus drei algebraischen Gleichungen
vom zweiten Grade mit zwei Variabeln : Crelle, t. xxvill. (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, yf + y^ 3 + ys a + 67ry 1 y 2 y 3 .
ARONHOLD, Zur Theorie der homogenen Functionen dritten Grades von drei
Variabeln: Crelle, t. xxxix. (1850), pp. 140159.
CAYLET, A Third Memoir on Quantics : Phil. Trans., t. CXLVI. (1856), pp. 627647 ;
[144].
ARONHOLD, Theorie der homogenen Functionen dritten Grades von drei Variabeln :
Crelle, t. LV. (1858), pp. 97191.
SALMON, Lessons Introductory to the Modern Higher Algebra: 8, Dublin, 1859.
CAYLEY, A Seventh Memoir on Quantics: Phil. Trans., t. CLI. (1861), pp. 277292;
[269].
BRIOSCHI, Sur la the'orie des formes cubiques a trois inde'termine'es : Comptes
Rendus, t. LVI. (1863), pp. 304307.
770]
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
343
HERMITE, Extrait d'une lettre a M. Brioschi: Grelle, t. LXIII. (1864), pp. 30 32,
followed by a note by Brioschi, pp. 32 33.
The skew covariant of the ninth order, which is y 3 z*.z? a?.a? y 3 for the canoni-
cal form a? + y 3 + z 3 + Qlxyz, and the corresponding contravariant if 3 . f 3 f 3 . f if,
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. 90128.
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 aa? + by 3 + cz a + Qlxyz rather than the
Hessian form y? + y 3 + 2" + Qlxyz, employed in my Third and Seventh Memoirs on
Quantics : for the form oaf + by 3 + cz 3 -f Qlxyz is what the general cubic
(a, b, c, f, g, h, i, j, k, 1) (x, y, z) s
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, ..., I), of (f, 17, f), 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 oaf + by" + of + 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 + 81 s 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 + SI 3 . As to the signification of this expression
abc + 81*, I call to mind that the discriminant of the form is abc (abc +
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 + 1*, instead of Aronhold's
8 = 4:abcl + 4Z 4 . 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 U, thus HU, PU, &c. ; in particular, I use U, H, P, Q
instead of Aronhold's f, A, Sj, Tj. 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 J,
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 4 , I wish the 400 to be regarded as denoting its
expressed value abcl +1*: if the same letter S is to be used in Aronhold's sense
to denote 4o6ci + 4i 4 , this would be completely expressed by the new definition
S = 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, b, c), (, 17, ?), (x, y, z).
770]
ON THE 34 CONCOMITANTS OF THE TEENAEY CUBIC.
345
Table of the 34 Covariants of the Canonical Cubic ax? + by 3 + cz 3 + Qlxyz and
the linear form %x + j\y + %z.
First Part, 10 Forms. Class = Order.
Current No.
6
7
10
2 T = 600 = a 2 & 2 c 2 - 20a6d 3 - 81.
3 A = 011= c + w + ?2.
4 = 222= af\- 1*?- - 2alriS]. . .
0'=422= x*[l(abc + 2l 3 )? + a (abc -
+ yz [GbcP? - 21 (abc + 2l 3 ) rf], . . .
0"=622= af[-(abc + 2l 3 Y^- + 12al* (abc + 2l 3 ) r,]. . .
+ yz [SQbcl^ + 2 (abc + 21 s ? rf]....
B = 333 = a? [a 2 (erf - b?)].. .
+ y*z [(abc + 81*) rf
8l 3
R = 533 = X s [3a*l- (ctf - b?)] . . .
+ y*z [- P (abc + 8l 3 ) rft + 4,bl (- abc + I 3 ) ? - be (abc - l
+ yz' [l> (abc + 8l) r,? + bc (abc - 10Z 3 ) ? - 4cZ (- abc + I s ) &-] ....
B" = 733 = a? [9a* (<)-&)]...
+ y-z [I (abc + 8l 3 ) (2abc + I 3 ) -rfC,
+ b (abc + 2l 3 ) (abc - Wl 3 ) ? + 6bcl* (- abc + I 3 ) ? 17] .
+ yz 1 [- I (abc + 8l 3 ) (2abc + 1 3 ) q?
- 6bcl- (- abc + I s ) & - c (abc + 2P) (abc -
B'" = 933 = a? [27aH< (cr, 3 - b 3 )].. .
+ fz [- (ate + 8l>) (abc - I 3 )- r,^+ MF (abc + 2Z 3 ) 2 ?-
12
13
14
C. XI.
+ yz 1 [(abc + 8l 3 ) (abc - 1 3 )- ^- + 27bcl* (abc + 21 s
Second Part, (4 + 4 =) 8 forms. Class = 0, and Order = 0.
Class =0.
U= 103 = aa? + bf + cz 3 + Qlxyz.
H = 303 = f (ao? + by 3 + cz 3 ) - (abc + 2l 3 ) xyz.
= 806 = (abc + 8l 3 )* (aW + 6y + c 2 ^ - 10 (bey
n = 1209 = (abc + 81 3 Y [by 3 - cz 3 .cz^-aa?. ax 3 - by 3 }.
44
346 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
Current No. Order =0.
15 P=
16 Q = 530 = (abc - 10?) (be? + car,' + ab? ) - 6J 1 (oabc + 4F) fr?
17 J* = 460 = &Vf + cW + a'&'(? -2(abc + 16P) (a^f + 6?'^ -r
- 24P (6c^ + carf + ab?) &$ - 24,1 (abc + 2P) V? 3 .
18 n . 1290 -(ate + 8P? {/-&. ap-cp.&p- aitf.
Third Part, (8 + 8 =) 16 forms. Class less or greater than Order.
Class less than Order.
19 J=
20 K = 514 = (abc + 81') {{[alx 4 - Zblxy 3 -
21 K'= n4, = (abc + 8l'){l-[(abc + 2l 3 )(ax 4 -2bxy 3 -2cxz 3 )-18bclyz*]...}.
22 E= Q25 = (abc + 8l 3 )
23 E' = 825 = (abc + 81 s ) { 2 (6y s - c* 8 ) [i (a6c + 2f) a? -
24 E" = 1025 = (aic + 8Z 3 ) {* (by 3 - cz 3 ) [(abc + 2l 3 )' a? + ISbctyz] . . .
+ rt (by 3 - cz 3 ) [- IZal* (abc + 21 s ) ^ + (060 + 2l>y yz]...}.
25 .17 = 917 = (abc + 8J 8 ) 1 {%(by 3 - cz>) [5ala* - blxy 3 - dxz* - 3bcy V]. . . }.
26 M'= 1117 =
Order less than Class.
27 / = 841 = (abc + 8l 3 y {xfa (erf - b?) + yr,b (a? 3 - cf) + z& (bg 3 - ai)%
28 K = 541 = (abc + 8l) [x [be? - 2cafr 3 - 2ab%? - 6a V? '] }
29 K = 741 = (abc + 8P) {x [I* (be? - 2ca& 8 - 2a6f f ) + a (abc + 2l 3 ) r,*?]...}.
30 E = 652 = (abc + 81') {x 1 (erf - b?) [2al? + afy? ]. . .
31 E'=
+ yz (ctf - b?) [41 (abc + 2l 3 ) f + a (abc-
32 E 77 = 1052 = (abc + 8l 3 ) {x> (cr,* - b?) [- Sal" (abc + 2l 3 ) ? + Qa'
+ yz(crf- b?) [(abc + 2l 3 y ? - Sal* (abc - 4Z 3 ) 77?]...}.
33 M= 1 1 1 = (abc + 81') {x (erf -b?) [(abc -8l 3 )?- a-cfr 3 - a'b
34 W = 97 1 = (abc 4- 8l 3 ) {x (erf - &?") [I* (7a6c + 8l 3 ) ? -
+ 4ai (abc - I 3 ) ?^+ a" (abc - 101 s ) if?]. ..}.
770]
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
347
To this may be joined the following Supplemental Table of certain Derived
Forms :
Current No.
35 R = 1200 = 64S 3 - T 2 = - abc (abc + 8Z 3 ) 3 .
36 C = 703 = - TU + 24ff = (abc + 81') {(- abc + 4,1 s ) (ay? + bf + cz 3 )
+ I8abclxyz}.
37 D = 903 = 8S 2 U - 3TH = (abc + 8Z 3 ) {V (5abc+ 4Z 3 ) (ax? + bf + cz 3 )
+ 3abc (abc - 10Z 3 ) xyz}.
39 =1130 = - 48SIP + TQ = (abc + 8l*y> {(abc + 2l 3 ) (be? + car, 3 + ab?)
= 1660 = 12 (abc + 81 3 )*F- 288STP- + 768/SlPQ - 8TQ 1
= (abc + 8P) 4
40
viz. these are derived forms characterized by having a power of abc + 8l 3 as a factor :
R is the discriminant ; C, D, Y, Z occur in Aronhold, and in my Seventh memoir
on Quantics [269] : * 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 formula? involving notations which will be explained, viz. we have
= 3 (be' + b'c - 2ff, . . . , gh' + g'h - af - a'f, . . . \X, Y, Z\X ', Y', Z') + TU'-.
13
14 fl = -fa Jac(7, H, ^).
18 n =-^[Jac](P, Q, F).
40= (bc-fV.., g h-af, ,
50'= 1 80.
60"= $ S 2 0.
7 B =-^Jac(T, 0, A).
8 B' =
9 B" =
10 B" =
19 J =-Jac(Z7, H, A).
27 J = HJac](P, Q' A).
20 K =-
21 ,8" = -
28 K = 3
29 K'
22 ^ =- t ^Jac(', 7, A).
23 E' = -
442
348
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
[770
24 E" =
30 E = -$J&c(K, U, A).
31 E' = -(S)E.
32 E" = - (#) S.
25 # = 3 ^Jac(D', , A).
26 JIT = - (8) M.
33 J? =-HJac](P, F, A).
34 J7' = (S) 37.
In explanation of the notations, observe that
U= ax' + by' + cz' + 6lxyz,
H = l*(ax* + by 3 + cz?) - (abc + 2V) xyz.
Hence, writing
6H = aV + b'f + c'z 3 + 61'xyz,
we have
a', b', c', l' = 6at>, 66P, 6cf, -(a6c + 2P).
And this being so, we write
X, Y, Z = aa? + Zlyz, by 1 + 2lzx, cz- + 2lxy,
&, 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
Z', F', Z" = ax* + 2l'yz, b'f + 21' zx, c'z- + 21' 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 QH.
Jac is written to denote the Jacobian, viz. :
Jac(C7, H, ) =
and in like manner [Jac] to denote the Jacobian, when the differentiations are in
regard to (f, 17, f) 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, &, &c., each operate directly on a function of (a, b, c, 1), the (', b', c, I') of the
symbol 8 being in the first instance regarded as constants, and being replaced ultimately
by their values ; for instance,
= a'bc+ab'c + abc', &abc = 2 (ab'c' + abc' + a'b'c), &abc
770]
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
349
In several of the formulas, instead of S or 6-, the symbol used is (8) or (S 2 ) ;
in these cases, the function operated upon contains the factor (abc + 8l 3 ) or (o6c + 8Z 3 ) 2 ,
and is of the form (abc+ 8P)(aU+bV+cW) or (abc + 81 3 ) 2 (a?U + abV + &c.) : the
meaning is, that the B or 8 2 is supposed to operate through the (abc + 8l 3 ) a, or
(abc + 8Z 3 ) 2 a 1 , &c., as if this were a constant, upon the V, V, &c., only ; thus :
(S).(abc+8l 3 )(aU+bV+cW) is used to denote (abc + 81 s ) (a&U + b&V + cSW). As to
this, observe that, operating with 8 instead of (8), there would be the additional
terms US (abc + 8Z 3 ) a + &c. ; we have in this case
S (abc + 8l 3 ) a, =a (la'bc + ab'c + abc' + 24W) + 8l 3 af,
= 24a 2 6c? ! - 24aZ 2 (abc + 2Z 3 ) + 48aP>, = ;
or the rejected terms in fact vanish. For (S 2 ) . (abc + 8l 3 )(aU+ bV+cW), operating
with S 2 , we should have, in like manner, terms US' (abc + 8l 3 ) a, &c. ; here
S 2 (abc + 8l 3 ) a = a'-bc + Zaba'c' + Zaca'b' + aVc' + ZWa'l' + l^att'-,
which is found to be = - 24a (abc + 8l 3 ) (- abcl + Z 4 ), that is, = - 24S (abc + 8l 3 ) a ; and
the terms in question are thus = 24S (abc + 8l 3 )(alI + bV + cW), viz.
(abc + 8l 3 )(aU+bV+cW)
being a co variant, this is also a co variant ; that is, in using (8 2 ) instead of 8 2 , we
in fact reject certain covariant terms ; or say, for instance, &E being a covariant,
then (8 2 ) E is also a covariant, but a different covariant. The calculation with (8)
or (S 2 ) is more simple than it would have been with 8 or S 2 . 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 + 8l 3 as a factor, showing how this factor presents itself.
Calculations for the 26 Covariants.
= 3 (be' + b'c - 2ff, . . . , gh' + g'h - af - a'f, . . .Z, T, Z\X', Y', Z') + TU\
= 3 ((be' + b'c) yz - Wa? , ..., 211'yz -(al' + a'l) a?, ... Qoa? + 2lyz, . . . $a V + 21'yz, ...)
13.
The whole coefficient of af is
- 6W + Ta\ = 36a 2 Z 3 (abc + 21 s ) + Ta-,
viz. the coefficient of a-af is
= 36f (abc + 2l 3 ) + a 2 b*c? - 20abcl 3 - 8l 6
= a 2 6 2 c 2 + Isabel 3 + 64Z 6
14.
H, V), =
X, X',
V V
* j * y
Z, Z',
350
Here
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
[770
YZ' - Y'Z = (6y> + Zlzx) (c'z' + Stay) - (cz' + 2lxy) (b'z> + 2l'xy)
= (be 1 - b'c) y'z> + (2bl f - b'f) xy'-Z (cl' - c'l) xz 3
= - 2 (abc + 81') x (by 3 - cz 3 ) ;
1 Jl \f/ - 1 ( ft-sy^ _ T//fJ^*"?/^ ^ "}flf^Y^sP}
Hence the whole is
= _ (abc + 81') {oV (by* - cz 3 ) + by (cz* -ax t ) + c 3 z t (ax 3 - by 3 )},
= (abc + 8l 3 )(by i -cz 3 )(cz 3 -ax>)(ax > -by).
is. n =
viz. if, in this calculation, we write
i.e. a, b, c,
a', b', c',
then
Here
or since
6lbc, -6lca, -6lab, -abc
- 10i 3 ) (be, ca, ab), -
= (be' - b'c) r,*? + 2 (bl' - b'l) fyf - 2 (cl' - c'l
be' - b'c = 0,
bl' - bl = - Glca . - I 1 (5abc + 4P) - (abc - Wl') ca (- abc +
= ca {61 s (5abc + 4,1 s ) + (abc - 4P) (abc - 101 3 )}
and the like for cl' cl, the expression is
= 2 (abc + 8l 3 )* (carf - ab?) f ;
and the whole is thus
= - i (abc + 81*)' {(car) 3 -ab?).bd ( F+ ...}
= - i (abc + 8l 3 )- {(car, 3 - ab?) [b-c^ - (abc + 161*) (b?> 3 + cfr, 3 ) + &c.]
+ (ab%* - be?) [c 2 Y - (abc + 16Z 3 ) (c?rf + ar, 3 ?) + &c.]
+ (beg 3 - cat} 3 ) [aW? - (abc + W 3 ) (arf? + b??) + &c.]}.
Here the coefficient of fV, inside the {}, is
ab'c 3 + 6c 2 (abc + IQl 3 ), = <$><? (abc + 8l 3 ),
770] ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
and consequently the whole is
= - (abc + 8l 3 ) 3 (bc-^rf -...),
= (abc + 8l 3 ) 3 {(erf - bg>) (a 3 - eg 3 ) (&" - oaf)}.
4. = (bc-f 2 , ...,gh-af,...$f, 77, O 2
= (bcyz - IV) f 2 + . . . + 2 (Fyz - ala?) ij+ ...
which are the terms of the final result
5 and 6. The S-process applied to the terms of just written down gives
' = | S0 = a? [- II '?- - (al' + a'l
351
substituting for a', b', c', I' their values, we have the corresponding terms of & and "
respectively.
7.
f = -JJac(Z7, e, A), =-
z, 8 2 ,
A term is X (rjd, fdj,), and if, in this calculation, we write
@=(^, B, C, F, G, H\x, y, zf, i.e. A = - 1*? -
then the term is
Here
and hence the whole term in ic 3 is = aV (crj 3 b% 3 ).
8, 9, 10. The coefficient of a?rf in B is a 2 c, and hence in SB, S-B, &B the coefficients
of this term are 2a'ac+aV, 2a /2 c + 4aa'c', 6aV, whence in
respectively,
D' D" Ty'f
Jj j JJ , Jj
the coefficients are
| (a"c' + 2aa'c), ^ ( a ' 2 + 2aa'c / ), ^ a' 2 c',
= 3i 2 a 2 c, 9 4 a 2 c, 27Z 6 a 2 c respectively.
^i, JL ,
ir r iT/rrii7A\ IV V
19 J = jJac(t/, .n, A) = $ Jf, 2 , r/
y > 3
a term is - \ ( YZ' Y'Z) %, where, as in a previous calculation,
YZ' - Y'Z = - 2 (060 + 8Z 3 ) (ft^/ 3 - C2 3 ).
352 OX THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
Hence, the whole is
- (abc + SI') {& (by* - cz>) + rjy (a? - oaf) + & (aa? - by 1 )}.
27. /=HJac](P, Q, A) =
if, as in a previous calculation
6P = a> + b; 3 + cf + eifrfc Q = a'p + by + c'
Here, as before,
(bij* + 2lgf) (c'{? + 21'fr) - (by + 21') (cp + 21fr) = 2 (oic + 8P) S (cai, 3 -
Hence, the whole is
= (abc + 81')* {xa (erf - ftp) + ynb (af - of) +
20. ^r
which, H being
= (aV + b'y* + c'z
and putting
8-(^, B, C, F, G,
= - 1 {(aV + 21'yz) (A + Hr, + G - (- obcl + l<) U (f*
+ (b'y 3 + Zl'zx) (HI- + Br, +
+ (cV + 2l'xy) (Gf
The whole coefficient of f is thus
H + (c'z 2 + 21'xy) G} - (- obcl + 1') Ux
+ (c'z 2 + 2,1' xy) (- Uy* + fax)} - (- abcl + 1 4 ) [ax 4 + bxy 3 + cxz* + 6la?yz],
and herein the coefficient of x 4 is
= f a'l 1 -al(-abc + 1 3 ), = 9al 4 - al (- abc + I 3 ), = (abc + SI 3 ) al ;
viz. we have thus the term (abc + SI 3 ) f . alx 4 of the final result.
21. K'=-(S)K, where K is of the form (abc + 8l 3 )(aU + bV+ cW); operating
with ($), we obtain (abc + SI 3 ) (a&U + bBV + cBW). Taking for instance the term of
K, (abc + 81') f [alx 4 - ZUxy 3 Zclxz 3 + Sbcy*z'], then, in operating with (8), the term be
may be considered indifferently as belonging to bV or cW, and the resulting term
of K' is
K' = -(S)K = - (abc + 8l 3 ) % [al'x 4 - Zbl'xy 3 - Zcl'xz 3 + Sbc'yV],
= (abc + SI 3 ) % [(abc + 21?) (ax 4 - Zbxy 3 - Vex?) -
770]
28.
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
353
K = 3 {d x d ( P + dyQd^P + dJ8d ( P} + QA ; viz. writing
= (A, B, C, F, 0, H$x, y, zf, A = - F? - Zol^ ... F = $ be
then this is
= 3 }[- 3bcl? + (- abc + 4P) r,^] 2 (Ax + Hy+ Qt)
+ [- Zcalif + (- abc + 4>l 3 ) g] 2 (Hx + By + Fz)
+ [- Sabl? + (- abc + 4l 3 ) 77] 2 (Gx + Fy + Cz)}
+ {(abc - 101 3 ) (beg 3 + carf + ob?) - 61* (5abc + 4Z 3 ) &} (c + rjy + &).
The whole coeflBcient of x is thus
= 3 {[- Sbcl? + (- abc + 4P) n ] (- W?
+ [- Scaly* + (- abc + 4P) ff] (ab? +
+ [- 3abl?+ (- abc + W) fr](ac<n* +
+ {(abc - 101 s ) (be? + ca%rf + abt;?) - 6Z 2 (o
herein the coefficient of f 4 is ISbcl 3 + (abc - 101 3 ) be, = (abc + 81*) be, giving, in the final
result, the term (abc + 8l 3 ) f . bcx*.
29. ^'=(8)^.
Here K is of the form (abc + 8l 3 )(aU+bV+cW), and we have
K' = $ (abc + 81') (aSU+ 68 V+ cS W).
sc [be'? -
= x [I 2 (be? - Zcafr* - 2a6f) + a (abc + 2>) if1\,
A term of aU + bV + cW is x [be? - 2caV - 2ab^ - 6a lrf{*], where be? may be con-
sidered as belonging indifferently to bV or cW; and so for the other terms. The
resulting term in $ (a&U + b&V + c&W ) is thus
which is
and we have thus a term of K'.
22. E = - ^g Jac (K, U, A):
K contains the factor abc + 8f, and if, omitting this factor, the value of K is called
A + Brj + C , then we have
Zd 2 C)(Xi,-Y%)},
and the term herein in ? is - ? (Zd y A - Yd,A), where A is
= alx* Vblxy 3 2clxz 3 + Sbcy^z* ;
c. XI. 45
354 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
viz. the coefficient of f is
1 + 2lxy) (- Qblxy 3 + Gbcyz') (by 1 + 2lzx) (- Gclxz* + Qbcy'z)}
-bc i yz t + Zbl i x 1 y*-
[770
Hence, restoring the omitted factor (abc + 81*), we have in E the term
(abc + 81') f (by' - cz>) [2JV + bcyz].
23, 24. E' = - i (S) ^, " = }(#) :
^ is of the form (abc + 8l*)(aU+ bV + cW), and, as before, in a term such as
(abc + 81 s ) f (by 3 - cz 3 ) (Zl'af + bcyz),
we operate with S or S 2 only on the factor 2te 2 + bcyz ; and in E' and J" respec-
tively, operating upon this factor, we obtain
- $ {Ul'a? + (be + b'c) yz], and \ {4JV + 26'c'yz},
viz. we thus obtain in E' the term
(abc + 8l 3 ) g> (by 3 - cz 3 ) [I (abc + 21 s ) a? - SbcPyz],
and in E" the term
30.
(abc + 81*) f (by 3 - cz*) [(abc +
ISbcl'yz'].
, X,
and, if omitting in K the factor abc + 8l 3 , we write K =
.4 =
which contains the term
, this is = -
~!z, where
A, X, \
B, Y, ,
C, Z, ,
- 77
= (aa? + Zlyz) (erf - b?) (21? +
Hence, restoring the factor abc + 81*, we have the terms
E . (abc + 81*) {* (erf - b?) [2ofp + a^ ] + yz (cr, 3 - b?) [4l>? + Zaltf]}.
31 and 32. E' = -^(B)E, E" = -$(&)E:
E is of the form (abc + 81*) (all + bV + cW), and we operate with 8 and S 2 on the factors
aW, &c.; viz.
8 (2al
2 (al r + a'l) p + Zaa'rf,
770] ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC,
and we thus obtain in E' the term
(abc + 8l 3 ) a? (erf - b?) [a (abc - 4/ 3 ) f - (
and in E" the term
(abc + 8l 3 ) a? (erf - b) [- 3aF (abc + 2l 3 ) f 2 + !
25. M=^Ja.c(U, , A): this, omitting the factor (a&c + 8Z 3 ) 2 of V, is
ix* + 2lyz, ax* (ax 3 - oby 3 bcz 3 ),
355
t, by 1 (by 3 - ocz 3 - oax 3 ), r/
cz 3 + 2lxy, cz* (cz 3 - oax 3 - oby 3 ), f
the coefficient of herein is
= \ {(bcyW + 2clxz 3 ) (cz 3 - oax 3 - 5by 3 ) - (bcy*z- + 2blxy 3 ) (by 3 - ocz 3 - oax 3 )},
= {bcy'z- (- 6by 3 + 6cz 3 ) + 2lx [- 6y + c 2 ^ 6 + box 3 (by 3 - cz 3 )]},
= (by cz 3 ) [oalx* blxy 3 clxz 3 Sbcyz*].
Hence, restoring the factor (abc + 8l 3 )*, we have the term
(abc + 8l 3 y . f (fa/ 3 - cz 3 ) [5alx* - blxy - clxz 3 - 3bcy"z*].
26. M' = -(S)M. Here M is of the form (abc + 81 3 ) S (a 1 U + &c.) ; and the 8 operates
through the (abc + 8Z 3 ) 2 a 2 , &c. ; we, in fact, have in M ' the term
- (abc + 8P) 2 .(by- cz 3 ) [5al'tf - bl'xy 3 - cl'xz 3 - Sbc'yW],
which is
= (abc + 8Z 3 ) 2 . f (by - cz 3 ) [(abc + 2?) (oax 4 - bxy 3 - cxz 3 ) + 186cPy V].
Slbc^ 3 + ( abc + 4<l 3 ) i), d$F, x
Slcarf + ( abc + 4P) , S^F, y
3ia6f 2 + ( abc + 4Z 3 ) ft, d$F, z
and the whole coefficient of x is thus
= {[Skew; 2 + (abc 4P) %] d ( F [3lab~ + (abc <
or substituting for jj dfF, fa d^F their values, this is
- 81 (abc
- 8/ (abc + W) f^? 2 ].
452
33.
JT
, f, A), =-
-f
- {Slab? + (abc - 4Z 3 ) ft] [aV^ - (abc + 16Z 3 ) (ar,^ 3 +
356 ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. [770
Collecting, first, the terms independent of abc 4J', and, next, those which contain
abc 4?, each set contains the factor crp bl?, and the whole is = crf bl? multiplied by
- Sla'bcrf? - 3aH (abc + 8P ) rf? - 121* (abc& + a'cfr' + a*b&) - 24aJ a (abc + 2P
+ (abc- 4:1') {a'cfr> + a'&f f - (abc + 161') ?
and here collecting the terms in 4 , %(crf + b?), f^f, and iff, each of these contains
the factor abc + 8P, and, finally, the term of M is
= (ate + 8P) (cif - b?) [(abc -
34.
Here M is of the form (abc + 8l')(aU'+bV + cW); and, operating with S through the
(abc + 8P) a, &a, we obtain in M' the term
(oic + 81') x (cr,* - b?) [(a'bc + ab'c + abc'- 2M) | 4 + &c.],
where
a'bc + ab'c + abc' - 24W = ISabcl 1 + 24P (abc + 2P), = 6P (Tabc + 81 s ),
and the term thus is
= (abc + 81') x (cr,* - b?) [(Tabc + 8l 3 ) I*? + ...].
This concludes the series of calculations.
Cambridge, England, 17 May, 1881.
771] 357
771.
SPECIMEN OF A LITERAL 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, 6 2 ; d, be, b 3 ; ..., 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 s=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, d~$x, yf, 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
b> c 1 cd ce
b*c b'd d"
& be" b*e
b*c bed
b'
c 3
b>d
we at once copy out the terms
etc.
a'd
abc
abd
ac 3
b'c
acd
ad?
bed
c 3
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 b*, the operation on the last letter gives
fc, and that on the last but one letter gives 6 5 . 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
012345 6
8 9 10 11 12 13 14 15 16 17 18 .
1123
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.
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
9
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, TO 18.
0.3
4.5
6.7
7.8
8.9
9
9. 10
10
10.11
c/
5 2 <7
bi
i
, C 4
bdg
c 5
1
4
5
6
11
<fe
6V
bcf
bde
ch
dg
I
}
> B e
bef
"
6 4 ce
1
(
^
6ce
bfP
C *d*
ef
I
V
,-7
cdf
Cf 1
6 r
1
i
I
c
d
a
I
c
/
e
c d
b'e
Vce
beg
bdf
I
I
( 5.2
7 c
tfe
6%
6V
b'e
b
t
I
"c
4
c
I
P
2 e
Wed
be 3
d
ly
I
6 2 cy
b'cd
6 4 c
cd
6 d
C 4
be
o d
2
5
c
1
8.7
6V
6 4 e
13 rt J
d 3
W^i
10
42
bey
L-J-
o c
o ca
t> g
ocde
C
C
u
(
I
V
6 7
6V
6V
bd*
v
J
>c
"
8
6"d
Wde
/
<?e
'6
r
/
3
(
d
22
6 6 c
bed 1
r
J
i
b'g
11
3
i
>"rf
7
i
6 8
c*d
e
Ih
6V
56
i
ic"
15
6A
g
W
e
g
6 s cfe
a
be
1
1
rc
f
i
I
df
30
b*ce
J
I
6Ve
I
bk
b
i
V
J
6V.
I
ch
6c rf
y
771]
OTHERWISE A PARTITION TABLE.
THE PARTITION TABLE, TO 18 (continued).
361
11
11.12
12
12.13
13
13.14
14
14
14
di
vf
beef
6 6 <2 2
bceg
bc 3 d 3
ffi
Vk
C 5
eh
6 5 ce
bdy
bVd
bcf*
<fd
If
Vcj
cV
fg
W
bde*
6V
MV
6%
6 2 m
b 3 di
W
by
6W
<?g
6 8 e
bdef
tfcg
bcl
Veh
b'ch
bci
6V
<?df
Vcd
be 1
t>W
bdk
b 3 fg
b*dg
bdh
Ve
cV
6V
eft
6V
bej
6Vt
Vef
beg
Vad
cd*e
bd
<?dg
6V/
bfi
Wcdh
V<?g
bf
6V
d*
6V
<?ef
Vcde
bgh
tfceg
b 4 cdf
<?h
bd
b*i
6 10 c
ccPf
b 4 d 3
c*k
6 2 c/ 2
b*ce*
cdg
6V
b 3 ch
6 12
Cfife 2
6Ve
cdj
b"-cPg
b'tfe
cef
6 9 c
b 3 dg
1 ^
d?e
bVd*
cei
tfdef
6V/
dy
6 11
b 3 ef
xo
Iftl
V}
bVd
cfh
6 2 e 3
bVde
de>
1 O
Vc*g
1U1
b*ci
6c 8
eg 3
bc 3 h
b 3 cd 3
b 3 i
U
Vcdf
n
b 3 dh
Vg
d*i
bc^dg
6Ve
b'ch
77
6 2 ce 2
6?n
Veg
Vef
deh
6c 2 e/
bVcP
Vdg
m
bWe
el
6 s / 2
b 6 de
dfg
bed 2 /
bc s d
b*ef
bl
be 3 /
dk
6VA
6Ve
fg
bcde*
c 7
b<?g
ck
bc*de
9
b*cdg
6 5 crf 2
/ 2
bd 3 e
Vh
bcdf
dj
bed 3
fi
Wcef
bVd
bH
cV
Vcg
bee 1
ei
c'e
gh
WPf
6V
Vck
C 3 df
Vdf
bd?e
fh
c 3 ^ 2
bH
6W
&y
Vdj
cV
6V
<?f
r
6%
bck
bc 3 g
Vce
b*ei
<?d*e
6V/
(?de
6%
My
bdj
bc*df
b'd*
Vfh
cd 4
Tfcde
cd 3
bcj
Vdf
bei
b<?e>
bVd
Vf
Vj
Vffl
b 4 h
bdi
6 4 e 2
bfh
bcd*e
bV
bey
b*ci
6Ve
Veg
beh
6 3 c 2 /
V
6f/ 4
b'e
bcdi
Vdh
6V^
b 3 df
Vff
6"ccfe
<?j
<f
b*cd
bceh
Veg
6Vrf
6V
<?i
6W
cdi
c'de
6V
bcfg
*>T
6V
6V/
cdh
Pee*
ceh
c*d 3
6 1( W
bd*h
6V/t
6V
Vcde
ceg
6Vrf 2
cfff
b*i
6V
bdeg
b 3 cdg
Vcf
b*d 3
tf*
6c 4 c^
d*h
Vch
6 n c
w-
b s cef
Vde
bc'e
ff
c 8
deg
Vdg
6 13
be*f
bWf
bVe
bcW
def
6y
df*
Vef
14
C 3 i
6 3 rfe 2
6 6 crf 2
c*d
e 3
6V
<?/
Ve*g
<Mk
bVg
6W
6V
Vj
b s de
Vk
b s cdf
135
<?eg
bWdf
6V
6 4 c/
6 2 ci
bWe
b*cj
6 3 ce 2
c 2 / 2
6Ve 2
6/
b'de
tfdh
6W
b*di
bWe
bn
ed*g
Vcd>e
6 8 ce
6Ve
Veg
6Vrf
Veh
6V/
cm
cdef
6\/ 4
6 8 rf 2
b'cd"
ft 2 / 2
6V
bYff
bVde
dl
ce 3
be 4 /
6W
bVd
6c%
6 7 /
bcH
Wed 3
ek
d 3 f
bc^de
6V
6C 5
bcdy
6 6 ce
bcdh
bc'e
fi
dV
bc*d 3
6 10
C. XI.
46
362 SPECIMEN OF A LITERAL TABLE FOR BINARY QUANTICS,
THE PARTITION TABLE, TO 18 (continued).
[771
14.15
15
15
15
15.16
16
16
16
16
b'cd
Pgh
bcdef
b"/ 3
6W
cdl
ce/ 3
crfV
Vft
6V
bc'k
bee 3
6VA
6V
cek
d'h
d'e
6Vt
b"d
bcdj
bd'/
b 4 cdg
Pg
tfj
d 3 eg
PI
Pcdh
6"V
beet
6eV
Pee/
PC/
cgi
d 2 / 3
b'ck
Pceg
6 w c
bcfh
c'h
b'd*/
b'de
eh 3
de 3 /
Pdj
Pc/ 3
6 14
beg 3
c a dg
b'de 3
6Ve
d 3 k
e 4
Pei
b*d 3 g
1 C
bdH
c 3 e/
bVg
b'cd 3
dej
b*m
P/h
Pde/
to
bdeh
c*dy
bVdf
bVd
d/i
Pel
b 4 g 3
6V
176
Wa
c 2 de 3
6 3 cV
6c
dffh
Pdk
Pc 3 j
6VA
P
be 3 g
cd 3 e
b 3 cd 3 e
6 10 /
e 3 i
Pej
Pcdi
b s c*dg
bo
be/ 3
d'
b 3 d t
Pee
efli
P/i
Pceh
Pc 3 ef
cn
fi
Pk
6V/
b'd 3
*<?
Pgh
o^cfo
PccPf
dm
c 3 di
Pej
bVde
bVd
/v
Pc'k
o d /I
Pcde 3
el
c 3 eh
b'di
b 3 c-d 3
6V
Pn
b 3 cdj
b 3 deg
Pd a e
/k
<?fg
O 6/1
6ce
6 u e
Wcm
b 3 cei
b 3 d/ 3
b 3 c*g
93
cd 3 h
Ojff
bc*d 3
b w cd
Pdl
Pc/h
6V/
bVd/
hi
cdeg
tfcH
c'd
6V
Pek
Peg 3
6Vt
bVe 3
Pn
cd/ 3
Pcdh
Vi
b l3 d
Pfj
b 3 d 3 i
bVdh
6Vd 2 e
bcm
cef
b 3 ceg
6 6 cA
6'V
6V
Pdeh
Pc 3 eg
6W 4
bdl
d a g
b 3 cf
b'dg
6 u c
6 2 A a
Pd/g
6V/ 2
bc"f
bek
d 3 e/
6 3 rf%r
b'e/
6"
bc 3 l
Pe 3 g
Ped'g
bc'de
j
de*
b a def
bVg
IB
bcdk
Pe/'
Pcdef
bc a d a
bgi
b'l
6V
Vcdf
bcej
bc a j
6 2 ce 3
c'e
bh*
b a ck
6VA
b"ce 3
231
bc/i
bc 3 di
PcPf
<fd 3
cH
b 3 dj
i 2 C 2 G&7
Vd*e
q
bcgh
bc 3 eh
Pd 3 e*
Pj
cdk
b 3 ei
b z c?ef
6V/
b P
bd 3 j
b<?fg
bc*h
b e ci
cej
b 3 /h
Pcd 3 /
6Vcfe
CO
bdei
bcd'h
bc 3 dg
Pdh
cfi
Pg 3
tfcde 3
6W 3
dn
bdfh
bcdeg
bc 3 ef
Peg
cgh
Pc'j
d
6Ve
em
bdff 3
bed/ 3
bcW/
P/ 3
d 3 j
Pcdi
be o
6Vd s
fl
b#h
bee 3 /
bcfde 3
6VA
dei
Pceh
bc*df
bVd
gk
be/g
bd 3 g
bcd a e
b'cdg
dfh
b 3 c/g
bc 3 e 3
be 7
V
b/ 3
bd 3 e/
bd 1
Pee/
dg 3
Pd 3 h
b<?d\
6 8 A
i 2
c 3 k
bde 3
<fg
Pd 2 /
e 3 h
Vdeg
bed*
Vcg
6 2 o
C 3 dj
c'i
c'df
b'de 3
fg
Pd/ 3
<?f
Vdf
ben
<?e.%
c s dh
c 4 e 2
P<?9
/ 3
6V/
c*de
6V
bdm
c 3 fh
c a eg
c a d 3 e
Pc 3 d/
b*m
6ct
c i d 3
b'c 3 /
bel
<?g 3
T
(?d*
6 4 cV
Pel
bc*dJi
b*j
b'cde
b/k
cd a i
b'k
Pcd 3 e
tfdk
bc?eg
b'ci
Pd 3
bgj
cdeh
<?def
Pej
b'd 4
Pej
be 3 / 3
Pdh
6Ve
bhi
cd/g
cV
Pdi
PC'/
P/i
bcd"g
b'eg
Pc 3 d 3
c 3 m
ce 3 g
edy
Peh
bVde
771]
OTHERWISE A PARTITION TABLE.
THE PARTITION TABLE, TO 18 (continued).
363
16
16.17
17
17
17
17
17
17
17.18
bVd 3
bVd
efi
def
C 2 <%
6cV
&>v
J7 /2
6 7 c^ 2
6Ve
6V
egh
<?f
c*df a
bed 3 /
bc*df
6VA
bVd
bVd-
b l *e
f*h
b'n
<?#f
bcdV
6cV
b e cdg
6V
bc*d
b u cd
f?
b 3 cm
cd 3 g
bd*e
bc a d"e
b*cef
tPg
c 8
W><?
6 3 o
b 3 dl
cd*ef
c s h
bc' 2 d*
b e d 2 f
6V
b s i
b a d
6 2 i
b 3 ek
cde 3
Mg
<?/
b e de*
b w de
Vch
6 12 c 2
Vdm
Vfj
d*f
c'ef
c"de
bVg
6Ve
Vdg
6 u c
Wd
b 3 gi
dV
<?d\f
(&
bVdf
6W 2
Vef
6 16
Vfk
6W
b*m
c s de >
Vk
6W
bVd
bVg
17
Vgj
bW
b'cl
cWe
6 6 9
6 6 crf 2 e
6V
tfcdf
X I
Phi
Vcdk
b'dk
cd s
b 6 di
6 5 ^ 4
6 12 /
Wee*
297
bc*m
Vcej
b'ej
m
beh
6V/
b u ce
bWe
r
bcdl
Vcfi
b'fi
b*ck
W9
bVde
6 n rf 2
6V/
bq
bcek
Wcgh
Vgh
Vdj
bWi
6W
bVd
bVde
cp
bcfj
b*d*j
bVk
6 5 et
Vcdh
6Ve
6V
bcd 3
do
bcgi
Vdei
b*cdj
by/i
b*ceg
6Vrf 2
6 I3 e
6Ve
en
bch*
Vdfh
b 3 cei
f>y
b*cf
b 2 C 6 d
b K cd
bVd?
fm
btfk
b*d ff *
b s cfh
b< c y
Vd*g
6c 8
b"c 3
bVd
gi
bdej
6VA
6V
b'cdi
b"def
6 9 i
b u d
bV
hk
bdfi
vfg
b 3 dH
b*ceh
6 8 e 3
b a ch
6 13 c 2
b'h
V
bdgh
by 3
Vdeh
Vc/9
6VA
b*dg
6 15 c
b*cg
Vp
be*i
bc 3 k
Vdfg
b 4 d*h
bVdg
b*ef
6 17
b'df
bco
befh
b(?dj
b 3 e 2 g
b*deg
bVef
V<?g
18
6V
bdn
beg 3
bc*ei
b 3 ef
b'df*
VccPf
Vcdf
J.O
6V/
bem
V*ff
bffh
bVj
6 4 2 /
b'cde*
b 7 ce>
385
Vcde
bfl
c*l
bcY
b*c*di
bVi
b 4 d 3 e
b 7 d*e
s
b 7 d 3
bgk
c*dk
bcdH
bVeh
bVdh
6 8 cV
6V/
6r
6V
bhj
c*ej
bcdeh
Wfg
bWeg
bVdf
bVde
c ?
bVd 3
6i 2
<?fi
bcdfg
b^cd^h
6 3 c 2 / 2
bVe>
bcd 3
dp
b*c*d
Ai
<?gh
bce*g
Wcdeg
b 3 cd*g
b 3 c*d*e
6Ve
eo
6V
cdm
ccPj
beef'
tfcclf
Vcdef
Vcd*
6Vrf 2
fn
Vg
eel
cdei
bd 3 h
6 2 ce 2 /
6'ce 3
6V/
bWd
gm
Vcf
rfk
cdfh
bd*eg
b*d*g
ray
6 2 c^e
6V
hi
b'de
wj
cdg*
6rf 2 / 2
b*d*ef
b 3 d*e*
bVd 3
6 10 A
ik
bVe
chi
ce'h
bd e y
b*de 3
b*c 4 h
6c 6 e
Veg
f
Wed*
dH
fff
be*
bcH
bVdg
6c 6 ^ 2
b a df
Vq
bVd
dek
cf*
c'j
bc*dh
bVef
C 7 d
6 9 e 2
bcp
6V
dfj
d 3 i
c a di
bc 3 eg
6W 2 /
b*j
6V/
bdo
6/
dgi
d\h
c 3 eh
6C 3 / 2
6V& 2
Vci
b*cde
ben
b l "ce
dh*
dVff
c 3 fff
bcWff
6 2 c^ 3 e
Vdh
b a d 3
bfm
b w d"
*)
1*9
c*d*h
bc'def
b*d*
Veg
bVe
bgl
462
364 SPECIMEN OF A LITERAL TABLE FOR BINARY QU ANTICS.
THE PARTITION TABLE, TO 18 (continued).
[771
18
18
18
18
18
18
18
18
18
bhk
befi
b'dej
d*ef
bc*deg
Vcdf*
6 4 c't
b>cd*f
bVff
bij
begh
Wdfi
rfV
icV/ 2
b 3 ce*f
b^dh
b'cde 1
b'cdf
<?o
bfh
Vdgh
b'n
b(?e>f
Vd*g
Vfeg
bWe
6'ce 2
cdn
w
6V
b'cm
bcd'g
b'd"ef
6V/ a
6V<?
6 8 rf 2
cem
<?m
Vefh
Vdl
bcd^ef
b s d<?
b'cd*g
b^df
6V/
cfl
<?dl
Peg*
b'ek
bcde>
6Vi
b'cdef
6Ve 2
V<?de
cgk
c*ek
bTff
Wj
bd'f
bVd/t
b 4 ce>
6Vrf 2 e
Vcd 3
chj
cVj
b<*l
b'ffi
bd 3 e>
b-e'eg
bWf
6W 4
6V
ci*
c'gi
bc>dk
Vff
ct
6V/ 2
bW
6V/
b'c*d*
d*m
c 2 A 2
bc*ej
bVl
c'dh
Vc'd'tg
6VA
bVde
bVd
del
cd*k
bc*fi
b*cdk
c*eg
bVdef
b 3 c 3 dg
b 3 <?d*
6V
dfk
cdej
b<?gh
b 3 cej
c'f*
6W
bVef
6Ve
b ll h
dgj
cdfi
bcdy
Vtf*
<?d*g
tfcd'f
bwf
bVd*
b w cg
dhi
cdgh
bcdei
b s c ff h
<?def
VccPf
6V* 2
bc 7 d
b w df
e'k
ce*i
bcdfh
bWj
cV
bWe
b 3 cd s e
c>
6'V
tf
cefh
bcdg*
b 3 dei
cWf
b<*h
b 3 d'
b'j
6V/
egi
cegr 2
bce*h
b s dfh
<?d*e>
b<?dg
Wg
b'ci
b'cde
eh*
Vf*9
bcefg
bW?
cd'e
bc'ef
bVdf
b*dh
b'd 3
/s
*j
bcf 3
bVh
d>
bJcPf
6Ve 2
b*eg
6Ve
M
d\i
bd 3 i
Vefg
6 8 m
6c 3 * 2
bVd*e
6 8 / 2
6Vrf 2
9*
djh
bd*eh
bT
bcl
b<?d*e
6Vd 4
6V/t
6Vrf
Vp
dy
bdyg
bVk
b s dk
bed 6
be'/
Vcdg
6V
Vco
de>h
bde'g
bVdj
bej
cV
bc*de
Vcef
bg
b*dn
defg
bdef*
bVei
bfi
<fdf
bc'd 3
Vtff
bcf
Vem
df*
b<?f
6V/7i
Vgk
<?e*
c 7 e
Vde>
b"de
vyi
*9
<*k
VcY
bWk
cWe
<*d*
bVg
6 10 c a e
Vgk
e>f*
<fdj
b*cd*i
Vcdj
c 8 ^
b*k
bVdf
b w cd*
Vhj
b'o
c?ei
Vcdeli
b 4 cei
VI
Vcj
6cV
6Vrf
b*i 3
Pen
<yh
Vcdfg
Vcfh
b*ck
b 7 di
Vcd'e
6V
b<?n
b'dm
<*<?
Wce*g
6V
b'dj
Veh
bW
6 U /
bcdm
b'el
<?d*i
tfcef*
Vd\
b'ei
Vfg
6V/
6 12 ce
bcel
Vfk
<?deh
bWh
Vdeh
b'fh
6Vi
bVde
6 12 d 2
bcfk
Vffj
<?dfg
bWeg
bWff
6V
bcdh
6Vrf s
6'Vrf
bcgj
b'/ti
<?<?g
ray
wv
6V/
b*ceg
6Ve
6'V
bchi
Vchn
cV 2
b*dey
b'ef*
b'cdi
6V/ 3
6Vrf 2
6 14 e
bdH
Vcdl
cd 3 h
6V
bVj
b"ceh
bd*g
bVd
b a cd
bdek
Week
cd*eg
bc*j
b 3 C 2 di
Vcfg
b'def
6V
6 12 c 3
bdfj
Vcfj
cd*f*
b<?di
bVeh
b*d*h
6V
b w i
b w d
bdgi
Vcgi
cdef
bfeh
V*fg
Ifdeg
6VA
b'ch
6'V
bdtf
6 2 c/t 2
ce 4
Wfg
6W%
b*df*
bVdg
b'dg
6 16 c
b#j
Vd-'k
d'g
boWk
IPcdeg
6V/
bVef
b'ef
6 18
772] 365
772.
ON THE ANALYTICAL FOKMS 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. xin. (1857), pp. 172 176, [203] given the solution of the first question;
viz. if (f> N denotes the number of the root-trees with N knots, then the successive
numbers <f> 1} <j> 2 , <j) 3 , etc., are given by the formula
... =(1 -#)-*' (1 - a?)-*(l -)-* ...,
viz. we thus find
suffix of </> 1 2 .3 4 5 6 7 8 9 10 11 12 13
0=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 101 I 2~~3 7 12 27 55 127 284 682
Bicentral 010113 4 11 20 51 108 267 619
Total 1 1 1 2 3 6 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, y9, 7, 8, e, etc.,
where of course a + $ + 7 + 8 + e + etc. = N 1. If N is even we may have, say
a = %N; and then /3-f 7+8 + e + etc. = i-W 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, /3, 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 ^ 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. a? in (1 - x)~^,
= %fa (fa + 1) + coeff. a* in (1 - #)-*,
o> = coeff. a* in (1 - )-* (1 -
= \fa (fa + 1) + coeff. of in (1 - a;)-*' (1 -
7 = coeff. of in (1 - #)-* (1 - a?)-*. (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=Q, the formula is
+ ifc*. (*. + 1) (*. + 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 + l). 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 ^>j'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
123456 7
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 formulas of the
paper of 1875.
368
[773
773.
ON THE 8-SQUARE IMAGINARIES.
[From the American Journal of Mathematics, vol. IV. (1881), pp. 293 296.]
I WRITE throughout 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
= 0, l' = 2 J = 3 J = 4 2 = 5 2 = 6 2 = 7 a 0,
123 = e,, 145 = e,, 167 = e 3 ,
246 = 6 4 , 257 = 6 5)
347 = e 6 , 356 = 6 7 ,
where = , viz. each e has a determinate value + or as the case may be ; and
where the formula, 123 = e,, denotes the six equations
23= e.l, 31= 6,2, 12= e.3,
32 = -e,l, 13 = -e,2, 21 = -e,3,
and so for the other formulae. The multiplication table of the eight symbols thus is
1234567
1
2
3
4
5
6
7
c 4 6
773]
ON THE 8 -SQUARE IMAGIN ARIES.
369
(05' + 0'5 + 2 14 + 6 5 72 + e 7 63
(06' + 0'6 + e 3 7H- 4 24 + e.35
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 + 7'7),
this is at once found to be =
(00' - 11' - 22' - 33' - 44' - 55' - 66' - 77')
+ (01' + 0'l+ 1 23 + 2 45 + e 3 67 )1
+ (02' + 0'2 + e 1 31 + e 4 46 + e 5 57 )2
)3
)4
) 5
)6
) 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
= (0 s + 1 2 + 2= + 3 2 + 4 2 + 5- + Q- + 7 2 ) (O' 2 + I' 2 + 2' 2 + 3' 2 + 4' 2 + 5' 2 + 6' 2 + 7' 2 ) ;
viz. the sum of the squares contains the several terms
,.,23. 45, 6,6328 . 67, ,431.46, 1 e 6 31.57, ,e 6 12.47, 1 e 7 12.56, e 2 e 3 45.67,
e 4 e 7 24.35, e 4 6 62.73, 2 e 7 14.63, 2 e 6 51.73, e 2 e 5 14.72, e 2 e 4 51.62, e 4 e 5 46.57,
e 5 e 6 25.34, 5 e 7 72.63, e 3 6 16.34, 3 7 71.35, e 3 e 4 71.24, e 3 e.,16.25, 6 6 7 47.56,
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
eie 2 = -e 4 e 7 = 6 5 e 6 ,
1 3 = - 4 6 = 5 6 7 ,
e,6 4 = -e 3 e 6 = - 2 7 ,
eifi= 6 3 7 = 2 6 8 ,
6i 7 = 2 4 = 6 3 5 ,
2 3 = - 4 5 = ,.
We may without loss of generality assume ^
become
+ = - 4 7 = 6 8 e 6 ,
= e 2 = e 3 = + ; the equations then
= - e 4 6 5 = 6 e 7 ,
= e a ,
=-e 4 ,
= e s ;
c. xi.
47
370
ON THE 8-SQUARE IMAGINARIES.
[773
and writing 0= at pleasure, these are all satisfied if e i = e i = e, = e 7 = Q. 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 e, = e, = f, = +, e t =t i =f, = e, = 0, 8 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 = - e, 0,
1.23 = 6,0, the next line 12.4 = 6,6,7, 1.24 = 63647, and so in other cases):
12.3 =
1 . 23 =
_ Cj
t ,
12.4 =
1.24 =
Ms
MJ
7
12.5 =
1.25
Cjy
- Ms
6
12.6 =
1.26 =
Cjy
- M4
5
12.7 =
1.27 =
- M
Mt
4
13.4 =
1.34 =
- M4
-M
6
13.5 =
1.35 =
-Mi
M7 7
13.6 =
1.36 =
M4
M7 4
13.7 =
1.37 =
Ms
- Ms
5
14.5 =
1.45 =
- i
- <2
14.6 =
1.46 =
Ml
M4
3
14.7 =
1.47 =
Ms
- Me
2
15.6= 1 1.56 =
- M4
-M7
2
15.7 =
1.57 =
- M
Ms
3
Iff. 7 =
1.67 =
-S
- s
23.4 =
2.34 =
Ms
- Ms ' 5
23.5 =
2.35 =
- fjtj
- M?
4
23.6 =
2.36 =
Ma
-M?
7
23.7 =
2.37 =
- Ms
-M.
6
24.5 =
2.45 =
- M?
-Ms
3
24.6 =
2.46 =
- c,
~~ *4
24.7 =
2.47-
M4
Mi
1
25.6 =
2.56 =
- Ms
M7
1
25.7 =
2.57 =
- < 5
_t 5
26.7 =
2.67 =
- M
-Ms
3
34.5 =
3.45 =
- Ms
Mi
2
34.6 =
3.46 =
- M
- M4
1
34.7 =
3.47 =
-
_ ee
35.6 =
3 . 56 =
- <7
_ t7
35.7 =
3.57 =
M7
- Ms
1
36.7 =
3.67 =
- M7
Ma
2
45.6 =
4.56 =
Ms
- M?
7
45.7 =
4 . 57 =
- Mj
- Ms
6
46.7 =
4.67 =
-M
- Ms
5
56.7 =
5.67-
-M7
Ms
4.
773]
ON THE 8 -SQUARE IMAGINARIES.
371
Write as before e 1 = e 2 = e 3 = + ; 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
- 47
- J
7
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 e 6 = 6 4 , e 7 = e s , e 7 =e 4 , that is,
e 4 = e 6 = e 7 = e 5 , which is inconsistent with the last equation of the system e e e 7 = +.
Hence the imaginaries 1, 2, 3, 4, 5, 6, 7, as defined by the original conditions, are
not in any case associative.
If we have e : = e 2 = e 3 = + and also e 4 = e 5 = e 6 = e 7 = 0, 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 5th, 1882.
47 2
372
[774
774.
TABLES FOR THE BINARY SEXTIC.
THE LEADING COEFFICIENTS OF THE FIRST 18 OF THE 26 Co VARIANTS.
[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 bindren algebraischen Formen,
Leipzig, 1872; viz. this is
Order
024 6 8 10 12
Deg.
1
2
3
4
5
7
8
9
10
12
15
f
A
i
H
i
p
(f, i)
T
B
(f, 0.
(f,
(H, i)
(*, t)>
(i, I)
(H, I)
A,<
(p, I)
((f, i), 0.
(/, ?)<
(f, 03
(*, *)
((/,), P) t
(/, 0.
(/,*),
((/, ^ 0.
((/, ), *>.
774]
TABLES FOR THE BINARY SEXTIC.
373
Or, using the capital letters A, B,..., Z to denote the 26 covariants in the same
order, the table is
024 6 8 10 12
9
10
12
15
B
E
M, = (C, Ef
Cf f A P2\4 W
"9 = (A, & ) I -*
U, = (,&)*
V, = (
H
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, C, G and E-, viz. writing
A = (a, b, c, d, e, f, g^x, y),
we have
", 4/8",
C
G
S, Coeff. a? = ae-
T, tf = a& - Zby + 3c0 -
U, x* = 2a'8 - /3' 7 + y/8 - 2S'a,
V,
Similarly the invariant W and the leading coefficients of X, Y depend on the coefficients
of A, G and E 3 ; and the invariant Z depends on the coefficients of G and E*.
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, of
B, x"
C,
, a?
E,
F,
G, of
H,
a+1
ay + 1
a6/- 6
ce + 15
ae + I
a"bd - 4
c 2 + 3
a c + 1
a6 2 - 1
acg + I
df -3
e* +2
ace +1
d 2 - 1
a6 2 e - 1
a6e 5
cd + 2
a s d + 1
6c - 3
o6 3 + 2
d'-lO
a6 2 <7 1
6cd + 2
(t6 2 d + 8
6c/ + 3
6de -1
c 3 - 1
6c 2 - 6
c 2 e -3
cd 2 +2
7,
K, of
L,
M, a: 2
N,
0,
aceg + 1
a 2 / 2 + 1 a 2 d</ + 1
o V + 1 a'V + 1
a\fg 1
a 2 crf#
e/ a 1
a 6/ - 10
ef - 1
de -1
dfg - 6
dey + 1
ce/ - 1
dV - 1
cd/ + 4
a6csr 3
a6 2 /-l
e'ff + 8
d/ 2 + 3
d' 2 f + 3
de/ + 2
c* + 16
bdf - 2
6ce - 2
e/ 2 - 3
e 2 / - 3
de 2 - 2
e> -I
d 2 e - 12
6e 2 + 5
6d 2 + 4
a 6 2 </ 2 - 1
aWfy + 1
a > 2 d</
cPWeg - 1
nio y i /j
tt oclj 4- lo
"c 2 / + 9
e 2 d 1
6c# + 6
6ee</ + 2
6V + 1
by + 1
6V + 9
cde - 17
a6 3 e + 3
bdeg - 34
6c/ 2 - 3
6c 2 y
6cd</ + 2
6c'/ - 12
d 3 + 8
6 2 cd- 6
6d/ 2 + 48
6dV - 4
6cd/ - 14
6ce/ - 2
6cde - 76
a"b 3 g + 2
be 3 + 3
bey - 18
bdef - 12
6ce 2 +11
6d 2 /-2
6d 3 + 48
6V/ - 6
c*eg + 18
6 3 + 15
6d 2 e + 1
6de 2 +2
(?e + 48
6 2 de + 2
c 2 / 2 - 45
c*dy + 1
c 3 / + 9
c*</ - 1
c 2 d 2 - 32
6c 2 + 6
cd' 2 </ + 4
c 2 e/ + 9
c 2 de 14
c 8 d/ + 2
6cd 3 - 4
cd/ + 78
c dy + 4
cd 3 + 6
cV + 1
ce 3 - 36
cde 2 - 21
a"b 3 cy
cd 2 e - 3
d 3 / - 48
tt 6 ~%~ o
6 3 d/ + 8
d 4 + 1
dV + 28
(jOJS^ _ 3
6 s e 2 - 9
a6 2 cegr
b*cdy+ G
6V/- 6
6 2 d'fy + 64
6 2 ce/+ 9
6 2 cde+ 1.6
b^def 144
b-dy+ 32
6 ! d 3 - 8
6V + 81
6 2 de 2 - 39
6c s e - 3
6c%- 96
6e 3 </ 3
6c 2 d 2 + 2
6c 2 e/ + 108
6c 2 d/- 66
bcdy+ 96
6cV + 18
6cde 2 - 12C
6cd 2 e+ 76
6d 3 e + 16
6d 4 - 32
c'y + 36
c 4 / + 27
c 3 d/ - 72
c 3 de -45
cV - 27
c 2 d 3 + 20
c 2 d 2 e + 96
cd 4 - 32
774]
TABLES FOR THE BINARY SEXTIC.
375
P,
Q,
R, a; 6
df*
e 3 y
e 2 / 2
a bcdg*
bcefg
be/ 3
bd*fg
bd<?g
be 3 /
cdeif
ce*
d'g
d 3 ef
6V/ 2
6W/
6V
bc'deg
bcde>
bdf
6dV
(teg
(?def
We*
cd'e
d>
+ 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
- 8
aw
- 1
*
+ 9
- 8
Cti OCO
+ 3
Oflffl
- 24
be 2 g
- 45
bef*
+ 66
<?fy
_ 2
+ 3
cdeg
+ 5
+ 48
cdf*
+ 6
- 12
c 2 /
- 7
- 51
#9
- 3
- 16
dfy
- 3
+ 36
de*
+ 4
- 8
a 6 3 2
_ 2
b^cfy + 4
+ 12
Wdeg 5
+ 192
ty-df*
- 6
- 48
b-ey
+ 7
- 144
b<?eu
- 5
-159
J c ya
- 6
+ 18
bcd*g
+ 7
48
bcdef
- 16
+ 24
ice 3
+ 23
+ 279
bd 3 f
+ 30
- 48
We 2
- 33
- 84
>fdg
- 1
+ 42
<??f
+ 36
+ 153
<?dy
- 37
- 36
cW
- 53
-399
cd 3 e
+ 79
+ 312
d 6
- 24
- 64
a"by</
2
b 3 ceg
+ 5
6 3 c/ 2
+ 6
b 3 d"*g
+ 2
- 224
b 3 def
+ 22
+ 144
bV
- 27
+ 54
bWdg
- 8
+ 336
6Ve/
- 39
- 108
tfcdy
- 50
+ 384
Wcde*
+ 107
-684
b*d 3 e
- 22
+ 144
b<?<i
+ 3
- 126
b<?df
+ 84
- 648
bcW
- 21
+ 432
bcWe
- 102
+ 564
bed*
+ 44
-288
C V
27
+ 270
c*de
+ 45
-450
cW
20
+ 200
376
TABLK.S FOR THE BINARY SEXTIC.
[774
Remaining Coefficients of C, E, G.
E G
G
-V
xy
/+2
be-6
cd+4
(ulg + 1
f 1
df ~ 1
6cp - 1
W/"- 8
V
6e 5 + 9
c 2 / + 9
Off+ 1
ce - 9
crfe - 17
d 3 + 8
</' + 8
*
-* 1
aeg + 1
fy+2
c/-6
fo + 4
af - 1
bdg- 3
fee/ + 3
<?ff + 2
rrlf 1
y 4
i
c 2 - 3
rf 2 e + 2
e 3 + 3
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 +
i
6/ +
2
ace -
19
ad 3 +
8
V -
6
bed +
44
<? -
30
*f
abg +
7
acf -
14
ade
14
by
bf,e
21
bd* +
112
<?d -
70
"V
aeg +
7
adf-
28
a# -
14
Vg +
14
bcf -
42
bde +
168
c 2 e -
105
*Y
adq
aef-
35
bcg +
35
bdf
be* +
105
y
105
*
aeg -
7
f-
14
bdg +
28
bef +
42
<?g +
14
cdf -
168
ce 2 +
105
y
*g-
7
beg +
14
bf
cdg +
14
cef +
21
dy -
112
de> +
70
*?
Off* -
1
9 ~
2
ceg +
19
ef +
6
d*g -
8
def-
44
e* +
30
*
l><f -
1
c f'l +
5
deg -
2
dr-
8
ey +
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. 379384, [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 quintic," 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)' 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=(l, 1, 1, 1, 1, 1, I) 1 to Q(x, y}\
B = (4) 6 (x, y), Invt.,
(7 = (3, 3, 4, 3, 3)= 4 to 8 (*, y?,
D = (2, 2, 3, 3, 4, 3, 3, 2, 2)' 2 to 10 (x, y)\
C. XI. 48
378
TABLES OF COVARIANT8 OF THE BINARY SEXTIC.
[775
E = (8, 8, 8V 8 to 10 (x, yf,
F=(T, 7, 8, 8, 8, 7, 7)> 6 to 12 (x, y?,
0-(5, 7, 7, 8, 8, 8, 7, 7, 5? 5 to 13 (, y) 8 ,
5 = (3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3) 3 to 15 (*, y)",
7 = (18y 12(, y), Invt,
J = (16, 16, 18, 16, 16) 1 10 to It (as, y) 4 ,
K = (14, 16, 16, 18, 16, 16, 14)' 9 to 15 (*, yf,
Z = (10, 13, 14, 16, 16, 18, 16, 16, 14, 13, 10? 7 to 17 (*, y) 10 ,
3f=(32, 32, 32) 14 to 16 (x, y?,
JV = (30, 32, 32, 32, 30)" 13 to 17 (x, y?,
= (25, 29, 30, 32, 32, 32, 30, 29, 25)" 11 to 19 (*, y) 9 ,
P = (58) 18 (x, y), Invt,
Q = (51, 55, 55, 58, 55, 55, 51)' 15 to 21 (*, y) 6 ,
R = (51, 55, 55, 58, 55, 55, 5l) 15 to 21 (*, y) 6 ,
8 = (94, 94, 94y 20 to 22 (*, yft
r=(90, 94, 94, 94, 90) 7 19 to 23 (*, y)\
U=(UT, 151, 147) 8 23 to 25 (x, y) 3 ,
F=(221, 227, 227, 227, 221) 25 to 29 (x, y) 1 ,
TT = (338)> 30 (a;, y), Invt.,
X = (332, 338, 332)' 29 to 31 (*, y) a ,
F = (668, 676, 668)" 35 to 37 (*, y) 2 ,
Z = (1636)" 45 (*, y), Invt.
A = ( Tfa y)>
of ofy aty* a?tf o?y' xy* y 6
0+ 1
6 + 6
c + 15
d+ 20
e + 15
/+6
? + l
<>, Invt.
off + I
bf - 6
ce + 15
d* - 10
+ 16
775]
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
fl / ^- \
ae + 1
o/+2
ag+ I
bff+2
Cgr + 1
bd-4:
be- 6
...
c/-6
rf/-4
c 2 +3
cd + 4
ce - 9
de + 4
e 2 +3
+ 4
6 9 6
/>=( <$*, y) 8
+ 4
+ 10
20 20 20 10
+ 4
y?
xy
y 2
a eg +1
a dg + 1
a eg +1
df -3
/ - 1
/ 2 -1
e 2 +2
cfbcg- 1
aObdg-3
afy - 1
bdf 8
bef+3
6c/ + 3
6e 2 + 9
c*g + 2
6rfe- 1
c 2 / + 9
erf/- 1
c fi 3
erfe - 17
ce 2 -3
erf 2 + 2
rf 3 + 8
rf 3 + 2
3 1 1
+ 5 26 7
2/) 6
a3 2 y 4
oc + 1
orf + 4
ae + 6
/+ 4
ag + 1
6^+4
cct + 6
dg + 4
e + 1
6 2 - 1
be -4
6rf + 4
be + 16
i/ + 14
c/ + 16
rf/ + 4
e/ -4
/s - 1
c 2 - 10
erf -20
ce + 5
rfe - 20
e 2 - 10
rf 2 -20
+ 1
v -
a bg ...
a eg +1
arf</ +2
a eg +1
a/jr ...
<?'
aJ/ ...
cf +2
df +2
/ - 2
/ 2 1
a% + 2
a*/& ...
ce + 1
rfe -2
e 2 - 3
o6c5r - 2
a"6rf(/ + 2
6/" 2 - 2
ceg + I
rf 8 - 1
afbf - 2
a& 2 <? - 1
J,^-!. 4
6e/- 2
crf^ 2
c/ 2 - 1
a& 2 e - 1
6ce + 2
&C/-2
6e 2 - 2
cV-3
C6/+2
rfV- 1
6crf+ 2
6rf 2 + 2
6rfe+ 4
cy -- 2
crf/+ 4
rfy+ 2
def+ 2
c 3 -1
c 2 rf -2
c 2 e 4- 2
crfe + 6
ce 2 + 2
rfe 2 - 2
e 3 -- 1
c? 8 3
rf 3 -- 4
rf 2 e-3
+ 1 2 3 2 1 +6 3
2 4 6 10 8
482
380
TABLES OF COVARIANT8 OF THE BINARY SEXTIC.
[775
u
II
<3
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a
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00
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"*
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o o : :
-*
oo
oo
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IN <M
(M
1
+
1 +
+ 1
-*
^
|
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o "O o o *^3
8
e
2
OO
o
OS
"* IM O O
(N OS "^ IO
OO
1 1 IM l-H
CO
+
1
1
-t- + 1 +
+1
8-
"B*
"
dT ^ J "S 1
e
8
OS
iO
CO IO O IO
o
I_H
CO O <M i^
t-
l-H
l-H l-H
IN
+
+
1
+ + 1 +
+1
*,i Q3 J* ''Sd
*X
"O
5*
^
-i -o t-o w
e
e
t
**<
O
f^> ^3 iO
o
<M
CO
oo co : IM
+
+
1
1 + "8 +
7i
&
^>
8
"i 3s -s t
e
8
e
eo
00
l-H
CO
IN l-H
CO
.j.
+
1
+ +
+1
^
"O
"8
2 S
"e
e
e
CO
10
2
IO
1t
+
1
+
+1
_
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Ss
8
e
8
r _ t
CO
IM
CO
+
i
+
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3 s
8
e
*
*
775]
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
381
a"g"
abfg ...
ceg + I
c/ 2 -1
ffg -1
def +2
e 3 -1
afPeg - 1
ay* + 1
bcdg + 2
beef - 2
bcPf - 2
bde 2 +2
cV -1
#df + 2
c 2 e 2 + I
cd 2 e - 3
d* + 1
3
+ 9
+ 142
a? eg
a 2 fg + 2
y + i
6^ 2 + 2
a eg*
f 2 + 1
a beg 10
abfy - 6
cfg - 10
dfg ...
a bdg
bf 2 - 8
Cjr 6
c?ey + 4
*9
be/ - 10
cdg + 4
c/ 2 + 6
#" + 16
/ !
#g
c/ + 26
rfV + 4
"/ 12
o6y + i
cdf + 4
#f - 8
cfc/ + 12
abyg - 8
bcfg - 10
ce 2 + 16
de> - 8
s 3 12
icegr + 26
6cfe^ + 4
d*e - 12
a6 2 rf + 16
a 6% + 6
6c/ 2 + 24
6<^ 2 + 16
aWcjr
tfef + 24
by 2
bd*g - 8
fee 2 / - 12
Wdf + 16
6cV - 12
bcdg + 12
6rfe/- 64
<?eg + 16
6V + 9
bcdf- 64
beef + 18
be 3 + 36
c 2 / 2 + 9
6c 2 / - 12
6ce 2 - 42
Wy- 96
c% - 8
crfV - 12
6crfe - 76
bd*e + 56
bde> + 60
c 2 e/ - 42
crfe/- 76
bd 3 + 48
c 3 / + 36
cV - 12
cd 2 /+ 56
ce* + 48
c 3 e + 48
c 2 rfe + 4
<?df + 60
cde- + 4
dy + 48
c 2 rf 2 - 32
erf 3 - 16
cV - 99
rf 3 e - 16
dV - 32
cd*e + 84
d> - 32
168
+ 263
+ 168
+ 142
xy 5
<*dy + 1
a 2 e<? + 2
a 2 fg
ay
abg 2
a eg 2 2
adg* - 1
/ - 1
/* - 2
a beg + 10
abfg
cfg - 10
dfg + 2
efff + 3
a beg - 3
a 6rf</ - 2
6/ 2 10
ceg
deg + 15
2 *
e y + o
/3 2
bdf - 2
6e/ + 2
cdg - 15
c/ 2 - 20
df- + 10
e/ 2 6
a6c</ 2 + 1
be 2 + 5
c 2 ^ 6
cef - 5
d 3 g
e 2 / - 15
a6y + 2
6rfy</+ 2
c 2 / + 9
cdf + 28
rf 2 / + 60
def + 60
a6 2 /</ + 10
fafy - 2
6eV- 9
crfe - 17
ce 2 - 26
rfe 2 - 40
e 3 - 40
bceg + 5
6rfey 28
6/ 2 + 6
rf 3 + 8
rf 2 e + 4
a6 2 rf# - 10
a6 2 e</ + 20
6c/ 2 - 30
1J/2 , QO
Oty + d
<?fff ~ $
a6V + 2
a6 2 <# + 6
6 2 e/ + 30
6 2 / 2
bd 2 g 60
be 2 f 6
cdeg+17
*V- 6
6 2 rf/- 32
6c 2 </ + 15
bcdg- 60
6rfe/" +110
c 2 e# + 26
cdf 2 - 2
6 2 rfe+ 2
6V + 36
bcdf- 110
6ce/
b<? 45
e/ 2 - 36
c 2 /- 6
bc*e+ 6
6c 2 /+ 6
bee' + 15
irf 2 /
c 2 rf^ + 40
cd?g - 4
d s g - 8
ierf 2 - 4
bcde 58
6rf 2 e + 40
6rfe 2 + 20
n / ^ ;
c / - 15
crfe/ + 58
rf 3 /+ 4
C 3 rf ...
6rf :J + 32
c 3 / + 45
c 3 */ + 40
erf 2 / - 40
ce 3 30
rf 3 ...
c 3 e + 30
c 2 rfe - 25
c 2 rf/- 20
crfe 2 + 25
rf 3 / - 32
e 2 rf 2 - 20
erf 3
C 2 2
d 3 e
rfV + 20
d 1
33
+ 146
+ 215
140
+ 215
+ 146
+ 33
382
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
[775
*
A
*
%
. rt ,- <M eo
_
,_
co
eo
: 1 + + + f
I
+
+
1
t Gl *V. ^k
t&5
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t
^
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1
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t
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^ X
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<M
co
01 A
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1 -H 1
r-M
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1 + + 1 +
1
+
+
1 +
1 + 1
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^^ **? ^5
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?L X
> V
C
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8 8
01 co co 01 01 co
Ol
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<M
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eo
co
r-*
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+
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01
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+ 1 + 1 1 1
1
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1
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s e
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775]
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
383
M=(
xy
a-C(? + I
W - 2
aV + 1
dfg - 6
tfg + 8
fff - 1
#9 + 8
f 3 - 6
a bdg* - 6
f j/ 3
o 6c</ 2 + 8
5^^r + 6
6 2 <? 2 1
6<//</- 20 6/ 3
bcfg + 6
6V - 24 cy + 8
bdeg- 34
6e/ 2 + 36
cdfg- 34
bdf>+ 48
ffff - 24
ce 2 */ + 18
6e 2 / - 18
cdeg + 76
ce/ 2
c'eg + 18
cdf + 36
rf% + 4
c 2 / 2 - 45
ce 2 / - 72
rf 2 / 2 + 64
ed'ff + 4
d'g - 32
de 2 /- 96
cdef + 78
<*y - 8
e 4 + 36
ce 3 - 36
t/e 3 + 24
a6 V - 3
d 3 f - 48
a6V 2 - 6
b*df ff + 48
rfV + 28
6V&+ 36
6Vgr - 45
a%y<j
b*deg+ 36
6V/ 2 ...
Vceg
6 2 f^ 2
bcyg 18
6 2 c/ 2 ...
6V/- 54
bcdeg+ 78
6 2 d 2 jr+ 64
bc?eg- 72
6crf/ 2 - 144
6 2 efe/~ 144
6C 2 / 2 - 54
6ce-/+ 108
6V + 81
bcvPy- 8
bd'ff- 48
bc'dg- 96
6cde/- 36
6cPe/+ 96
6cV+ 108
6C6 3 +216
6tfe 3 - 72
6cd 2 /+ 96
bd*f+ 128
c% - 36
6cc/e s - 126
6^V- 192
c 3 / 2 + 81
bd 3 e + 16
<?dg + 24
c 2 ^V+ 28
cV + 36
c'e/ +216
c 2 ^e/- 126
<?df - 72
c'dy- 192
cV - 27
cV - 27
c 2 <fe 2 - 378
c<Pf + 16
c 2 c? 2 e + 96
C (i 3 e + 464
2 e 2 + 96
cd 4 - 32
d* 128
rf 4 e - 32
+ 9 8 +1
182 180 + 136
497 1120 + 551
+ 688 +1308 688
884
TABLES OF COVARIANTS OF THE BINARY SEXTIC.
[775
v
V - l
a'df
a?eg 3 + 1
w ...
e /y i
4& + 4
efy + 3
/V 1
a beg"- + 1
deg + I
*9
f* - 3
a bdg* 4
*/V l
df* + 3
ef - 3
a6c7 2 - 3
6/c, + 4
cdff 3 1
*f 3
a6y + 1
bdfg ...
6/ 3
ce/y 2
a b 3 fy + 1
6c/^ 4
be'g -15
<*g*
c/ 3 + 3
bceg + 2
bdeg 16
be/ 3 + 18
cdfg + 16
rf% + 4
6c/* - 3
6rf/ ...
c'/y + 15
c#g - 22
*v - i
6rfV - 4
bey + 18
cdey ...
ce/ 2 + 6
rf*/ 2 - 6
We/ -12
cfy + 22
cdf- - 36
rf 2 e<? + 8
/ + 3
6e* + 15
c?/ 2 + 3
ce 2 / + 9
rf 2 / 2 -32
a6%' - 3
c*dy + I
cefy - 8
d'g
rf 2 / +36
U&/Q *4- 3
c*e/ + 9
crf/ -48
d*ef + 24
4 - 12
/>2^"3
erf 2 / + 4
ce 8 + 12
rfe 3 -12
a6V + 3
6cy + 3
ede* - 21
rfy +32
a&y + 3
b-dfg ...
6crf^ + 12
rf s + 8
rfV - 12
6'V^ - 18
6Vy 3
bce*g 9
ab>eg 3
a6 3 /<,
Vdeg + 36
b'ef"
6C6/ 2 - 9
ay
tfceg - - 6
b s d/ 3 ...
b<?fg - 18
bd'ey - 4
b'cdg + 6
6V" -
6V/ -27
bcdeg + 48
6rf 2 / 2 - 32
6'ce/ + 9
6 2 rf 2 # + 32
6c 2 e^ - 9
6crf/ 2 ...
6rfey + 66
6vy + 32
tfdef
6c 2 / 2 + 27
6e 2 / - 18 6e 4 - 27
6W - 39
6V - 27
JcrfV - 24
bd\j - 32
<?fy -15
6cV - 3
bc*dg 36
6crf/ ...
6rf ! e/ + 32
<?deg + 21
bc*df - 66
6c 2 / + 18
6ce 3 + 27
6rfe s - 12
c^ + 39
6cV + 18
bed 1 / - 32
6rf 3 / ...
c 3 e<? - 12
cV/ - 18
6cd 2 e + 76
bcde* + 84
6rfV - 12
c 3 / 2 +27
crfV - 8
bd* - 32
6rfe - 32
c?dg + 12
C 2 rf s <7 + 12
cd'e/ - 76
c 4 / + 27
cV + 12
(?ef -27
<?def - 84
crfe 3 + 45
c*rfe 45
(r-rf/ + 12
c'rf 2 / + 12
cV + 45
rf 4 / + 32
c'rf 5 + 20
cV - 45
cW ...
erf/ + 32
rfV - 20
c'rf 2 e + 20
crf 3 e
erfV - 20
erf 4
d> ...
rf 4 . ...
775]
TABLES OF COVARIANTS OP THE BINAEY SEXTIC.
385
a?y
afy 3
or 1 /
xy 7
!/</
a y
a 2 bg
aV 1
a?dg*
v + i
2 //
y
abg 3
a 2 beg
*6/? .-
#& - 3
/ 7
dfg + 4
efg + 5
/ 2 ^ ' 1
o6egr 2 + 3
a6/<? 2 ...
tig*
t/ y
cdg
cegr - 1
e?e<? + 10
e 3 ^ + 7
f 3 - 5
a 6o^ 2 - 4
kTg- 3
cegr 2 + 1
deg 2 ...
6/ 2 ...
c/ 2 - 2
df + 2
e/ 2 10
a beg 2 - 5
befg + 18
crf^ 2 - 10
/V- i
df*g ...
cef 1
rf 2 ^ + 3
e 2 / - 9
a by + 1
bdfg ...
6/ 3 14
cefg + 22
dy - 3
ejg ...
rf 2 / + 3
def + 6
a 6% + 3
*efc - 18
6e 2 ^ +10
c y - 7
c/ 3 - 12
rfe/Jr + 14
ef* ...
de 2 - 2
e 3 - 6
bceg - 22
bdeg 16
6e/ 2 - 5
c^jr + 16
fffff + 19
rf/ 3 - 8
Myg 2 ...
a Vdg ...
a6 2 e$- + 1
6c/ 2 - 2
6rf/ 2 + 28
eft -10
ce*g - 1
rfe^ - 24
e s g - 9
bceg 2 + 1
Vef + 1
6 2 / 2 + 2
6^V- 19
bey + 4
crfeg' ...
cef - 1
rfe/ 2 - 4
e 2 / 2 + 6
&</V- 1
6cV ...
bcdg - 14
fcfe/+ 34
c 2 e^ + 1
cdf +80
dV - 13
e y + 9
a6 2 e/ + 2
bd 2 g 2 - 3
bcdf - 14
6ce/ ... be 3 + 3
c 2 / 2 + 24
ce 2 / -55
rf 2 / 2 + 38
aWdg 2 - 2
&yv- 2
bdefg+ 14
bee 2 + 11
6rf 2 / -18 c 2 ^ + 24
ed*g +13
rfV -
rfe 2 /- 62
6 2 e# + 2
6crf0 2 - 6
6rf/ 3 - 8
6rf 2 e + 1
6rfe 2 + 26
cy + 4
cefe/- 6
ePef - 65
e 4 + 30
6 2 / 3
6ce/y ...
be 3 g - 9
c 3 / + 9
cV + 9
c^/- 58
ce 3 - 37
rfe 3 +50
a&V+ 10
6C 2 / + 9
6c/ 3 + 6
6e 2 / 2 + 6
c 2 rfe -14
<?df 4-10
cde" - 42
d 3 / - 52
a6y + 5
6 2 c%- 28
6crf/^- 34
6rf 2 /^+ 18
c 2 rf^ 2 + 2
erf 3 + C
CV 4-13
rf 3 e + 38
rfV + 58
6 2 c/^+ 5
6V^- 24
6ce 2 ^ 4
6rfeV- 10
c 2 efg-\\
af&cg
crf 2 e -53
afb 3 eg +12
a&% + 14
6 2 <%- 80
6 2 e/ 2 + 42
6ce/ 2 + 57
6rfe/ ! -20
c 2 / 3 + 9
Vdf + 8
rf 4 +24
6 s / 2
Wceg + 1
Vdf* ...
6c 2 /g-- 4
bd?eff+ 58
6e 3 / + 12
crf 2 /^- 1
6V - 9
a6% + 8
b*cdg+ 4
^c/ 3 - 42
6V/ + 60
6crfejr -i- 6
6rf 2 / 2 + 16
c y + 6
crfe 2 ^+14
6V/- 6
&V - 6
&"<*-/- 57
bWg- 38
bc*eg + 55
bcdf 2 ...
6rfey-110
c 2 rf/^- 26
crfe/ 2 - 16
Vcde + 16
6 2 cV- 6
fcW/- 16
6 2 */ ...
6c 2 / 2 - 60
bee 2 /- 18
6 4 + 45
c 3 e 2 ^ - 13
ce 3 / + 3
6 2 rf 3 - 8
6 2 crf/+ 20
6W+ 30
6V + 36
bcd?g+ 55
bd s ff + 52
<?fy - 3
c 2 / 2 + 21
d 3 eg 6
ic 3 * -- 3
6 2 ce 2 - 21
Wg - 9
6c<%+ 62
bcdef . . .
6d 2 e/- 66
c 2 c%+ 42
cd?eg+ 53
rf 3 / 2 + 8
6c 2 rf 2 + 2 6 2 rf 2 e- 2
b<?df+ 110
6cV+ 18
bee 3 -60
bde* + 30
c 2 rf/ 2 - 30
crfy 2 + 2
rfV/- 2
c'rf
be 3 / -12
6c 2 e' + 12
60^7+ 66
6cZ 3 / ...
c s e^ + 37
c 2 * 2 /- 12
ccfey-52
rfe 4
6c"rf+52
bcffe- 87
6c(fo 2 -126
6rfV+60
c 3 / 2 - 66
cd 3 g - 38
ce 4 + 15
6crf - 28
6rf 4 + 16
6rf 3 e + 24
c'dg -50
c 2 rf 2 ^- 58
crf 2 e/+ 87
d'g -24
c*e -15
c 4 / + 45
<*g - 30
<*ef +60
c 2 def+ 126
crfe 3 - 40
rf 3 / + 28
(rW 3 + 10
c 3 ^ + 40
<?df - 30
c 2 ^ 2 /- 15
cV - 60
rf 4 / - 16
rfV - 10
<?d* - 10
c 3 * 2 + 60
c 2 cfe 2 ...
erf 3 /- 24
rfV + 10
c 2 rf 2 e- 15
cd'e ...
crfV+ 15
cd*
d"
rf 4 e
C. XI.
49
386
TABLES OF COVARIANT8 OF THE BINARY SEXTIC.
[775
P=(
'' ' '/ ' *
a6W + 4
(mO /(f
e/i? - 12
aWceg*
/ 3 + 8
C/tf i .
6V^ 2 - 3
Cv Cr H- 1
cdfg
defy - 6
ce'g + 30
df* + 4
ce/ s -24
<?g + 4
e? 2 e^ 12
e'/> 3
dy* - 24
a 6V
<fey + eo
/"</
e 4 - 27
a 6 cdg' 6
6 <?fg +6
ce/gr + 18
c 2 ^^ 42
c/ 3 - 12
c 2 ^ 2 + 60
d^& + 12
cV/ - 30
de'g - 18
cd 3 g + 24
def* ...
cd 2 e/ - 84
e 3 / + 6
crfe* + 66
6V<^ + 4
d*f + 24
c^dfg 18
dV - 24
cVgr - 24
6Vegr +12
<?ef + 30
c 4 / 2 - 27
cd\g + 54
(V - 8
erf 2 / 3 - 12
c'rfe/ + 66
cdV - 42
(r'e 3 - 8
ce 4 + 12
^y _ 24
d*g - 20
(PcPe* 39
</'/ + 24
cd 4 e + 36
dV - 8
d - 8
775]
TABLES OF COVAEIANTS OF THE BINARY SEXTIC.
387
51
55
55
>, v)
58
55
55
51
efg
"X :
a 2 6 e/
:
01 6 y . . .
o efg . . .
dfg* ...
(v b do , , ,
/Q , . .
F
a 2 6d/
/V -
6ce/
deg* + 5
e 2 / + 2
fa -
O 2 6c/
tfg
6cd/ - 5
c/V ...
d/V- 5
e/ 2 5r - 4
abcff 3
dfg ...
f 3
ce/<7 + 15
dy ...
e]fg 5
/ 4 + 2
dfg* ...
*g ...
w/ - 2
c/ 3 - 10
de/fir + 20
e/ 3 + 5
(j O"^ . . .
ey + 2
ef
cdy^ + 6
d*fg + 20
d/ 3 -20
a& 2 /<7 2
o c/t/ . .
/V - 4
bVfg - 2
eefy + 8
defy - 25
e 3 ? -20
6 ce/ - 15
de</ 2 - 6
/ 4 + 2
cdeg + 5
ce/ 2 - 10
de/' - 10
e/ 2 + 20
efg + 15
d/ 2 <jr + 6
6V 2 // ...
cdf + 6
d 2 e0 - 10
e 3 / + 15
a 6V
d 2 / -20
<?fg + 6
cde/ 5
ce 2 / - 7
d 2 / 2 + 18
a6 2 d/ + 5
/V
defy + 90
e/ 3 - 6
cdfg+ 5
d'g - 3
de 2 / - 22
efg - 15
6 cd^ 2 - 20
d/ 3 -50
V s <?e.(j i 8
ce 2 /^ + 5
dV - 3
e 4 + 12
/ 3 + 10
ce/s' ...
$g 35
c 2 /^ + 8
ce/ 3 - 5
de 3 + 4
a 6 2 c/ + 4
6 cy + 5
c/ 3 + 20
e 2 / 2 - 15
cdy+ 10
dy + 3
a ay
dy^r 6
cdfg - 90
<Pfg ...
iVdjr 2 + 25
cdefg+ 4
dVJr- 7
6 2 c& + 4
efy - 8
ce 2 ;? + 40
de'g + 40
c 2 e/^ - 40
cd/ 3 - 24
d 2 /- 1 - 2
de<7 5
e/ 2 + 10
ce/ 2 + 40
de/ 2 - 20
c 2 / 3 + 15
cefy - 8
defy + 1
d/ 3 - 6
ab<?fg - 6
d% + 50
e 3 / - 20
cd?fg- 50
&?/*+ 18
de 2 / 2 + 8
e 2 / + 7
cdejr 4
d 2 / 2 + 10
6cy + 20
cde 2 ^+ 45
d 3 /^ - 22
e 4 / - 3
6c 2 e !/ - 5
cd/ 2 - 68
de 2 / -40
c*dfff- 40
cde/ 2 + 5
d~e*g + 14
a6y
c 2 / 2 - 6
ce 2 / + 76
e 4 - 15
<Vg ...
ce 3 / ...
d 2 e/ 2 + 42
6 2 c// ...
cdV+ 7
d*g + 22
6V/<, +35
c 2 e/ 2 - 20
d% - 5
de 3 / - 46
de/ - 6
cde/ - 16
J2-^" i QQ
a e/ + uo
(?deg 45
cd'eg ...
d 3 / 2 + 50
e 5 + 12
df*g + 6
ce 3 +23
de 3 - 58
C 2 d/ 2 ...
cdy s + 60
dVf 65
"W
% + 6
d 3 / + 30
6Ve<, + 8
cV/-65
cde 2 /- 20
de 4 + 20
622 1 A
ee<7 + 10
e/ 3 - 6
dV - 33
c 3 / 2 + 42
cdV+ 5
ce 4 + 20
a6 3 e/ +10
c f*g - 10
6 c 2 e/ + 7
bVdg - 1
c 2 d 2 gr- 14
cd 2 e/+ 65
dV ...
f-g - 10
dy - is
yv- 7
cV/ + 36
c 2 */- 82
cde 3 -45
d 3 e/ - 20
6 2 cd/ +10
dffg + 68
cd 2 /+ 3
cd 2 /- 37
cV - 44
d 4 / -20
dV ...
cefg - 40
d/ 3 - 32
cdefg+ 16
c 2 de 2 - 53
cdf/+ 12
dV - 20
a6 3 d^ 2 + 20
c/ 3 +30
e^ 42
cd/ 3 - 22
492
388
TABLES OF COVARIAXTS OF THE BINARY 8EXTIC.
[775
51
55
(continued).
58 55
55
51
<&/ + 79
a6cdV +122
MV -
a'Vefg - 20
aWd'/g - 10
ai*e'/^ + 24
rt^ce 8 ^ 86
d t 24
d' - 44
eJ/<7 + 50
/'
de>g ...
6c J ^ + 22
ce 2 / 2 + 39
a'Vfg 2
a6y - 2
V -15
6 2 cV - 20
def + 10
<?efy - 76
d 8 /^ - 30
tfceg + 5
&V0 + 6
e/ 2 - 30
ce(^ + 20
'/ ...
c'f 3 + 54
ffl$Q J_ Q]
C/ 2 + 6
'/;/ - 24
bV/g -15
ce?g + 20
C^ Q 15
cc^ - 38
d'e/ 2 + 50
d*g + 2
df> + 32
cdeg 5
ce/ 2 ...
c*dfg+ 40
cde'g + 82
d<?f -- 84
efo/ + 22
y 54
cdf* - 10
d 2 e<? - 60
eVy + 65
cdef - 50
e 8 +27
27
JVe0 - 18
ce 2 / + 60
d'f ...
c 2 e^+ 60
ce 3 / + 6
Wdg> - 4
6W? - 8
c 2 / 8 - 24
d'g -50
cfe 2 / + 40
cd*eg- 65
d s eg 12
c 8 ^ _ 23
cV - 39
cd'g - 42
d'e/ - 10
e 4
crf 2 / 2 + 10
rf 3 / 2 + 64
c 3 / 3 + 27
cd*f - 50
cdef+ 50
tfe 3 + 30
ficfy^r + 20
ccfe 2 /+ 10
ePe 2 / - 82
<?d?fg + 33
cde* + 107
ce 8 + 54
6 Ay ...
c*deg+ 20
ce 4
de* + 30
c 2 </eV+ 53
d'e - 22
d*f - 64
c 3 / 2 ...
C 2 ^>_ 40
<*V + 20
6cy - 12
<?def*- 107
6cV + 3
dV + 32
c 2 ^ + 65
c 2 e 2 / ...
tfef - 10
<?dfg + 58
cV/ + 21
<*df + 84
ic 3 ^ + 46
<?def- 10
cd 3 g + 20
rfV + 5
c 3 e 2 ^ + 44
c? s e$r - 79
c s e 2 - 21
c"e/ - 6
cV -30
cd 2 e/ ...
6^ +15
cV 2 - 54
cc/f/ 2 + 22
<**"- 102
c 2 rf'/+ 82
cd 8 / + 10
crfe 8 - 40
<?deg+ 45
c*d*eg- 122
cd*e*/+ 102
erf 4 + 44
<?d#- 112
crfV - 75
rfy ...
c 3 ^ 2 - 30
c 2 ^/ 2 - 32
cde 4 - 45
Vtff 27
cd'e - 34
d*e + 40
rf'e 2 + 20
c'e 2 / + 30
c^e 2 /+ 112
d*g + 24
c 4 efe + 45
d> + 32
6c% - 20
6V 4 e^ - 20
c*d 3 g+ 20
cV - 30
<i 4 e / _ 44
<?d* - 20
jogfy _ 1 2
cV ...
c 4 / 2 ...
cWc/f 75
cof 4 ^ + 44
d'e* +
c*df - 30
c 8 ^ 2 / 1 5
c 3 ^ ...
er'^-SO
crf s e/+ 34
cV + 30
c 3 * 2 + 50
<?def+ 40
cd'f - 40
crf 2 e 8 - 30
cWe + 30
eWe - 25
c s e 3
cc?V + 25
d"f - 32
c*d' 20
cd*
C 2 <i 3 /- 20
d'e
dV + 20
cc? 4 e
d e
-to
50
60
776]
389
776.
ON THE JACOBIAN SEXTIC EQUATION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvni. (1882),
pp. 52 65.]
THE Jacobian sextic equation has been discussed under the form
(z - a) 6 - 4a (z - af + 106 (z - a) 3 -4c(z-a) + 56 2 - 4ac = 0,
(see references at end of paper), but the connexion of this form with the general
sextic equation has not, so far as I am aware, been considered. And although this
is probably known, I do not find it to have been explicitly stated that the group
of the equation is the positive half-group, or group of the 60 positive substitutions
out of the 120 substitutions, which leave unaltered Serret's 6-valued function of six
letters.
Invariantive Property of the Jacobian Sextic.
Taking z a as the variable, and comparing the equation with the general sextic
equation
(a, b, c, d, e, f, g$>-a, 1) = 0,
we have
a, b, c, d, e, f , g
= 1, - a, 0, \b, 0, - fc, 56 2 - 4ac ;
the Jacobian equation is thus an equation
(a, b, c, d, e, f, gfa, y) 6 = 0,
for which c = 0, e = 0, ag-f 9bf- 20d 2 = 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, gfa, y) = 0,
linearly transformable into the form
(a, b, 0, d, 0, /, g\x, y) = 0,
390
ON THE JACOBIAN 8EXTIC EQUATION.
[776
where ag + 9bf 20d* = 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 6bf+ loce
and that B is the determinant
a, b, c, d
b, c, d, e
c, d, e, f
d, e, f, g
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. B =
(7 =
D =
ag + 1
ad*g-l
a 2 dy + 1
a y + i
bf - 6
&y + 1
d/ 3 + ^
d 2 - 10
bd]f 2
(t bd fo + 12
Idy - soo
d* + 1
d'g - 20
df*g - 2500
a6 3 cty 2 + 4
/ - 3125
,,6 3 / 3 + 8
a 3 b 2 fy 15
Wd?f- 24
bdtfg* - 4800
6d 4 / + 24
bd/ 4 g - 7500
,, d 8
dy + 30000
d a / 3 g + 50000
aWdg 4 - 2500
,,b 3 /y - 410
,,b^dy"y- 171300
b*df 5 240000
,,6^ 4 // + 780000
,,bd 3 /* + 1200000
dy - 1000000
d 5 / 2 - 1600000
a b'dfg 3 - 7500
", b 3 dy + 50000
b 3 d*f 3 g + 83200
oTby 3125
Wdfg* - 240000
,,6y 5 - 331776
..ft^/jr' + 1200000
6 4 rf 2 / 4 + 1843200
b 3 dy - 1600000
6W/ 8 - 2560000
776]
ON THE JACOBIAN SEXTIC EQUATION.
391
It is clear that these are all functions of ag, bf, c 2 and a 2 / 8 + fe 3 ^ 2 , say of at, /3, 8 and <.
In fact, A and are functions of a, /3, 8; C contains two terms, coefficient 4, which
are = 4 \/8 . </> ; A contains two terms
-3125 (a 4 /* + &y),
which are = 3125 (< 2 2a 2 $ 3 ); and also several pairs of terms, each which pair contains
the factor </>. We thus have
A =
B =
(7 =
A =
a + I
aS - 1
a 2 8 + 1
a 6 + 1
ft- 6
P* + 1
a/38 + 12
a 4 /? 30
8-10
/3S - 2
aS 2 - 20
a 4 8 300
S 2 + 1
/3 s +8
a 2 /? 3 + 5840 (=6250- 410)
0"8 - 24
a"/3 3 15
/3S 2 + 24
a 3 y38 4800
8 s - 8
a'S 2 + 3000
4> v/8 + 4
a^S 171300
a 2 /3S 2 + 780000
a'S 3 - 1000000
a/3 4 11520
a/J'S + 83200
1? - 331776
^8 + 1843200
P>8* - 2560000
^8. a 3 - 2500
a/3 - 7500
a8 + 50000
/S 2 - 240000
08 + 1200000
S 2 - 1600000
tf. - 3125
We have ante, the relation a + 9/9-20S = 0, and using this to eliminate a, we have
A, B, C, A as functions of ft, B, <f> (that is, of bf, d* and aV + fcy). Effecting the
substitution, we find the values of A, B, C without difficulty. As regards the value of
A, this is
= - 3125< 2 + 2<j)K V8 + terms without <f>,
392
where
ON THE JACOBIAN 8EXTIC EQUATION.
[776
2K = - 2500 ( 81/9 3 - 360/38 + 4008 3 )
7500 (- 9/9"+ 20/98 )
+ 50000 ( - 9/98+ 20S 3 )
240000 ( i j y )
+ 1200000 ( /98 )
- 1GOOOOO ( 8*),
or, reducing and dividing by 2,
A" = - 3125 (60/3 2 - 240/38 + 256S 3 ).
The calculation of the terms without tj> is much more laborious, but they come out
= - 3125 (GO/3 3 - 240/38 + 256S 3 ) 3 8.
Hence the value of A is
A = -3125{ < 2
+ 20 (60/9 2 - 2408 + 25G8 2 ) V8
+ (60/3 - 24008 + 2568 3 ) 3 8),
A = - 3125ft,
h = + (60/9 3 - 240/SS + 2568 2 ) V8,
= a 3 / 1 + &y + (Wb'df - 24Qbd'f
The values of A, B, C, and the foregoing value of h then are
say this is
where
that is,
A =
B =
C =
A =
/3-15
P + I
/Q 8 + 8
/8 s V8 + 60
8+10
PS + 7
/3 a 8+ 51
j88 ^8 - 240
S 2 - 19
/JS+ 84
S 2 V 8 + 256
8 3 - 8
<#> + 1
<^>^/8 + 4
We may, if we please, regard /8, 8, </> as irrational invariants of the sextic, viz. A, B, C
being rational and integral functions of (3, 8, <, we have conversely /9, 8, < irrational
functions of A, B, C; and then the equation for h, say
.. V(- A) = + V8 (60/9 3 - 240S + 2568')
is the invariantive relation which characterises the Jacobian sextic.
776] OX THE JACOBIAN 8EXTIC EQUATION. 393
The expression for A in terms of A, B, C, D is
& = A>- 375A'B - 6254*0 + 3125 A
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 b = 0, /= ; and also its value in terms of
a, ft, B, <f>. These are
tfbdf'g 12
2 - 90
- 48
,,b#f<? +246
bd 3 / 4 + 480
rfy - 258
d'f - 432
- 12
12
+ 168
+ 240
- 168
-228
</"</
,,bd'f
,,d"
- 48
72
+ 480
+ 552
-432
-976
+ 336
+ 408
- 248
SV
,,0
ft
- 1
+ 2
12
72
90
+ 168
+ 552
+ 5
+ 246
+ 240
-976
-258
- 168
+ 336
-228
-408
-240
12
48
-480
- 432
The Group of the Jacobian Sextic.
The solution of the Jacobian sextic equation depends upon that of a quintic; in
fact, calling the roots z^, z , z lt z^, z t , z t , then there exists a quintic having the roots
c. xi.
50
394 ON THE JACOBIAN 8EXTIC 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., >Jh. 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 m z^.z t z, ,z 4 z lt &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 z a z, . z, z, . z t 2, , &c., are thus rational func-
tions of the coefficients of the sextic ; each such rational function is a 12-valued function
of z m , z,, z lt z,, z,, z t , invariable by all the substitutions of a group of 60 substitutions;
and therefore also every like 12-valued function of the roots z^, z,, z lt z t , z,, z t 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,
z^, z , z,, z it z 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, af. 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,
adf be,
aebfd,
afedb,
afced,
acdfe,
aefdc,
adecf,
afdbc,
adcfb,
abfcd,
acbdf,
bdcef,
bcfde ,
bedfc,
bfeed.
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, abcde, abc, &c. denote cyclical substitutions. It is easy to verify
that each of these substitutions does in fact merely permute B, G, D, E, F; thus
B C D E F
abcde on ab.ce. df, ac . bf . de, ad .be . ef, ae . bd . cf, of . be . cd
= bc . da. ef, bd . cf . ea, be . cd. of, ba . ce . df, bf.ca.de
= + ad . be . ef, ae . bd . cf, af . be . cd, - ab . ce . df, ac . bf . de
D E F B G,
which (expressed as a cyclical substitution) is = BDFGE, 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.
502
396
ON 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 B*, C', IP, E*, F'; 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, b, 0, d, 0,f, </$*, 1) = 0,
ag -f 9bf 20(2" =0, I effect upon it the Tschirnhausen transformation
X = -az>-6bz'*-10d;
which, it may be remarked, is a particular case of the Tschirnhausen-Hermite form
X (az + b) B + (a* + 6bz + 5c) C + (az> + 6bz* + I5cz + Wd) D
+ (az 4 + 6hz 3 + 15cz* + 20dz + lOe) E + (az* + Gbz* + locz 3 + 20ck 2 + loez + of) F.
Writing for convenience Y=X+lOd, Z=X-10d, this is
az 3 + Qb2 1 - . +7=0,
and we thence have
Gbz 3 . + Yz . =0,
. + 7z* . . = 0,
- Zz* . -6fz -0=0,
- Z* . - 6fz* - gz . = 0,
-Zz* . -Gfz 3 - gz* . . =0,
or, eliminating, the resulting equation is
az> +
a, 66,
a, 66, . Y, .
a, 66, . Y, . .
Z, -, 6/, g
Z, . Qf, a, .
2, . Qf, 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] OX THE JACOBIAX SEXTIC EQUATION,
and those of 123, &c., to the matrix
397
z,
6/,
9
. Z,
6/
ft
z, .
6/,
ft
.
,
The several terms are
123 .
456
H a 3 . g :>
-124.
356
6a 2 6 . - 6fgt
+ 125 .
346
+ .-36/V
-126.
345
-a"F . g-Z-^lQf 3
+ 134.
256
+ - 36a6 2 .
- 135.
246
- a 2 F . -g-Z
+ 136.
245
+ - 6a6 F . - QfgZ
+ 145.
236
+ 6a6 F . - QfgZ
-146 .
235
o . - seyxz
+ 156.
234
+ - a F 2 . - gZ*
-234.
156
-a 2 F-2166 3 . ,fZ
+ 235.
146
+ Qab Y . QfgZ
-236.
145
- - 366 2 F . 36^
-245.
136
- 36&'F .
+ 246.
135
+ aF 2 g Z^
- 256 .
134
- 66 F 2 . 6/Z 2
+ 345 .
126
+ _ a Y* -g Z-
- 346 .
125
- 66 F 2 . - 6/ 2
+ 356.
124
+ .0
-456.
123
F 3 Z 3 .
Hence, collecting and reducing, the equation is
= Y 3 Z 3 .
+ YW . (Sag + 726/)
+ YZ . (Say + 36agbf+ 12966 2 / 2 )
+ F . 216a 2 />
+ Z .- 2166V
+ ay - 36ay&/,
398
ON THE JACOBIAN 8EXTIC EQUATION.
[776
where F, Z denote X + lOd, X-lOd respectively, and consequently YZ = X* - IQQd*.
Hence, writing as before a, yS, 8, <f> to denote ag, bf, d* and a 2 /* + 6y respectively, the
result finally is
1
a + 3
a' + 3
a 2 / 3 + 216
^8 + 2160
0+72
a/3 + 36
6y -216
a 3 + 1
8 -300
aS - COO
a" 36
F + 1296
a'S 30
08- 14400
a/38 - 360
8" + 30000
aS 2 4- 30000
/TO + 12960
08" + 720000
S 3 - 1000000
i) 6 =o,
where observe that the coefficient of the term in X is 216 (a 2 / 8 - fry), = 216 V(</> 2 -4a 2 /3 3 ).
We have as before ag + 9bf 20rf" = 0, that is, o + 9/9 208=0; and using this equation
to eliminate a, also in the constant term writing its value for <f> in terms of h,
the new equation is
(- 60/9 2 + 240/9S - 256S 2 )
-5 x
5x
5x
_.'_*-
_
,
1
ft- 9
/3 s - 243
- 21 ^A
A^/8 + 432
8+48
08-1872
/3 s 729
8 s +3840
/3"8 + 4184
0S 2 - 11520
8 3 + 8292
where
A = {h + (- 60/8 2 + 240/9S - 256S 2 ) *JS}* - 4 (- 9/3 + 20S) 2 /9 s
= h* + 2h JS
& -60
/3S + 240
S 2 -256
It is to be sho\vn 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&r, 1)' = 0,
a = l, c = 2d, e = - 96/4-
776]
ON THE JACOBIAN SEXTIC EQUATION.
399
I consider the general quintic (a, b, c, d, e, /$#, 1) 5 = 0; taking the roots to be
x-i, # 2 , x 3 , x 4 , x s , and writing
& = 12345 -24135,
fa = 13425 - 32145,
fa = 14235 - 43125,
< 4 = 21435-13245,
<k = 31245-14325,
6 = 41325-12435,
where 12345 is used to denote the function
= (x^ + x,p 3 + x^ + XtX 6 + ovi) V(20),
(this numerical factor V(20) being inserted for greater convenience), then the equation
whose roots are fa, fa, fa, fa, fa, fa, which equation may be regarded as the resolvent
sextic of the given quintic equation, is
a 6 x
-5 4 x
5a 2 x
- V /(Q). 2
+ 5
1
ae
-1a?df
+ 1
+ Ia 3 c/ 3
-4bd
+ 3V
- 2a?def
+ 3c"
&c.
+ &c.
D = a 4 / 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. 263276, [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$a>, I) 5 .
Many of the terms thus vanish, and the equation assumes the form
a'x
-5a 4
5a 2
-aVD
+ 5
1
ae + 1
aV + 3
+ 1
a 3 cf 2 + 1
c 2 +3
ac 2 e- 2
aV + 1
c" +15
aW-11
ac 4 e + 35
c 6 -25
and then if, as before,
or say
a = 1, c = 2d, e = - 96/+ 36d 2 , f 2 = 216A,
a = 1, c = 2 VS, e = - 9/3 + 368, f 2 = 216A,
[This Collection, vol. iv., p. 321.]
400 ON THE JACOBIAN 8EXTIC EQUATION. [776
this becomes identical with the foregoing Tschirnhausen-transformation equation; thus
ae + 3c" = - 9 + 368 + 128, = - 9
8 + 48;
and similarly
3a'e s - 2ac 1 e -I- 15c = &- + 243,
8 - 1872,
8= + 3840.
So for the constant term, + la'cf 2 gives the term 432/t */&, and + la'e 3 , &c., give the
remaining terms 729/3", &c. of the value in question.
It only remains to verify the equality of the coefficients of X,
216 VA = VD or 46656A = D.
Here D, the discriminant of the quintic (a, 0, c, 0, e, f^x, I) 5 , from the general
form (see my Second Memoir on Quantics, [141], or Salmon's Higher Algebra, third
edition, p. 209) putting therein b = 0, d = 0, is
D = a*f 4 + 1,
a'e" + 256,
a'c'ef - 1440,
a 2 c 2 e 4 -2560,
ac'f 2 +3456,
ac 4 e s + 6400,
and writing for a, c, e, f their values 1, 2/S, 9( /9 + 4S), 216/t, the value becomes
D = (216) 2 . A 2 .
+ 432A V8 . 12960 ( - 48) 2
+ 34560 (/9 - 48) 8
+ 55296 S 2
- 256 . 9" . ( - 48)"
-10240 .9'. (/9-48)<S
- 102400. 9 3 . 08-48)' 8 s .
The whole divides by (21 6) 2 , and we thus obtain
A = A' + 2AV8. 60 (-/9 + 48) 2 . + (- / S-4S) 3 . 324(/3-4S) 2
- 240 (- /3 + 48) 8 + 1440 (/3 - 48) 8
+ 256 & + 1600 8",
which is, in fact, equal to the foregoing value of A.
776]
ON THE JACOBIAN SEXTIC EQUATION.
401
The conclusion is that, starting from the Jacobian sextic
(a, b, 0, d, 0, / g$z, I) 6 = 0,
where ag + 9bf 20d- = 0, and effecting upon it the Tschimhausen-transformation
X = - az s - 6fo 2 - 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$, 1)' = 0,
where
c = 2d, e = - 9bf+ 36rf 2 , f = V(216A),
and, A being the discriminant of the Jacobian sextic, then
h = - - V(- A), = a 2 / 3 + b 3 g- + 60W/ -
O y O
As to the subject of the present paper, see in particular Brioschi, "Ueber die
Auflosung der Gleichungen vom funften Grade," Math. Annalen, t. xm. (1878), pp. 109
160, and the third Appendix to his translation of my Elliptic Functions, Milan, 1880,
each containing references to the earlier papers.
C. XI.
51
402 [777
777.
A SOLVABLE CASE OF THE QUINTIC EQUATION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvm. (1882),
pp. 154157.]
THE roots of the general quintic equation
(a, b, c, d, e,f$x, 1)' =
may be taken to be
--+ B+ C+ D+ E
(AI
- -,, + m t ,, + a? H + m* n + u>
,, + co 3 ,, + co + co* + o> 2
-,, + w a ,, + w 4 ,, + o) + o> 3 ,,
- + w + a>- + <a s + co* ,
where w is an imaginary fifth root of unity; and if one of the four functions B,
C, D, E is = 0, say if E = (this implies of course a single relation between the
coefficients), then the equation is solvable.
Writing x = -- , we have
(a, b, c, d, ,/)(-* lY=(a', 0, c', d', e',f'^, 1)*,
\ u> /
where
a' = a,
ac' = ac - If,
a"d' = a*d - 3abc + 26 s ,
a'e' = a*e - 4a 2 bd + Qab"c - 36 4 ,
a'/' = a 4 / - 5a 3 be + Wab"-d - 10a6 2 c + 4& 6 ,
and the roots of the new equation
(a', 0, c', d', e',/'$, 1)' =
777]
have
difficulty
A SOLVABLE CASE OF THE QUINTIC EQUATION.
403
have the above-mentioned values, omitting therefrom the terms ; we find without
Of
2-, = -BE -CD,
>.*:
a'
-,=- B 3 C - &E* + BCDE + BD* + C 3 E + C-D 2 - DE\
Q
f
J - = -B* + 5B 3 DE - 5B-C*E - S&CD" + bBC 3 D + 5BCE 3
a
-C s + 5CD 3 E - 5CD"-E- -D"- E\
and hence, when E = Q, we have
9 c rn
Z-,- -LL>,
a
^ =
a'
$
or, as these may be written,
5BC*D -C*-
= CD,
a
a a-
f
equations which imply a single relation between the coefficients a', c', d', e', f.
Supposing this satisfied, we may attend only to the first three equations; or, writing
for convenience,
' O
c * / i\
a*
1
Ct Ct
the equations are
-SfC).
512
404 A SOLVABLE CASE OF THE QUINTIC EQUATION. [777
The first equation gives C = -K, and substituting this value in the other two
equations, we have
7 + BI> + eD = 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 Z); and omitting this factor,
the result is
D>, 7", -&D ;=0.
D', T 2 , -SD',
7, 0, - Z>,
y, 0, - Z>, 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
Hence, collecting, and changing the sign of the whole expression, we obtain
SD"> _ (2 7 So + 7 '0 + 0>) D'o + (_ ys + 8780 + S>) 'fD' + y>0 = 0,
a cubic equation for D\ 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, S, 6 their values in terms of the original coefficients, the
equation for IP becomes
+ + a? (ac - fry (ae - 4bd + 3c 2 ) (aZ)) 10
(- 16 (ac - 6 2 ) (a?d - 3a6c + 26 s } 2 ]
| 28 (ac-6 2 ) 3 (a ! d-3a6c + 26 3 ) \
+ 4 (ac - b*y> < + 1 2a 2 (ac - 6 2 ) (a 2 d - 3abc + 26') (ae - 4bd + 3c 2 ) 1 (aD)
U 8 (a'd - Sabc + 2b 3 ) 3 J
- 128 (ac - fry {a" (ae - 46d + Sc 2 ) + (ac 2 - fry] = 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. 226229.]
IT seems desirable to present in a more developed form some of the results of
the foregoing paper.
Thus, if the equation (a , a,,..., a n $, 1)" = of the" order n has n v equal
roots, where v is not > \n 1, then we have i/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
20 + 2, 2^ + 3, ..., n-m.
The signification is
1
i/f (r, v + 1, m) = r . . f-^+- 3 <*>m a r +m
V *)
+ (r-4)
1
' [r - 1]" +2
1 2
[ r _ 2T>+ 2
r+m-i
Thus, when v = 0, the condition is
r .rl
1
-0,
ADDITION TO MR HUDSON S PAPER
406
that is,
satisfied when the equation has all its roots equal.
[778
0,
The values of m are 0, 1, 2, ...,n 2, and those of r are 2v + 2, 2v + 3, ..., n-m;
in particular, if m = 0, the values of r are 2, 3 ..... n, and the corresponding conditions
are
Oott, - Oi s = 0,
a a s - a^a, = 0,
flofln Oi<*>n-l = 0,
and so for the different values of m up to the final value n 2, for which r = 2,
and the condition is
-ian - 0*n-i = ;
we have thus, it is clear, the whole series of conditions included in
0,
a n _j
j, a, a^-i, a n
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
1
r .1.
- (r- 2) . 2 .
r.r-l.r-2
1
that is,
I "*
= 0;
r-l.r-2 r-l.r-3r-2.r-3
or, what is the same thing,
(r - 3) a m a r+m - 2 (r - 2) a m+l a r+m ^ + (r - 1) a m+2 a r+m _ 2 = 0,
where n = 4 at least, and m, r have the values
0, 1, 2 n-4
r =
4 4
' ^>
5, 5
thus, when n = 4, the only values are m = 0, r = 4, and the condition is
0.
778] "OX EQUAL ROOTS OF EQUATIONS." 407
Similarly, when v = 2, the condition for n 2 equal roots is found to be
r-l.r-2.r-3 r-1 .r- 3.r-4" r r-2.r-3.r- 5 r-3.r-4.r-5~
or, what is the same thing,
r - 4 . r - 5 . a m a r+m
-3.r-2.r-5. a m+1 a r+n ^. l
+ 3.r-l.r-4. a m+2 a r+m _ 2
. r 1 . r 2 . tt m +3 Br+ms = 0,
where w = 6 at least, and m, r have the values
m 0, 1, ..., n 6
r =
6, 6,
7, 7
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 ty (r, v + I, m) = 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 = 1, 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 + 1, ?)) = 0.
408 [779
779.
[NOTE ON MR JEFFREY'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 /ca s /9 = u 2 , or, as this is afterwards written,
2i<x*y = aa? + 2bxy + cy* + 2ex + Zdy + \,
is, I think, introduced without a proper explanation. Say, the form is a?y = z* (*$X y, zf,
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, z = 0), z = for the tangent at the cusp, and x =
an arbitrary line through the cusp ; then the line z = besides intersects the curve
in a single point, and, if y = is taken as the tangent at that point, the equation
of the curve must, it can be seen, be of the form
(<B + ftA)y = *(a, b, c,f, g, h\x, y, zf.
The conic (a, b, c, f, g, h^jx, y, z)* = touches the quartic at each of the two inter-
sections of the quartic with the arbitrary line # = 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 x + ^6z = 0, there exists such a conic, viz.
writing x instead of x + $0z, the form is o?y = z^(a', b', c',f, g', h'Qx, y, z) 2 , and the
new conic (a', ...$#, y, z)* = Q has the property in question. This is the adopted form,
and it thus appears that in it the line x = 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 # = 0, y = 0, z = 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 formulae. For the
binary seventhic
(a, b, c, d, e, f, g, h\x, y) 7 ,
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
fa 1
Jo
abdh - 4
beg - 2
bf + 6
c*h + 3
cdg - 2
cef - 6
tPf +10
de 2 5
a6 s cft
Vdg + 20
6c 2 - 15
6cd/-24
bee" -30
We -10
c 3 / +27
C 2 de - 45
cd 3 +20
2. 10
2 . 2
3
3
.3
. 9
ac + 1
6-l
ach
dff
aWh
beg
bdf
6e 2
y
cde
d 3
+ 2
- 7
+ 5
- 2
+ 7
+ 22
-25
-27
+ 45
-20
Deg-order.
Deg-weight.
C. XI.
52
410
ADDITION TO MR HAMMONDS PAPER.
[780
m.
rv.
2.6
2.4
8.
8.
7
7
ae + 1
bd-4
c 3 + 3
./-//
abg
cf
de
aWf
bee
bd?
+ 1
- 7
+ 9
5
+ 12
-30
+ 20
r
2.2
2.6
3. 11
3. 5
ag+ 1
bf- 6
c + 15
d-10
off +1
a be -5
cd +2
a6d + 8
6c-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 - 3c") (Qbf- loce + Wffi) ;
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
ce 1
cPe
aV>h
Peg
b'df
6V J
bcj
bcde
bd?
b*ef
Ifde
b'c'e
Wed?
5 . 11
I.
III.
IV.
n.
a 3 eh
+ 1
+ 1
fff
a*bdh
- 1
- 4
- 4
+ 1
beg
- 2
- 7
- 5
b/ 2
+ 6
- 6
<?h
+ 3
+ 3
+ 2
cdg
- 2
+ 2
- 7
cef
- 1
- 6
+ 9
+ 15
+ 5
&
+ 3
+ 10
- 10
de'
- 2
- 5
- 5
ab*ch
- 4
Vdg
+ 20
+ 28
+ 8
+ 7
6V
+ 1
+ 57
+ 12
- 45
- 5
6c s i;
-15
- 21
- 6
+ 7
bcdf
-14
-24
- 36
- 12
+ 22
bee 3
+ 11
-30
- 30
-25
bd?e
+ 1
-10
+ 40
+ 50
c 8 /
+ 9
+ 27
+ 27
-27
c'de
-14
-45
- 15
+ 30
+ 45
cd 3
+ 6
+ 20
- 20
-20
oPb*h
- 2
b*cg
- 7
b 3 df
+ 8
- 48
- 48
-22
6 s e a
- 9
+ 25
bWf
- 6
+ 36
+ 36
+ 27
Vcde
+ 16
+ 120
+ 120
-45
b'd 3
- 8
- 80
- 80
+ 20
13
3
90
- 90
hcW
+ 2
+ 60
+ 60
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 seminvariants
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. 103108. Read Jan. 11, 1883.]
I CONSIDER the cubic equation (a, b, c, cx, l) s =0. It is shown (Serret, Cours
d'Algebre supdrieure, 4th ed., Paris, 1879, t. n. 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 - c 2 , \ = V^f,
n = & - 4- A C, = aW + 4ac 3 + 4MJ - 36 2 c 2 - Qabcd,
_
(values which give a + S = 1, aS y3y = 1, and therefore also
a 2 + aS + B- + Py = 0,
which is the condition in order that the function </>#, = -' , may be periodic of the
third order, <px = x), then, u being a root of the equation, say (a, b, c, d$u, 1) 3 = 0, the
other two roots are
OM + /3
and
''
Su-/3
where observe that, by the change of \ffl into - JSl, a, /8, 7, 8 become 8, /9, 7, a ;
so that the last-mentioned value <^ -I M is, in fact, the value obtained from <M by the
mere change of sign of the radical.
522
412 ON THE AUTOMORPHIC TRANSFORM ATION [781
It is to be observed that, if we have between two roots u, v of the equation
(a, 6, c, d^x, 1Y = 0,
ff^tt _l_ jO
a relation v = - f , where at, /9, 7, 8 have given values, this implies in the first place
a relation between o, 6, c, d (and the given values of a, /8, y, 8), and it implies more-
over that u, and consequently also v, are not any roots whatever, but two determinate
roots of the equation; viz. u, v will be each of them expressible rationally in terms
of a, 6, c, d and a, /9, 7, 8. And if, in order that (o, 6, c, d) may remain arbitrary,
we consider a, /9, 7, 8 as given quantities satisfying the relation which exists between
these quantities and (a, b, c, d), then in general we still have u, v determinate roots
of the cubic equation. But in the foregoing solution u is any root whatever of the
cubic equation.
To examine how this is, starting from the equations
(a, b, c, d$w, 1)>=0, (a, b, c, d$v, 1) 3 = 0, B-
we have
o (w - v 3 ) + 36 (u? -v>) + 3c(u-v) = 0,
and therefore
a (u 1 + uv + v*) + 3b(u + v) + 3c = 0,
that is,
(au + 36) v + au 2 + 3bu + 3c = ;
or, writing herein for v its value,
a (aw + /8) + (awM- ) (71* + 8) (aw + 36) + (yu + 8) 2 (aw 2 + 3bu + 3) = ;
that is,
a (au + /3) 2 + ay (au 3 + 3bu?) + 7" (au 4 + 3bu + 3cw 2 )
+ (08 + 7) (aw 2 + 36w) + 278 (aw 3 + 3&u" + 3cw)
+ 8 (aw +36) + 8 2 (aM s + 36w +c) = 0;
or, reducing by the equation au* + 36u 2 + 3cw + d = 0, this is
a (au + &y+ ay (- Sou - d) + 7 2 (-
+ (a8 + @y) (aw 2 + 36w) + 78 (- d)
+ /38 (aw + 36) + 8 2 (aw 2 + 36w + 3c) = 0,
and, collecting the terms, this is
w j a (a 2 + a8 + S 2 + 7)
+ u [a (2a/3 + /3S) + 36 (aS + 8- + 7) - 3ca 7 - dy*}
+ a/3 2 + 36/98 + 3cS 2 + d (- ay - 2 7 S) = 0.
We can, from this equation, and the equation au 3 + 36w a + 3cw + d = 0, eliminate w, thus
obtaining a relation between a, 6, c, d, a, /9, 7, 8; and, this relation being satisfied, the
two equations then determine u rationally in terms of these quantities.
781]
OF THE BINARY CUBIC FUNCTION.
413
We may without loss of generality assume atS - 0y = 1 ; and, this being so, if we
then further assume a+8= 1, then we have
tt 2 + 08 + fr + firy = 0,
which is, as above appearing, the condition for (fix = 0. The foregoing equation in u
thus becomes
u {a$ (a - 1) - 36a s - Scay - dy' 2 }
+ (a/3 3 + 36,98 + 3c8 2 - dy (8 - l)j = ;
a linear equation giving (in a simplified form) the like results to those given by the
quadric equation; viz. substituting iu 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/8 (a - 1 ) - 36a 2 - 3ca 7 - d^ = 0,
a/3"
equations which are, in fact, satisfied by the foregoing values of a, /3, 7, B, as may be
verified without difficulty.
It is to be remarked that if, instead of the root u and the equation v = - S ,
yu + 8
r\ /O
we consider the root v and the equation u = ; then, instead of a, /3, 7, 8, we
yv + a
have B, /3, 7, a, and the corresponding equations are
dy* =0,
equations which are also satisfied by the foregoing values of a, /3, 7, 8. And the four
equations, together with aS 7 = 1 and a + 8=1, are more than sufficient to determine
the foregoing values of a, /8, 7, 8.
But we further verify without difficulty that the foregoing values of o, y8, 7, 8 give
identically
(a, b, c, dJicuc + Py, ^x + 8y) 3 = (a, b, c, d^x, y) s ;
or the formulae lead to an automorphic transformation of the binary cubic (a, b, c, d$x, y) 3 .
And conversely, starting from the notion of the automorphic transformation of the binary
cubic, we ought to be able to obtain the foregoing formulae.
For greater convenience, I write the equation of transformation in the form
(a, b, c, dQax + Py, yx + Sy) s = - 6 (a, b, c, d$x, y) 3 ;
414 ON THE AUTOMORPHIC TRANSFORMATION [781
the equations to be satisfied by a, /9, 7, S, 6 then are
oo + 6 . So^ + c . 307 s + drf = -a0,
aa'/9 + 6 (a'S + 2a/9 7 ) + c (2a 7 8 + #/) + dy'B = - be,
aa/9 5 + 6 (2aS + &y) + c (aS* + 2/9 7 8) + cfyS* = - cd,
off + b . S/^S + c . 3/98* + dS" = - d6.
Writing a8 7 = V , and as before 1 for the discriminant, the theory of invariants
gives fi V^nfl 4 . We are considering the case of the general cubic function (a, b, c, d~x, yj,
for which H is not = ; and we have therefore V ' d 4 = 0, or, what is the same thing,
we may write V = q*, Q = <f, where q is arbitrary.
It is to be shown that a + 8 is = q or 2q, the latter value giving the trivial
solution ax + fty, yx + &y = (x, y). The proper solution thus corresponds to V = q 1 , a. + 8 = q,
that is,
= 0, or
the condition for the periodic function <f>'x x = 0.
For this purpose, from the foregoing equations eliminating a, b, c, d, we have
3a 7 2
0;
a/3* ,
/8 s ,
an equation which may b written
D + 6 (123 + 234 + 341+ 412) + &> (12 + 23 + 34 + 41 + 13 + 42) + 0"(1 + 2 + 3 + 4) + 0* = 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 =o 3 , a 2 8 + 2o 7 , a8 2 + 2/3 7 S, 8 s ,
12, 13, 14, 23, 24, 34 = {a, a 2 (aS + 3/3 7 ), (a'S 2 + 8/3 7 + /3V),
(a'8 2 + aS/9 7 + /Sy), S 1 (aS + 3/9 7 ), S'} V ,
123, 124, 134, 234 = (a 3 , a (08 + 2/87), 8(aS + 2/37) ; 8 s } V,
D = V 8 ,
and the equation thus is
+ 6" [of + a'S + aS* + 8 3 + 2o/S 7 + 2/3 7 8]
+ ffi V [a 4 + a 3 8 + aS" + 8 4 + 3a 3 y37 + 3/3 7 8 2 + 2a 2 S 2 + 208/87 + 2/Sy]
+ V'[a 3 + 8 3
+ V = 0.
781] OF THE BINARY CUBIC FUNCTION. 415
Putting herein a + & = m, aS = n, 7 = n - V , it is found that n disappears altogether
from the equation; viz. the resulting form is
2 V (m 4 -3Vm 2 +2V 2 ) + 0V s m(m 2 -2V) + V 6 = 0,
or, what is the same thing,
V 6 = 0.
Putting for 0, V, their values q 3 , q\ the equation divides by q s , and omitting this
factor it becomes
m 4 + 2m?q 3m 2 <f - 4mg 3 + 4 j 4 = ;
viz. this is
{(m-q)(m+2q)}* = 0,
or we have
m = q or 2q ; that is, a + 8 = q, or a + & = 2q.
Writing, as before, A, B, (7=ac-6 2 , ad be, bd-c 2 , we deduce from the foregoing
equations
PA = V s [Aa* + B . ay + Oy 2 ],
PB = V 2 [A . 2a/3 + B (a& + 7) + C . 2 7 S],
+ B.08
which are, in fact, the equations for the automorphic transformation of the Hessian
(A, B, C\x, yf. And, writing herein 9, V = q 3 , q\ the equations become
A (a 2 - 3 2 ) + Bay + C . 7 2 =0,
.A2a/3 +B(aS + /3y-q>)+C . 2yS =0,
A/3* +B0S
From the first and second of these we have
A :B : = 7 2 (V +^ 2 ) : - 2 7 aV +278^ : a 2 V - ( a 2 +
or, writing herein for V , q* the values q*, q 2 (aS - /3 7 ), the three expressions divide by
2yq-, and we have
A : B : C = y : 8- a : -/3.
Combining these values in the first place with the equation a + 8 = 2^, we may write
a, & 7, S = -q-pB, -2pC, 2pA, -q+pB,
where p is to be determined. Substituting in the last of the three equations, we have
A . 4p 2 C 2 - 2pBC (-q+pB)+C (- 2pqB + p>B?) = 0,
that is,
p>C (4, AC - 5 2 ), =-p 2 .Cfl, =0,
and the form (a, b, c, d~$x, yf being arbitrary, neither G nor O is = ; whence p = 0, and
the values are a, y3, 7, 8 = 5, 0, 0, q, that is,
(a, 6, c, <f- #, - 2y) 3 = - q 3 . (a, b, c,
a trivial result.
416 AUTOMORPHIC TRANSFORMATION OF THE BINARY CUBIC FUNCTION. [781
But, combining the same values with a + B = q, we have
and then, substituting in the third equation, we have
A . VC" - 2pC (q+ P B) + C(- f?' + pqB + ?&) = 0,
that is,
C {(4 A C-
or, omitting the factor C, and introducing the foregoing notation X 2 = -, this is
' - q* = 0, or say p = q.
For the unimodular substitution aS /3y=I, we must have q"=l: but, the transforma-
tion being
(a, b, c, d$*x + &y, yx + &y)* = - q* . (a, b, c, d$x, yY,
to make this strictly automorphic, we must have 9" = 1, and the two conditions are
satisfied by q = 1 ; we then have p = ._ ; and the coefficients are
_- 20 -2A -
*, ft 7. . =
which are the values given at the beginning of the paper, and which belong to the
automorphic transformation
(a, b,*c, dQcuK + fSy, yx + SyY = (a, b, c, d~$x, y) 3 .
The & priori reason for the periodicity-equation <f> 3 x = 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 fy, must leave the factor unaltered, and it must
thus be a periodic substitution <f> a x = x. But it was interesting to see how the condition
for this, cf + 8* + ot8 + 7 = 0, comes out from the consideration of the equation
(a, b, c, d$ax + py, <yx + &yy> = (a, b, c, d$x, 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. 139142. Read March 8, 1883.]
THE Memoir referred to, published in the Me'moires de I'Academie, 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 De"blai,
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 1 }. He then considers
the normals 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 A a, Bb must not cross each other, for if they did, drawing the two lines
A
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 DES DEBLAIS ET DBS REMBLAIS." 419
a right line ; and, assuming that some one element of the filament ED 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 which 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)dz= Ix'dx' - jxdx,
the equation dx' = dx meaning x' = z + a discontinuous constant. In the actual case of the filament, the formula
i, if rdr=/dr / , then
/V - r) r dr= jr"' dr 1 - fr* dr.
532
420 ON MONDE'S "MEMOIRS SUR LA THEORIE DBS 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
/ 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. 222229. Bead May 10, 1883.]
CONSIDERING the three cones,
(^ + X).Y 2 + (<? + X)F 2
*= 0,
(p + v) Z 2 + (q + v) F 2 + (r + v) Z* = 0,
where
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 (x, 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 (a;, y, z)
being an arbitrary line on the first cone, we can find (x', y', z'} and (x", y", z") such
that
+ (r + X)* 2 =0,
(p + /*) a;' 2 + (q + /*) y' 2 + (r + /*) z"> = 0,
(p + v) tf" 2 + (q+v) y" 2 + (r + v) z"* = 0,
x"x +y"y +z"z =0,
x x' + y y' + z z 1 = 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. [78:5
and consequently
(p + v) (yz 1 - y 7 *)' + (q + v) (zx r - z'x)* + (r + v) (xy 1 - x'y? = 0.
It is to be shown that this equation is implied in the remaining first, second, and
third equations ; t for, this being so, (x, y, z), (x' t y', if) satisfy only these equations ;
or (a;, y, z) are any values whatever satisfying the first equation. The other two
equations then determine (x, y', /), 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
(P + ") [(2/ J + z') (y' + *") - x*x'*\ + (q + v) [(z* + x 3 ) (z* + x' 3 )
+ (r + v) [(of + y*) (x 1 * + y') - z V s ] = 0,
or, what is the same thing, in the form
= ;
for, comparing in the two forms, first the coefficients of a?x'*, these are
v and -2
which are equal in virtue of p + q + r + \ + /j, + v=0; and comparing next the
coefficients of y-z'", these are
p + v and - (r +fi)-(q + X),
which are equal in virtue \>f 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 , 6 , c 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 6 aQ d c respectively : then we may write
= (l, ,
ca
/
p+ v, q+ v, r + v= i
for, starting from the first set of values, we have the second set if only
' """" ~~ ~~~ " il ^~ ~f"i~A ^~ T-J/-M
"V\ r e thence obtain
783] ON MR WILKINSON'S RECTANGULAR TRANSFORMATION. 423
and, in order to the identity of the two values of 0, we must have
that is,
(abc - b 2 ) (ABC -A*}- (abc - a 2 ) (ABC - fl") = 0,
or, reducing,
(a 2 - b 2 ) ABC - (A 1 - B?) abc + A*fr - BW = 0.
But
hence the whole equation divides by a 2 6 2 , and, omitting this factor, it becomes
,
which is a known relation between the elliptic functions of the arguments a , b , c
connected by the equation a + 6 + c = 0. Similarly, for <f>, we have
, , , c a C A
= + *>-te = + AB-BC'
and, comparing the two values of <j>, 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 , b , c connected by the equation a + 6 + c =0) contain a singly infinite
system of rectangular axes.
Considering an argument /, and denoting its sn, en, dn by /, f, F respectively,
we have, for an arbitrary line on the first cone, the values
y, z =
.C M>J-a,BC.F.
In fact, substituting in the equation of the cone, we obtain the identity
and if we determine M by the condition that tf + y^ + z* shall be = 1, then we have
1 = M 2 {k'*Aa. + &4bcf* - *BCF>},
where the coefficient of M 2 is
424 ON MR WILKINSON'S RECTANGULAR TRANSFORMATION.
which is easily shown to be
= fctt' J &c(a'-/'),
so that the values of x, y, z are
[783
and, similarly taking the arguments g t , /, and denoting their elliptic functions by
g, g, G, h, h, H, we have for a system of arbitrary lines in the three cones
respectively, the values
x, y, z =
x', y, z =
V/fc-'Cc
V^bc ./
ifCab . A
-J-&BC .F
V-
a* -/")
-r- >Jk>k'*ca (6 1 - g 1 )
these values being such that of + y 2 + z 2 , a/ 2 + y' 1 + z"', ft" 1 + y" 1 + a" 2 are each = 1. The
radicals in the first line would be more correctly written, and may be understood as
meaning k' >JA Va, k *JA \/b Vc. * "J & V-B *JC, 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 = g h ; similarly the condition for the third and first lines
is satisfied if 6 = /i / > and we then have a + 6 = <7o /o ; that is, c<,=g<,f<, or
c l> =f g l) , which is the condition for the first and second lines; hence the arguments
o<>, &o, Co, /o> fft, h being such that
A, - g 4- a c = 0,
-/. . +/+&<, = <),
S'o -/o + C = 0,
6 - c . =0,
or, what is the same thing, a , 6 , c , / , ^ , h 9 being the differences of any four
arguments a, /9, 7, S, the foregoing values of (#, y, ^), (', y', z'), (x", y", z") will
satisfy the equations
a? +y a - +z* =1,
x'" +y'* +z'* =1,
x"x + y"y + z"z = 0,
x x' + y y' + z z 1 = 0,
for the transformation of a set of rectangular axes. These are, in fact, Mr Wilkinson's
expressions, the a c , &, c , / 0) # , /i c being his tp,p q, qt, 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 X 2 + F 2 + Z* = 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 + X, &c. of the equations of
the three cones satisfy the equations
1, q + \, q + fJ., q + v, (q + X) (q + p.) (q + v)
1, r +\, r + (i, r + v, (r + X) (r + /j.) (r + v)
This is obviously the case for each equation such as
| 1, p+\, p + /t | = 0;
and any equation containing the fifth column is at once reducible to
| 1, p, p a +p 2 (\ + fji + v) =0,
that is,
| 1, p, p 3 1 + (A. + /i + ") 1, p, p 2 | = ;
or, dividing by | 1, p, p* \, this is p + q + r + \ + fi + v = 0, the equation connecting
the coefficients.
Hence, representing the three cones by
p Z" + q F" + r Z" = 0,
and the absolute by
y2 I "17*2 i 72 f)
C. XI. 54
426 ON MR WILKINSON'S RECTANGULAR TRANSFORMATION. [783
the coefficients p, q, &c., are connected by the equations
i. P- P'- P"> PP'P"
i- q, q', q", qq'q"
I, r, r', r", rr'r"
;
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
The equations | 1, p, p" \ = and | 1, p, pp'p" \ = 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 : rp : 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 + Ty + 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"X a + r"p"Y ! '+p"q"Z 2 = 0.
Suppose, in general, that we have an arbitrary line (x, y, z) and an arbitrary
plane oX + pY+yZ = 0, 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+0Y + yZ = ()
with the cone p'X 3 + q'Y* + rZ 3 = 0: the equation of the pair of planes is
(aX + /3Y+yZy(p'a? +q'f
+ (ax +/3y + yzy(p'X* + q'Y*
-2(aX + pY + yZ)(ax+l3y + yz) (p'Xx + q'Yy+ r'Zz) = 0.
In the present case, the plane aX + 0Y + yZ = 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
y* +
z-) (p'Xx + q'Yy+ r'Zz) = 0,
783] ON MB WILKINSON'S RECTANGULAR TRANSFORMATION. 427
where the quantities (x, y, z), as belonging to a line on the first cone, satisfy the
condition pa? + qy- + rz* = 0. The equation may be written
(a, b, c,f, g, h\yZ-zY, zX-xZ, xY-
where
a, b, c,f, g, h=q'z* + r'y\ r'o?+p'z\ p'y* + q'a?, -p'yz, -q'zx, -r'xy,
and, as before, pa? + qy- + rz* = ; 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 / y'X 3 + r""Y i +p"q"^ = is
(q"r"o? + r"p"f + /Y'* 2 ) (q"r"X* + r"p"Y* +p"q"&) - (q"r"xX + r"p"yY + p"q"zZ)* = ;
viz. omitting a factor p"q"r", this equation is
(p", q", r", 0, 0, O^yZ-zY, zX-xZ, xY-yX)* = 0.
And it is to be shown that this is equivalent to the former equation ; viz. writing
yZ zY, zX xZ, xYyX=\ fj,, v, then that the two equations
(q'z^ + r'y 3 , r'a? + p'z*, p'y* + q'x>, -p'yz, -q'zx, -r'xy$\, p,, z/) 2 = 0,
are equivalent to each other.
We have identically \x + py + vz = 0, and thence also
(\x + p.y + vz)[(p' - q'-r')\a;+ (-p' + q - r')/j,y + (-p - q
where, on the left-hand side, the terms in /*i/, v\ and \/t are
= Zp'yzpv 2q'zxv\ Zr'xy\p,.
Hence the first equation may be written
[q'z* + Sf + (p'-q'-^) a?} V + (VV +p'z* + (-p' + q'- r') f\ pf
and it is to be shown that this is equivalent to
p"K+q"p? + r"S = Q;
viz. that we have p" : q" : r" =
q'z 2 + r'y* - ( p' - q' - r') x 2
: r'a? + p'z 1 -(-p' + q'~ r') y"
: p'f + q'x* -(-p'-q' + r') z\
where pa? + qy 1 + rz* = 0. Writing the equation in the form
p" : q" : r" = A : B : C,
542
428 ON MB WILKINSON'S RECTANGULAR TRANSFORMATION. [783
we have
A = q'z* + r'f - p'a? + q'x' + r'a?
= - pa? + (cf + r') (x> + y* + *') - q'y- - r'z*.
By what precedes, we have an identity of the form
a? + y* + z* = a (pa? + q'y* + r'z>) + (pa? + qy* + rz*),
where, determining a from the equations 1 = q'a. + qfi, 1 = r'a + rfi, we find
but pa? + qy* + rz* = 0, and the relation thus is
3? + f + z* = a(p'a? + q'y'+r'z*) ;
hence
A = {(q + /) a - 1} (ffaf + q'y 3 + r'z*),
or, substituting for a its value, this is
and, forming the like values of B and G, the relations to be verified become
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. 337.]
SINCE our last 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
risumi 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 Republic 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 appeai-s 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 OVTUX; ov-ra, 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 iiegative 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 spacje 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
552
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, and 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 n-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 f(x) = 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 Gepm&rie, forming a short supplement
to his Traite de la Mtihode, 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 a; 2 + 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 = it is the real magnitude a, and in the case a = 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 (x, 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 a; 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 *Jx, sin a;, or log a;, 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 <f>(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
<f> (x + iy) = X + i 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) = ; 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-dimensional 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, t) 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+ 1) coordinates
(<r, y, e, ...). 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. xvn.), 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
562
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 M6canique celeste, or of the whole of the Mdcanique 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 congrueuce 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 V2, and so consider numbers of the form a + b V2, (a 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 + 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 atfxiipa and K&JZXK, 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. 430347):
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 Thecetetm, 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. 287212), 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 " Eupij/ca," 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] BBITISH 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 e* or exp. x, 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 Philosophies
naturalis Principia Mathematics 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 P n , 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 re-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 " Me"moire 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
Eider's remark in the paper " Sur une contradiction apparente dans la throne 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
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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 Plucker's great work, the Theorie der
algebraischen Curoen (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 (Plucker'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 Biemann in the memoir
" Theorie der Abel'schen Functionen," Grelle, t. LIV. (1857). For a curve of a given order,
reckoning cusps as double points, the deficiency is equal to the greatest number
4( 1) (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 3/> 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 curve 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 n-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 modem 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, y, x) has been treated in this manner, with the most interesting results;
the function so determined for all values of the parameters (a, /3, 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
572
452 PRESIDENTIAL ADDRESS TO THE [784
infinite series of ascending powers, convergent for all finite values, real or imaginary, of
the variable x or I/a; c, and so having the one essential singular point x = 00 or x = c,
as the case may be : the memoir is published in the Berlin Abhandlungen, 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 e*. 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 Thdorie des Fonctions
Elliptiques (1825 1828); also Jacobi's Fundamenta nova theories 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 vX, the square root of a rational and integral quartic function of a
variable x; for this he substituted a radical A$, = Vl k? sin 2 <f>, and he arrived at
his three kinds of elliptic integrals F<f>, E<j>, H<f>, 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, 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 formulas containing q, a tran-
scendental function of the modulus defined by the equation q = e~" K ' 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 , @ 3 , @ 4 ; these are the so-called theta-functions, or, when the distinction is necessary,
the single theta-functions. Finally, Jacobi using the transformation sin <f> = 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 <s>: and the elliptic functions sn, en, dn are
expressed with these, or say with 9 lf 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 connected with it by the equation q = e'""
(so that (a is in fact =iK'/K), was led to consider the three functions <f>a>, i/r, ^ea,
which denote respectively the values of v 4/ T, \/k' and \/kk' regarded as functions of o>.
784] BRITISH ASSOCIATION, SEPTEMBER 1883. 453
A theta-function, putting the argument = 0, and then regarding it as a function of o>,
is what Professor Smith in a valuable memoir, left incomplete by his death, calls an
omega-function, and the three functions $a>, -fra, %e0 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 a; + 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 [ lt 2 , @ 3 ]^- 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 Abelian functions the date 1826 1832. Abel gave what is called
his theorem in various forms, but in its most general form in the Mdmoire sur une
propriete gdnerale dune clause tres-6tendue de Fonction-s 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 x by any
algebraical equation F(x, y) = Q 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
F(x, y) = 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<j> + 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 generates 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 X(M, v), XJ(M, v) each
of two variables, and gives in effect an addition-theorem for the expression of the
functions X (u + u', v + v'), \ (u + u, v + v') algebraically in terms of the functions X (u, v),
\(u, v), X(u', v'), \(n', tO.
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 \(u + v) can be expressed algebraically in terms of \(u) and \(v). This is of
course perfectly correct, sn (w + v) is expressible algebraically in terms of sn u, sn v, but
the expression involves the radicals Vl sn a w, Vl-& 2 sn a M, Vl sn 2 y, Vl ^sn 5 !); 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 p 1, and these may be considered as the ratios of 4P 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 = Z, which is hyperelliptic, the two extremely
valuable memoirs, Gopel, "Theoria transcendentium Abelianarum primi ordinis adum-
bratio laeva," Crelle, t. xxxv. (1847), and Rosenhain, "Memoire sur les fonctions de
deux variables et a quatre pe'riodes qui sont les inverses des inte'grales ultra-elliptiques
de la premiere classe" (1846), Paris, Mdm. Savans fitrang. 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 Grosse: 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 + 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 p-tuple theta-function, as the
sum of a jp-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^ 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 TraM des Equations Numdriques
(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
#" 1 = 0: throwing out the factor x 1, the equation becomes an equation of the
order n I, 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 Traite" 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
Arithmetics (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, yf, 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 aP 1 = :
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
a* 1 s* 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
at 1 = 0, we may take the equation a? 1 = : 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 %, r)i, ...,r) e are the e periods, each of them
a sum of f roots, of the equation x n 1 = 0, then the complex numbers considered
are the numbers of the form a^ + a 2 ?; 2 + ... +a e ri e (a^, a 3 ,...,a e positive or negative
ordinary integers, including zero) : f 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.jP, we cannot have p 3 = a, 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 p 3 = 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 p" = 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 Rummer'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 a 2 + 3 = 0, we have as a whole
number the apparently fractional form \ (1 + i V3) 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 = (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 5 2 numbers obtained by giving to each of the numbers a, b, the values 0, 1, 2, 3, 4,
successively. And so in general, the consideration of an irreducible congruence F (x) =
(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 4- ci- + . . . + /h' 1 *" 1 , where
a, b, &, ...&c., are ordinary integers each =0, 1, 2, ... , p 1.
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 \ greater than 3, of the equation
= z\ has 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 oc+iy, where
C. XI. 58
458
PRESIDENTIAL ADDRESS TO THE [784
t is the before-mentioned imaginary or V - 1 of ordinary analysis. The letters i, j, &c.,
are such that every binary combination i 1 , 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\ 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 ({" = 0, or i 2 = 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,
Warren, 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 18G2 ; 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 x, 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 " (Canib. 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 Abzdhlende Ge&metrie (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 ft, v, of the conies which satisfy four given conditions;
questions relating to cubics and quartics were afterwards considered by Halliard 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.
582
460
785.
CURVE.
[From the Encyclopedia 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 kiuematical 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
called the origin (the axes are usually at right angles to each other, and for the
.V
y
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 ; and similarly y as positive or negative according as it
is in the sense Oy or Oy' from ; or, what is the same thing,
In the quadrant xy, or N.E., we have
x'y N.W.
xy' S.E.
T'II' S W
x y >s. VY .
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 CURVE. [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
a? y"
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 tlie
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 = 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, ax+by + c=0, or say (*$#, y, 1)=0; or by an equation of the second order;
ax* + 2hxy + by- + 2fy + 2gx + c = 0, say (*$#, y, 1) 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=x 3 , y = x (x- 1) (x-2), y = (x - 1) ./.t - 2, &o., he should proceed to
trace such curves as y = (x-a) (x-b) (x-c), y = (x - a) *Jx - l>, &c., 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 x* + y s - 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 + 1) (m + 2) 1, that is, \m (m + 3) ; and a curve of the order m may be
made to satisfy this number of conditions ; for example, to pass through 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'.c+ b'y + c'= 0, viz. the original equation is satisfied if either of these is satisfied.
Each of these la.st 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
a~ ;i 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 = m ; in particular, if the curve
of the order n is an arbitrary line (n. = 1), 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 mth 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 m = 2, if the hyperbola soy - 1 = be cut
by the line y = , the resultant equation in x is #c - 1 = 0, and there is apparently
only the intersection (x = 3 , y = fit) ; 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 y = /9; the two intersections are the point (=o< y = @)> and tne P oint at infinity
on the line y y9.
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 Geom&rie 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 Qfom&trie, was Sir Isaac Newton's Enunwratio
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] CUKVE. 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 poiut 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 Hue 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 in., 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 propriette
projectives (1822), where we find (p. 49) the statement, "deux circles place's arbitraire-
inent sur un plan...ont idealement 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 7, /; 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 ge"ne"rale 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 e'le'mens de la science de l'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 geometric plane, a chaque
the'oreme il en re"pond ndcessairement un autre qui s'en de"duit en e"changeant simple-
ment 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 e'changer entre eux pour passer d'un
theoreme a son corre'latif " ; 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 the'oremes,
&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 with 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
the order m, or say the class of the curve, is in general and at most =m(m 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'te's 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 in by an equation (*]#, y, z) m =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 (*$#, y, l) m = 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 ge'ome'trie sur deux principes
geneVaux de la science, la dualite et I'homographie," which forms a sequel to the
" Apercu 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).
Pliicker 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 ;
and corresponding dually thereto, we have the point as a curve of the order and the
class 1.
Pliicker 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 (, t], ), 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 gx + i)y+z = Q to be the equation of a
line, we say that (f, j], ) are the line-coordinates of this line. A linear relation
%+br} + 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 + brj +cf=0 expresses
that the equation %x + rjy + & = 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
592
468 CURVE. [785
the point (a, b, c). Thus (, ij, f) are the line-coordinates of any line whatever; but
when these, instead of being absolutely arbitrary, are subject to the restriction
ul- + bij + c = 0, this obliges the line to pass through a point (a, b, c); and the last-
mentioned equation of + by + cf= 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 (*$#, y, z) m =
is the relation which is satisfied by the point-coordinates (x, y, z) of each point of
the curve; and similarly the line-equation (*$, t), )" = is the relation which is
satisfied by the line-coordinates (, 17, ) 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.
(1. The stationary point, 2. The double point, or node ;
Point-singularities 4 . *
I cusp, or spinode ;
T . . , ., . (3. The stationary tangent, 4. The double tangent :
Line-singularities 1 a
( 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, 05 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 (*$#, y, z) m = 0, or general curve of the order m, has
double tangents and inflexions; (2) presents itself as a singularity, for the equations
4(*$#, y, z) m = 0, d y (*\x, y, z) m = Q, d z (*$a;, y, z) m = 0, implying (*$x, y, z) m = 0, are
not in general satisfied by any values (a, b, c) whatever of (#, 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,
class,
8 number of double points,
t cusps,
T double tangents,
K ,, inflexions ;
and this being so, Pliicker's " six equations " are
(1) n= wi(m-l)-28-3K,
(2) i = 3m (m - 2) - 68 - 8/e,
(3) r = m (m - 2) (m 2 - 9) - (m> - m - 6) (28 + 3*) + 28 (8 - 1) + 68/c + f K (K - 1),
(4) TO= n(n-l)-2r-3i,
(5) =3(n-2)-6T-8t,
(6) 8 =n(n-2)(n 2 -9)-(ft 2 -n
470 CURVE. [785
It is easy to derive the further forms
(7) t-K = 3(-w),
(8) 2(r-S) = (n-m)(n + m-9),
(9) m(m + 3)-S-2ic =n(n + 3)- T- 2t,
(10) (m - I ) (m - 2) - 8 - * = (n - 1) (n - 2) - T - t,
(11, 12) m ! -28-3/f = i 2 -2r-3f, = m + n,
the whole system being equivalent to three equations only: and it may be added that,
using o to denote the equal quantities 3m + 1 and 3n + K, everything may be expressed
in terms of m, n, a. We have
* = a 3w,
i =a 3m,
28 = m 2 m + Sn 3a,
2 T = n 2 - H + 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, T, i, B, 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 -J-nt (m + 3) S 2 is that of the number of the disposable con-
stants in a curve of the order m with & 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 l) S 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 m 2 28 3 = n" 2r 3t, = m + n, present themselves in the theory
of envelopes, as will appear further on.
With regard to the demonstration of Plticker'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 m, having the given numbers 8 and K of nodes and cusps. We can only
use the general equation (*$#, y, z) m =G, say for shortness =0, of a curve of the
7/i.th 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, i, 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 = 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 l, 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, and
the value of n is n = m(m 1) 28 3.
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 it in regard to the variables (a;, y, z) ; H = 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
= (]8 + 8/c, and the number of inflexions is t = 3m (in 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 II = 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)(m 2 9) ; but if the given curve has a node
then there is a diminution = 4 (m a m 6), and if it has a cusp then there is a
diminution = 6 (TO" m 6), where, however, it is to be noticed that the factor
(/' TO 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 (m 2 m 6) (28 + 3) is
too great, and it has in fact to be diminished by 2 {28(8 1) + 68 + | (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)(m 2 9) &c.;
or dividing the whole by 2, we have the expression for T given by the third of
PHicker's equations.
It is obvious that we cannot by consideration of the equation u = in point-
coordinates obtain the remaining three of Pliicker's equations ; they might be obtained
472
CURVE.
[785
in a precisely analogous manner by means of the equation v = 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, T, i respectively.
To complete Plucker'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 Cayley, 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, r double tangents, and t inflexions. So that, in fact, Plucker's equations
properly understood apply to a curve with any singularities whatever.
By means of Plucker's equations we may form a table
m
n
8
K
T
t
1
_
1
2
2
3
6
9
4
1
3
3
1
1
12
28
24
10
1
16
18
9
1
10
16
8
2
8
12
7
1
1
4
10
6
2
1
8
6
3
4
6
5
2
1
2
4
4
1
2
1
2
3
3
1
785] CURVE. 473
The table is arranged according to the value of m; and we have t=0, n = l, the
point; m = 1, n = 0, the line; ??i = 2, n = 2, the conic; of m=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 = 6, =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 Pliicker, 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 Curven dritter Ordnung" (Crelle, t. xxvin., 1844); in the latter of
these the points of inflexion are obtained as the intersections of the curve u =
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
"^ + y 3 + & + 6te/2 = ; 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)(m a 9) for the number of double tangents of a curve
of the order m was obtained by Plucker only as a consequence of his first, second,
fourth, and fifth equations. An investigation by means of the curve II = 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'dlimination et la the*orie 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. XI. 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. CXLVIII.,
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 11=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 Curven, 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
= represented by means of this form ; and' the theory may be very briefly referred
to. A curve u = 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' = 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 .4=0, B = 0, &c., then we must
have between the new coefficients the like relations, A' = 0, B' = 0, &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 = Q, &c., is said to be invariantive, but it does not follow that
A, B, &c., are of necessity invariants of M. Similarly, if we have a curve U=Q derived
from the curve w = in a manner independent of the particular axes of coordinates,
then from the transformed equation u' = deriving in like manner the curve V = 0,
the two equations 17=0, E7' = 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, V' = 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 = ; 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 Pliickerian 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 TMorie des fonctions analytiques
(1798) ; it is there remarked that the equation obtained by the elimination of the
parameter a from an equation f(x, y, a) = and the derived equation in respect to a
is a curve, the envelope of the series of curves represented by the equation f (x, y, a) =
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, t, being connected by the
Pliickerian 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 S 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 = m 2 28 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 = n 2 2r 3t (= m? 28 3/c, inasmuch
as each of these numbers is as was seen = m + n). At any one of the m 2 28 3
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 n- 2r 3t 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 3t common tangents are the tangents to the variable
curve at the m- 28 3/e points respectively, and the envelope is at the same time
generated by the m? 28 3 points, and enveloped by the n- 2i 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
7 3 28 3* points and n 2 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
602
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 m 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. 477
after the curve of that name, a limacon. 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 1 cut an arc oaf, so
that the two segments ab, ba' lie on opposite sides of the line, then projecting the
figure so that the line fi 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 we thus see that the two hyperbolic legs belong to a simple inter-
section of the curve by the line infinity. Next, if the line 1 touch at b the arc aa'
so that the two portions ab', ba lie on the same side of the line 1, then projecting
the figure as before, the tangent at b, that is, the line 1 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 H is a tangent at an inflexion,
passes through a crunode, or touches one of the branches of a crunode, &c. ; thus, if
the line 1 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
y = a?, 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 = o?. 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 10", 13, 22,
and 22*. 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 cnmodal 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. I.). 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 au
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 appenrs
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 diffe'rentes
formes des courbes planes du quatrieme ordre" (Math. Ann. t. VII., 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] CURVE. 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 Pliicker, in the memoir " Ueber solche Puncte die bei Curven einer
hb'heren Ordnung den Brennpuncten der Kegelschnitte entsprechen," (Grelle, 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 (X, 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 X 2 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 X 8 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 eveiy 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 = 34-3 + 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. xi. 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 at, 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', Y', 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 M' = 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 = and the point (of, y', /) on
the curve u' = 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 = there corresponds a single
point (a;', y', z'} determined by the equations x' : y' : z = X : Y : Z ; from these equations
and the equation u = eliminating x, y, z we obtain the equation u' = of the trans-
formed curve. To a given point (x ', y', z') not on the curve u' = 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, Y = 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 =0. 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' = there corresponds a single
point of the curve u = ; 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 S + >c of nodes and cusps is at most = (m 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 !)(TO 2) - 8 , is the " Geschlecht," or "deficiency," of the curve, say this is
= D. When .0 = 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 : 6 : <j> 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 6, <j>, that is, of 6 and
a certain irrational function of d.
In particular, if D = 0, that is, if the given curve be unicursal, the transformed
curve is a line, $ 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 6 must be of the same order m.
If D = 1, then the transformed curve is a cubic ; it can be shown that in a cubic,
the axes of coordinates being properly chosen, <j> 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 0, and of the square root of a quartic function of 9.
And so if D = 2, then the transformed curve is a nodal quartic ; </> can be ex-
pressed as the square root of a sextic function of 6, 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 0, and of the square root of a sextic function
of 6. But when D = 3, we have no longer the like law, viz. <f> 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 0* . 1 &&, and writing
612
484 CURVE. [785
6 = sn u, the radical becomes = en u . da u ; and we have expressions of the form in
question.
It will be observed that the equations a/ : 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 (a/, 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,
Y = 0, Z = had no common intersections, correspond n 1 points (x, y, z). If, however,
the functions are such that the curves X = Q, F=0, Z=0 have k common inter-
sections, then among the n 2 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 = n 2 k ; and in
particular if &=n. 2 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' = have n"' 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, zx = 0, xy = 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 2 1 of common intersections can only occur by reason of common
multiple points on the three curves; and assuming that the curves X = 0, 7 = 0, Z=0
have a, + Oj + a, + . . . + _! common intersections, where the a, points are ordinary points,
the a, points are double points, the et 3 points are triple points, &c., on each curve, we
have the condition
a, + 4a 2 + 9a 3 + ... + (n- I) 3 a,,-, = w 2 - 1 ;
but to this must be joined the condition
a, + So, + 6a s + . . . + i ( - 1) (n - 2) a^ = J n (n + 3) - 2,
(without which the transformation would be illusory); and the conclusion is that
i> <*a, , o-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, Y=0, Z=0 are unicursal. The transformation
may be applied to any curve M = 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 the'orie de ces courbes."
The theorem is as follows : if in a unicursal curve two points have an (a, ft)
correspondence, then the number of united points (or points each corresponding to
itself) is = a + /9. 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, ft) correspondence between the two
points is given by an equation of the form (#$#, l)"(<f>, 1/=0 between their para-
meters 6 and <f> ; at a united point <f> 6, and the value of is given by an equation
of the order a + ft. 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 ty each q times, &c.. in such manner that the points (P, P'),
the points (P, Q 1 ), &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 (ft, ft') correspondence with a number =b
of united points, and so on, the theorem is that we have
p(a-a-a.') + q(b-ft-ft')+... = 2kD.
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
(wi 1, m1) correspondence, and the value of k is =1, hence the number of united
points is = 2m 2+2.D; the united points are the points of contact of the tangents from
and (as special solutions) the cusps, and we have thus the relation n+/e=2m 2 + 2Z) ;
or, writing D = % (m l)(m 2) & K, this is n = m(m 1) 28 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 t y) and (x, y') ; for example in the imaginary circle w 2 + w' 2 = (a + bif, the corre-
spondence is given by the two equations of - y' 2 + #' 2 y' 2 = a 2 b 2 , ay + afy 1 = 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 (Gomptes Rendus, t. LVIII., 1864, et seq.) refer to the
conies which satisfy given conditions. There is an earlier paper by De Jonquieres,
" Theoremes gdneYaux 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 fyi (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 (/i, v) of such a system of conies, viz. /A 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 (/*, v) contains 2i/ /* line-pairs (that is,
conies, each of them a pair of lines), and 2p 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. CLVIII., 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 (4p) 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 (Qp), (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 caracttristiques des systemes elementaires des courbes planes du
troisieme ordre (Paris, 1871). Thus, considering the several cases of a cubic
No. of coasts.
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 (4p), (3p, II), Sic. 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 (p, 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 /t, 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 pf 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, P 2| ... be rational and integral functions of the coordinates (x, y, z) of the orders
? m,,... respectively, we have the curve Pf'P,**... = 0, of the order m, =a 1 in 1 +a. 2 m.,+ ...,
composed of the curve P l taken o t times, the curve P 2 = taken cu, times, &c.
Instead of the equation P, a 'P 2 a2 ... =0, we may start with an equation u = 0, where
M is a function of the order m containing a parameter 6, and for a particular value
say = 0, of the parameter reducing itself to P^'P/*.... Supposing indefinitely
small, we have what may be called the penultimate curve, and when = 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 P l = 0, P a = 0, respectively ; a summit S is a point
such that, drawing from an arbitrary point 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, 0-, . . . ; they are thus in some measure
arbitrary points as regards the ultimate curve PfiP t a . . . = 0.
It may be added that we have summits only on the component curves P, = 0, of
a multiplicity !> 1 ; the number of summits on such a curve is in general =(ai i a I )'m 1 -.
Thus assuming that the penultimate curve is without nodes or cusps, the number of
the tangents to it is =m? m, = (a l m^ + o^m^->r ..,) 2 (a 1 m 1 + a 2 m 2 +...), taking P 1 = to
have $! nodes and KI cusps, and therefore its class n^ to be =m 1 2 mi 2^ 3/Cj, &c.,
the expression for the number of tangents to the penultimate curve is
= (i s - i) m* + (a 2 2 - a,) m 2 2 + . . . + 2a 1 a 2 TO 1 m 2 + . . . + ^ (n t + 2S, + 3*0 + a, (?i 2 + 2S 2 + 3 2 ) + . . .
where a term 2a 1 o 2 m 1 m 2 indicates tangents which are in the limit the lines drawn to the
intersections of the curves P 1 = 0, P 2 = each line Za.,^ times; a term a I (ni + 2$ 1 + 3 1 )
tangents which are in the limit the proper tangents to P l = each cti times, the lines
to its nodes each 2! times, and the lines to its cusps each '3^ times ; the remaining
terms (a^ a^ m^ + (af a,,) m 2 2 + . . . indicate tangents which are in the limit the lines
drawn to the several summits, that is, we have (a^ e^) mf summits on the curve
P, = 0, &c.
There is of course a precisely similar theory as regards line-coordinates; taking
II! , IT 2 , &c., to be rational and integral functions of the coordinates (f, 17, f ), we con-
nect with the ultimate curve II^'Il.^... = 0, and consider as belonging to it certain
lines, which for the moment may be called "axes," tangents to the component curves
785] 'CURVE. 489
IT, = 0, II 2 = 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' . . . P^ . . . = 0, where L t = is a right line, Pj = 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!*' ... H!"' ... =0, where A 1 = 0,... are the equations of the summits in question, II, = 0,
&c., are the line-equations corresponding to the several point-equations PI = 0, &c. ; and
this curve will contain as part of itself axes not exhibited by this equation, but which
are the lines LI = 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 a 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
ra 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, V=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 = 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. VIII. (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 z= d ,
a' x + V y + c' z = df ,
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 = and the whole coefficient of z becomes
= ; 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 x, and on the right-hand side a constant term ,
the coefficient of x has the value
a (b'c" - b"c') + a' (We - 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 3 2 , it is called a
determinant of the third order. It is to be noticed that the resulting equation is
a , b , c x= d , b , c
a', b', c'
a", 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
M -a,
a, b
= a
b'
-a'
b ,
a', V
a,b,
c
= a
& t
c'
+ a'
b", c" +a"
b,c ,
a', b',
c'
b",
c"
b, c
b',e'
a", b",
c"
a , b
, c
, d
mm
b'
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', c', d'
a", b"
, c", d"
b'", 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.
622
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 2 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 =J.1.2...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
= alb'c" - ab"c + a'b"c - a'bc" + a"bc - 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', 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 n- 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 = ; 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 + b"y + c"z-d", b", c"
d , b , c
d', b', c'
d", b", c"
it appears that this is
-00,6,0 +y
b , b , c
+ 2
c , b , c
a', b', c'
b', b', c
c', b', c'
a", b", c"
b", b", c"
c", b", c"
viz. the second and the third terms each vanishing, it is
= x
a , b , c
- d , b , c
a', b', c'
d', b', c'
\
a", 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 = ; that is, the linear equations
give
x a , b , c
a', b', c'
a", b", c"
which is the result obtained above.
d , b , c
d', V, c
d", b", c"
= 0;
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 + @y + yz = $ ;
a like process shows that, the equations being satisfied, we have
a , b , c , d
a, b', c', d'
a", b", c", d"
= 0;
or, as this may be written,
a , 6 , c , d
a', 6', c', d'
" I// // J//
a , o , c , a
a , , c
a', b', c'
a", b", c"
= 0;
which, considering 8 as standing herein for its value cue + y3y + <yz, is a consequence of
the original equations only. We have thus an expression for ox + {3y + yz, an arbitrary
linear function of the unknown quantities x, y, z; and by comparing the coefficients
of a, y9, 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
but these are
respectively.
1
J
1
)
1
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"
b , c , d
I
c , d , a
>
d , a , b
I
b', c', d'
c' , d' , a
d', a', b'
b", c", d"
c", d", a"
d", a", b"
le same thing,
=
b , c , d
c , a , d
a , b , d
b', c', d'
c', a', d'
a', b', d'
b", c", d"
c", a", d"
a", b", d"
786]
EQUATION.
495
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', a"), (ft, ft, ft"), (y, y', 7")
(a , b , c)
=
a , b , c
.
, # , 7
(of, b', c')
a', b', c'
OC jo ^
(a", b", c")
J> ft j
a", b", c"
a", ft", 7"
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 + ba.' + ca", (a, b, c)(ft, ft', ft"), that is, aft + bft' + cft",
(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 (3 3 =) 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/37-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 aic-determinant
last written down ; the sum of all other factors + afty" is the a$7-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, b
that they enter only through the determinants of the second order t
a t o
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'" ,
mi -LIIII nn jnn nn
a , o , c , a , 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 + ab' . 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
8. Any determinant
af, b'
formed out of the elements of the original determ-
iimnt, 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 1, then such determinant is called a. first minor; the number of
the first minors is = n 1 , 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
a, h, g
h, b, f
a, v, - fj,
-v, b, X
(J,, - \, C
0,
- v,
P;
v,
0,
-X,
X
are respectively symmetrical, skew, and skew symmetrical.
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 x" p 1 x n ~ 1 + ... p n = 0, where p lt p z ,...,p n are regarded as known,
and x as & quantity to be determined ; for shortness, the equation is written f(x) = 0.
The equation may be numerical ; that is, the coefficients p lt p 2 , . . . , p n are then
numbers, understanding by number a quantity of the form a + fti (a and /3 having
any positive or negative real values whatever, or say each of these is regarded a&
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, p it p^,-..,p n
are real positive or negative quantities, and a root a, if it exists, is a positive or
negative quantity such that a" p 1 a n ~ I + ... p n = 0, or say, f(a) = 0. The fundamental
theorems are given in the article Algebra, sections x., xni., xiv. ; but there are various
points in the theory which require further development.
It is very useful to consider the curve y =/(#), or, 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? 6a? + \\x 6 '06 =0,* then the curve y=a? 6o? + lI<K 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 x n is +1, then when x=cc, y = +cc; but when x = oo , then
y - + ac or - x , according as re 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 y9, a are any two values of x (at > j3, that is, a nearer +00), then if / (/3),
f(o) have opposite signs, the curve cuts the axis an odd number of times, and
therefore at least once, between the points x = ft, x = a; but if /(/8), /() 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, /(/3), /(a) having opposite signs, there are
between the limits /3, 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. xi. 63
498 EQUATION. [786
root; but /03), /() 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 ft, 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 >a (that is, * nearer to +00) f(x) is always +,
and y9 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 B 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 < S ; 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 Traitt de la resolution des Equations Nume'riques*; 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), f,(x),...,f n (x)
of the degrees n, n\, n 2, ...,0 respectively, the last term f n (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 > /3 (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 /3 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 176768 and 177071.
786] EQUATION. 499
between the two limits. In particular, taking ft, a = - oo , +00 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 8 ; 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 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 & 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; f([3) and /(a) have therefore opposite
signs. Suppose any intermediate value is 6 ; in order to determine by Sturm's theorem
whether the root lies between /3, 0, or between 0, a, it would be quite unnecessary
to calculate the signs of f(Q), f (0), / 2 (#), ', only the sign of /(#) is required: for,
if this has the same sign as /(/9), then the root is between /9, 6; if the same sign
as f(a), then the root is between 0, a. We want to make 6 increase from the inferior-
limit y3, at which f(0) has the sign of f(f3), so long as f(6) 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 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 Homer'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
632
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 + j- ...,
~T C ~T*
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 x = 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 + 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 + A the correct value,
then f(y + h) = 0, that is,
' "
and then, if h be so small that the terms after the second may be neglected,
y (7) + hf' (7) = 0> that is, h = 3jnr-4 , or the new approximate value is x = j--;
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 - - 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 (7 is nearer to X than
was 0; and, starting with G' 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 C; and in this
.2 ltDD/a/\!C' C ,
N
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 G, such that OG 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
ON, 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, GP,
and joining the points P, Q by a line which meets the axis in Z)', 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', G', 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 = @, 00 = a, we have
approximately
whence, if the root is /3+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 /(a) +f (a) which presents itself in Newton's method. The
difference is that with Homer the integer part of this quotient, is taken as the
presumptive value of A, 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
CL
accurately, but as nearly as we please, any positive magnitude whatever; so an equation
ax + b = gives x = , which (approximately as before) represents any negative
U
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 # 2 + 1 = 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 2 + 1 = ;
and then taking a and b any real numbers, we form an expression such as a + In,
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 x?+px + 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? + pa? + qx + r = 0, with numerical coefficients, not satisfied by any numerical
value of x, 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 + cf (a, b, c numbers a + (3i of the kind heretofore considered), first we should
be thrown back on the quadric equation a? + px + q = 0, 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 + /3i)
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 x n p l x n ~ l + ...+ p n
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
o" -pitt"- 1 + . . . + p n = 0, or say /(a) = ;
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)/, (x), where fi(x) is a function a;"" 1 q 1 x n ~*+ ... + <?n-i of the order
n 1, with numerical coefficients q lt q 2 ,-..,q n -i-
In general, a is not a root of the equation _/i (x) = 0, but it may be so i.e., /i (x)
may contain the factor x a ; when this is so, f(x) will contain the factor (x a.) 2 ;
writing then f(x) = (x aff t (x), and assuming that a is not a root of the equation
f t (x) = 0, x = a is then said to be a double root of the equation f(x) = ; 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)* F (x), a. being a positive integer which
may be =1, (x a) a the highest power of a; a which divides f(x), and F{x) being
of course of the order n a, then the equation F (x) = 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) a (x &y <I> (x),
$ a positive integer which may be =1, (x bf the highest power of x b in F (x)
or f(x), ' and <& (x) being of course of the order n a. 0. The original equation
f(x) = is in this case said to have a roots each = a, /3 roots each = b ; and so on
for any other factors (x c)t, &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) = has all its
roots unequal.
If the coefficients p,,p t> ... 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) + t'<J> (x) = 0, where F(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) = and <I> (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 + /3t being a root, then a fii 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, ft 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
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, &e., 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 + pi: see Lagrauge's Equations
Numrique.
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 x n a = (a = at + @i, 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 wth
root, of real numbers, and by the aid of a table of natural sines and cosines*. For
writing
a + &i= Va 2 + /S
there is always a real angle X (positive and less than 2ir), such that its cosine and
_ o
sine are = -^-- - and : respectively ; that is, writing for shortness Va 2 + /8 2 = p,
v a 2 + p 1 V a 2 + p a
we have a + fti = p (cos \ + i sin X), or the equation is x n = p (cos X + i sin X) ; hence
(X X\ - / X X\
cos-+isin- = cos X + i sin X, a value of x is = \/ p ( cos - + isin - ) .
n n> \ n n/
The formula really gives all the roots, for instead of X we may write X + 2s7r, s a
positive or negative integer, and then we have
x = \/ p (i
+ 2S7T . .
cos - + i sin
n
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 n = a, then assuming x = 8y, the equation becomes y n 1 = 0, which
equation, like the original equation, has precisely n roots (one of them being of course
= 1). And the original equation x n a = is thus reduced to the more simple
equation x n - 1 = ; and although the theory of this equation is included in the pre-
ceding one, yet it is proper to state it separately.
The equation a;" 1=0 has its several roots expressed in the form 1, w, o> 2 , ..., a> n ~\
where to may be taken = cos + isin ; in fact, <o having this value, any integer
TV 71
power o* is = cos ' "" + i sin -^ , and we thence have (&>*)" = cos 2-irk + i sin 2-rrk, = 1,
n n
that is, <* is a root of the equation. The theory will be resumed further on.
* The square root of o + j3i 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 a" regarded
as an n-valued function ; any one of these being denoted by \/a, then the series of
_i
values is x/a, w\Xa, ..., u> n ~ t \^a; or we may, if we please, use \/a instead of a* as
a symbol to denote the n-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
j), = 2a, p, = ^.ab,..., p n = abc...;
i.e., regarding the coefficients p lt p^ ..... p n as given, then we assume the existence of
roots a, b, c,... such that p l = 2a, &c. ; or, regarding the roots as given, then we write
PI, pi, &c., to denote the functions So., 'S.ab, &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 a? px + q = Q
has a root a; = $ (p + Vp 2 4>q), we mean that writing this value for x the equation
becomes an identity, { (p + ^p- - 4g)) 2 -p {% (p + Vp" - 4^)} + q = ; 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 b ; 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
a" p^o;"' 1 + ... =x a . x b . &c.,
or the equivalent equations p^ = 2a, &c., we find
a n -p l a n ~ l + ...=0,
b n -p t b n - 1 + ...=0;
(it is as satisfying these equations that a, b, ... are said to be the roots of x n p 1 # n ~ 1 +...=0) ;
and conversely from the last-mentioned equations, assuming that a, b,... are all different,
we deduce
p 1 = '2a, p 2 = 2a6, &c.,
and
#* p l af l ~ l + ... x a.x b . &c.
Observe that if, for instance, a = b, then the equations a p^a n ~ l + . . . = 0, 6" pib n ~ 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 a) a was a factor of <8" pt*~ l +"&o ; but by
considering b as the limit of a + h, h indefinitely small, we obtain a second equation
no"- 1 - (n - 1) ptd"-* + ... =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 />, = Sa, &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 2a 2 , 2a 3 , &c., are rational and integral
functions of the coefficients.
642
508 EQUATION. [786
The process originally employed for the expression of other functions 2a a 6^, &c.,
in terms of the coefficients is to make them depend upon the sums of powers : for
instance, 2a6" = 2o"2a s - 2a*+* ; but this is very objectionable ; the true theory consists
in showing that we have systems of equations
Pi = 2a,
Jp, = 2a6,
2a6c,
+ 3!
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 Sa&c, Sa 5 6, 2a 3 . Hence in
each system the root-functions can be determined linearly in terms of the powers and
products of the coefficients :
p 3 ,
ptfs - 3p 3 ,
P! S - 3p,p, + 3p,,
and so on. The older process, if applied consistently, would derive the originally
assumed value 2a&, =p t , from the two equations 2a=p 1( 2a 2 = pf 2p 2 ; i.e. we have
22a& = 2a . 2a - 2a 2 , = pf - (pf - 2p a ), = 2p s .
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 x n a function of the form in question ; multiplying each side
by x, and on the right-hand side writing for x n its foregoing value, we have x n+1 , 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
^7 - = I(x), or say <j> (x) i|r (x) I (x) = 0, where <f>, ty, 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
<j>(x)-ty(x)I(x) = M (x)f(x). And it is, moreover, clear that the equation M^ = I (x)
fO)
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 ~ a <0 (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 by, (a - c) 2 , (a d) z , (b - c) 2 , (b - d)*, (c - d) 1 , 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 <(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 I ; and, retaining a; in place of any
one of the roots, the problem is to find y from the equations x n p^x n ~ l + ... = 0, and
y = Af x n ~ 1 + NtX"- 2 + ..., 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 4- 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 / T r 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 unsymmetrical 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 + &)* + (c 4- df.
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 a6 + 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 + &)* + (c + d? to
be = 370, and not any other value ; that is, (a + bf + (c + df, 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,
d a 6 = 2 determines a to be = 1 and b to be =2; but if the roots had been 1, 2, 4, 16
then a-b = W 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,
ti = ab+cd, y l = (a + b) 3 + (
t 3 = ad + be, y 3 = (a + dy + (b + c )" :
then
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 t lt t 2 , t 3 are all known; then the equations being
linear in y l , y 3 , y 3 these can be expressed rationally in terms of the coefficients and
of ti, tt, t,', that is, y lt y a , y 3 will be known. But observe further that y l is obtained
as a function of t lt < 2 , t 3 symmetrical as regards 2 , t 3 : it can therefore be expressed
into b, b into c, c into d, d into a. Substitutions may, of coarse, be represented by single letters a, ,i, . . ;
/-=!' , =1, is the substitution which leaves the letters unaltered. Two or more substitutions may be com-
(abcd
pounded together and give rise to a substitution; i.e., performing upon the primitive arrangement first the
substitution ft and then upon the result the substitution a, we have the substitution a/3. Substitutions are
not commutative; thus, <t|3 is not in general =/3a ; but they are associative, ap .y=a . (iy, so that afiy has a
determinate meaning. A substitution may be compounded any number of times with itself, and we thus
have the powers a 2 , a', . . , &c. Since the number of substitutions is limited, some power a 1 " 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 = J.1.2.3..n. 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 ^ and of t^+t a , t 2 t 3 , and thence as a rational function of tf,
and of ti + tz + ta, ^2 + Ms + Lt 3 , ^Lt,; 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 L or ( 3 ) being known, y^ 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 = 0, we can by the assistance of the
sign \/( ) 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
x = I \p ^p--4q],
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
z- = J (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
i_
of radicals, that is, by aid of the sign y/( ) or ( ) m , 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 x 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 Vp 2 lq) t
substituting for p, q their given numerical values, we obtain for x an expression of
the form x = a + /3i V-y + Si, where o, 0, j, 8 are real numbers. This expression sub-
stituted for x in the quadric equation would satisfy it identically, and it is thus an
algebraical solution ; but there is no obvious a priori reason why vy + 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 VY + Si has such a value calculable by means of the radical
expressions \/{"S'f + S- 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 806 = 5 an d o? + b 3 = r, equations which give (a 3 -6 s ) 2 =r" ^q 3 , a
512 EQUATION. [78*5
quadric equation solvable by radicals, and giving a 3 - b 3 = Vr 2 - fa', a, two-valued
function of the coefficients: combining this with a' 4- b* = r, we have a" = (r + Vr* -
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 ? = ^, we have, as may
be shown, a + b a three-valued function of the coefficients ; and a; = a 4- 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 Sab = q, in its original form, but use only
the derived equation 27a 3 > 3 = <f, implied in, but not implying, the original form.
An interesting variation of the solution is to write x = ab (a + 6), giving
3r
a s b' (a* + b s ) = r and 3a?b 3 = q, or say a 3 + 6 3 = , a 3 b* = %q; and consequently
o = ( r + Vrz-fa 3 ), b 3 = (r - Vr 2 -
i.e., here a 3 , b 3 are each of them a two-valued function, but as the only effect of
altering the sign of the quadric radical is to interchange a 3 , b 3 , they may be regarded
as each of them one-valued; a and b are each of them 3-valued (for observe that
here only a 3 ?* 3 , not ab, is given) ; and ab (a -t- 6) 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 + Si, where y and 8 are real
numbers (it will do so if r 2 -^q 3 is a negative number), and then we cannot by
the extraction of any root or roots of real positive numbers reduce v/y 4- Si 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 3 = a, solvable by the extraction of a cube root, but) to a
cubic of the form 4a? 3# = a, corresponding to the equation 4 cos 3 6 3 cos 6 = cos 30
which serves to determine cos when cos 30 is known. The theory is applicable to
an algebraical cubic equation ; say that such an equation, if it can be reduced to
the form 4a? 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
ab + 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 fc-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 + \/p 2 4gO, sub-
stituting for p and q their values, and observing that (a + by 4a6 = (a by, this
becomes x = ^ [a + b + */(a 6)"j, the value being or b according as the radical is
taken to be +(a b) or (a 6).
So in the cubic equation of pa? + qx r 0, if the roots are a, b, c, and if o>
is used to denote an imaginary cube root of unity, to 8 + o> + 1 = 0, then writing for
shortness p =a + b + c, L = a + <ob + <o-c, M=a + aPb + <ac, it is at once seen that LM,
C. XI. 65
514 EQUATION. [786
L* + M*, and therefore also (' M*f are symmetrical functions of the roots, and con-
sequently rational functions of the coefficients : hence
is a rational function of the coefficients, which when these are replaced by their
values as functions of the roots becomes, according to the sign given to the quadric
radical, = L 3 or M 3 : taking it = L 3 , the cube root of the expression has the three
values L, taL, ofL ; and LM divided by the same cube root has therefore the values
M, io"M, eoM; whence finally the expression
- M 3 ?)}
+ LM+ j/$ (L' + M* + V(D - M 3 ?)}]
has the three values
that is, these are = a, b, c respectively. If the value M 3 had been taken instead of
L 3 , then the expression would have had the same three values a, b, c. Comparing
the solution given for the cubic a? + qx r = 0, it will readily be seen that the
two solutions are identical, and that the function r 2 -fc(f under the radical sign
must (by aid of the relation p = which subsists in this case) reduce itself to
{L 3 M 3 )* ; it is only by each radical being equal to a rational function of the
roots that the final expression can become equal to the roots a, b, c respectively.
28. The formulae for the cubic were obtained by Lagrange (1770 71) from a
different point of view. Upon examining and comparing the principal known methods
for the solution of algebraical equations, he found that they all ultimately depended
upon finding a " resolvent " equation of which the root is a + tab + ta"c + <a 3 d + ... , co
being an imaginary root of unity, of the same order as the equation ; e.g., for the
cubic the root is a + a>b + uPc, co an imaginary cube root of unity. Evidently the
method gives for L 3 a quadric equation, which is the " resolvent " equation in this
particular case.
For a quartic the formulae present themselves in a somewhat different form, by
reason that 4 is not a prime number. Attempting to apply it to a quintic, we seek
for the equation of which the root is (a + a>b + &> 2 c + a> 3 d + &> 4 e), o> an imaginary fifth
root of unity, or rather the fifth power thereof (a + a>b + o> 2 c + ia 3 d + to'e) 1 ; this is a
24-valued function, but if we consider the four values corresponding to the roots of
unity a>, &> s , a) 3 , to 4 , viz. the values
(a + to b + a> 2 c + ea 3 d + o> 4 e) 6 ,
(a + tfb + &> 4 c + CD d + ft> 8 e) 5 ,
(a + a> 3 b + o> c + oa'd + a> 2 e) 5 ,
(a + (o*b + o> 3 c + tfd + os e) 6 ,
any symmetrical function of these, for instance their sum, is a six-valued function
of the roots, and may therefore be determined by means of a sextic equation, the
786] EQUATION. 515
coefficients whereof are rational functions of the coefficients of the original quintic
equation ; the conclusion being that the solution of an equation of the fifth 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, as already mentioned, does
not admit of solution by radicals ; but the equation of the sixth order, Lagrauge's
resolvent sextic, is very important, and is intimately connected with all the later
investigations in the theory.
29. 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 that they are not so expressed
in terms of other arbitrary quantities as to make a solution possible. If the
coefficients are not all arbitrary, for instance, if some of them are zero, a sextic
equation might be of the form af + bx* + ca? + d = Q, and so be solvable as a cubic;
or if the coefficients of the sextic are given functions of the six arbitrary quantities
a, b, c, d, e, f, such that the sextic is really of the form
(a? + ox + b) (x* + ex* + do? + ex +/) = 0,
then it breaks up into the equations a? + ox + b = 0, #* + ceo 3 + do? + ex +f= 0, and is
consequently solvable by radicals ; so also if the form is
(x -o)(x- b) (x -c)(x- d) (x - e) (x -/) = 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 x n l=0 present an instance.
30. It has already been shown how the several roots of the equation x n 1 =
) O o_
can be expressed in the form cos -"-+i sin . but 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 to, w 2 , . . . , u> n ~ l are all of them roots ;
x" 1 contains the factor x 1, and it thus appears that a>, to 2 ,..., w"" 1 are the n 1
roots of the equation
x n ~ 1 + a;"--+ ...+ + 1 = 0;
we have, of course,
2 + ... + a, + 1=0.
It is proper to distinguish the cases n prime and n composite ; and in the
latter case there is a distinction according as the prime factors of n are simple or
multiple. By way of illustration, suppose successively n = 15 and n = 9 ; in the former
case, if a be an imaginary root of a? 1 = (or root of a? + x + 1 = 0), and /3 an
imaginary root of of 1 = (or root of x t + x a + a? + x + l=Q), then w may be
taken = a/9; the successive powers thereof, a/3, a 2 /?, /3 3 , a/3 4 , a 2 , ft, a/3 2 , a 8 /3 3 , /3 4 , a,
652
516 EQUATION. [786
oV9, ft 1 , a/8 3 , a'/S 4 , are the roots of of 4 + x l *+ ... +x+l = 0; the solution thus depends
on the solution of the equations a? 1 = and of 1=0. In the latter case, if a
be an imaginary root of a? 1 = (or root of x? + x + 1 = 0), then the equation a? 1 =
gives a?=I, a, or a*; af=\ gives #=1, a, or a 3 ; and the solution thus depends on
the solution of the equations a? -1 = 0, a? a = Q, a? a? = 0. The first equation has
the roots 1, a, a s ; if /9 be a root of either of the others, say if /8 s = a, then
assuming <a = /3, the successive powers are /9, /3 s , a, a/3, a/9 2 , a 2 , a 2 /3, a 2 /?, which are
the roots of the equation of + x ? + ... + 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 03, we have r, r 2 , r 3 ,...,i Mr ~ l
as the (n 1) roots of the reduced equation
then not only r n - 1 = 0, but also 7-"- 1 + r"- 2 + . . . + r + 1 = 0.
31. The process of solution due to Gauss (1801) depends essentially on the
arrangement of the roots in a certain order, viz. not 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 g n ~ 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 2 ,..., g n ~*, each divided by n, leave (in a
different order) the remainders 1, 2, 3,..., n 1; hence giving to r in succession the
indices 1 , g, g*, ..., g""*, we have, in a different order, the whole series of roots
.,.:.:. r n-l
',' I ','
In the most simple case, ?i = 5, the equation to be solved is
here 2 is a prime root of 5, and the order of the roots is r, r 3 , r*, r 3 . The Gaussian
process consists in forming an equation for determining the periods P,, P 3 , = r + r*
and r 3 + r 3 respectively, these being such that the symmetrical functions P l + P 2 , PiP*
are rationally determinate : in fact,
P, + P, = -1, P 1 P s = (r + r t )(^ + r > ), = r> + r> + r + r 7 , =r 3 + r t + r + r*, = - 1.
PI, PI 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 PI, r 2 and r" for P 2 ; the symmetrical functions of these are then rationally
determined in terms of P a and P 2 ; thus r + r* = P 1 , r.r*=\, or r, r* are the roots
of v? P,M +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, r 3 , r 3 , r", r*, r 6 . We may form either 3 periods each of 2 terms,
P,,P>,P 3 , =r + i*, r* + r*, r' + r",
respectively; or else 2 periods each of 3 terms, P lt P i = r + r J ' + r t , r^r^r 5 respec-
tively ; in each case the symmetrical functions of the periods are rationally determinable ;
thus in the case of the two periods P, + Pj = 1, PjP a = 3 + r + r 2 + r 3 + r 4 + r 5 + r, =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 4 = P 2 , r.r s .r t =l.
The theory was further developed by Lagrange (1808), who, applying his general
process to the equation in question, x n ~* + a,'"'" 2 + . . . + x + 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 + eab + o>*c + ...)"~ 1 was in this case a given function of <a with integer
coefficients. Reverting to the before-mentioned particular equation x t + a? + a? + x+l = Q,
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 o>, a root of the equation w 4 1 = (whence co is =!,!, i, or i, at
pleasure), and consider the expression
(a + cab + a) 2 c + co 3 dy.
The developed value of this is
a 4 + b* + c 4 + d* + 6 (aV + bW) + 12 (a?bd + frca + c-db + d*ac)
+ a> {4 (a?b + frc + (fd + d 3 a) + 12 (tfcd + frda + c?ab + d*bc)}
+ co* {6 (a'fc* + b*c? + c?d? + dW) + 4 (o 3 c + b s d + c 3 a + d*b) + 24,abcd}
+ co 3 (4 (a*d + b 3 a+(^b + d*c) + 1 2 (a'bc + Vcd + (?da + d*ab)} ;
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, o, b, c, d denote the roots r, r", r*, r 3 of the special equation, then
the expression becomes
+ a> {4(1 + 1 +1 +l)
+ ta* (6 (r + r 2 + r* + r 3 ) + 4 (r s + r* + r 3 + r )}
-I- a> 3 {4(r + r a + r 4 + r 3 )+ 12(r 3 +r
viz. this is
a completely determined value. That is, we have
(r + car 1 + wV + tuV) 4 = - 1 + 4<a + 14w 2 - 16&> 3 ,
which result contains the solution of the equation. If &> = !, we have
which is right; if &> = 1, then (r + r 4 r 2 r 3 ) 4 = 25; if co = i, then we have
{r-r t + i(r>-r i )} t = -15 + 2Qi; and if a> = -i, then [r-r^-i (r 2 -r 3 )) 4 = - 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 a;"" 1 + a^"" +...+a; + l=0, to a root of w 71 " 1 1=0, and <? a prime root of
(j n ~ l = 1 (mod. n), then
(r + o>r* + . . . + a)"- 2
518 EQUATION. [786
is a given function M,, + M ,&> + ...+ Mn-, ta n ~ t with integer coefficients, and by the
extraction of (n l)th roots of this and similar expressions we ultimately obtain r
in terms of o>, which is taken to be known ; the equation #* 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 ouly ; 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 a? 1 " 1 + . . . + x + 1 = 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, 6x, &-x,..., nr ~ l x (6x being a rational function of x,
and n x = 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, dx, Q^x being
any two of the roots, we have 06^ = d$x, the equation will be solvable algebraically.
It is proper to refer also to Abel's definition of an irreducible equation : an equation
<f>x=Q, 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 fas 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 x t + x* + a? + as+l = 0; the left-hand side has here no rational
divisor, and the equation is irreducible; but this function is = (a? + %x + Vf - \ a?, and
it has thus the irrational divisors x* + J(l +Jl)x+ 1, a; 2 + (1 -J5)x+l ; and these,
if we adjoin the radical J5, 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
The fundamental theorem is the Proposition I. of the "Memoire sur les conditions
de resolubilite 1 des Equations par radicaux " ; viz. given an equation of which a, b, c, ...
are the ra 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 <f>x = Q, then <f>x 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 a^~ l + ... + #+1 = 0, n a prime number,
a, b, c,...,k = r, >, r , . . . , rv n ~ ,
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 a 2 6 + 6 2 c + c 2 a 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
x lt ,,...,, invariable by the substitutions #*+!, for x t , 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 supilrieure, 2nd ed. 1854; 4th ed. 187778.
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 of + 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 Auflosung der Gleichungen vom
filnften Grade," Math. Annalen, t. xm. (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 Substitutimis et
des Equations Alg&rriques, Paris, 1870. The work is divided into four books book I.,
786] EQUATION. 521
preliminary, relating to the theory of congruences ; book II. is in two chapters, the first
relating to substitutions in general, the second to substitutions defined analytically, and
chiefly to linear substitutions; book ill. 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 in., 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 Rummer'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. 818824.]
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 if
or sinw, 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 w, 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=<j>(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, ...) = <(?, w, ...); 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, x 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 = x + iy.
To a given value x + iy of u there corresponds in general for v a value or values
of the like form v = x' + iy', and we obtain a geometrical notion of the meaning of
the functional relation v=fu by regarding x, y as rectangular coordinates of a point P
iti 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 = x + iy and u' = x + iy' respectively ;
and, according to a locution above referred to, the point f might be regarded as a
function of the point P ; a given value of ux + iy is thus represented by a point
P in the plane II, 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=x' + 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 = x + iy.
Known Functions.
1. The most simple kind of function is the rational and integral function. We
have the series of powers w", u 3 , ... each calculable not only for a real but also for a
complex value of u, (a; + iy)- = x- iy- + 2ixy, (x + iyY = x 3 3xy- + i (3a?y y 3 ), &c., and
thence, if u, b, ... be real or complex numbers, the general form a +- bu + cu- + . . . + ku m ,
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 ia the algebraical function, and in particular
the algebraical function expressible by radicals. To take the most simple case, suppose
j.
(m being a positive integer) that v m u; v is here the irrational function =u m .
Obviously, if is real and positive, there is always a real and positive value of v,
calculable to any extent of approximation from the equation v m = u, which serves as
the definition of u m ; 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 u m ; and that for their determination we require the theory of the so-called
662
524 FUNCTION. [787
circular functions sine and cosine; and these depend on the exponential function expw,
or, as it is commonly written, e n , which has for its inverse the logarithmic function
log w ; these are all of them transcendental functions.
2. In a rational and integral function a + bu + cw 2 + . . . + ku m , 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 + cu? + ... extending to infinity, non constat that such an
expression has any calculable value, that is, any meaning at all ; the coefficients
o, 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,
u
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 exp?;, 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 a; 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 = cosy + 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 ycosz sin y sin z,
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 denning 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 tr denoting as usual the number
3'14159..., 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 u m . 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 d + i sin 0),
where, writing */& + y' to denote the positive value of the square root, we have
r = Vic 3 + v 3 , cos 6 = . sin 6 = -. ,
and therefore also
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 6 is the inclination of r to the positive
part of the axis of x ; to fix the ideas may be regarded as lying within the
limits 0, IT, or 0, IT, according as y is positive or negative ; 6 is thus completely
determinate, except in the case, x negative, y = 0, for which 6 is = TT or TT indifferently.
And if u = x + iy, we hence have
,- i /
u m = (x + iy) m = r m I
cos -
^
where r m is real and positive and s has any positive or negative integer value what-
_i
ever: but we thus obtain for u m 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
f)lt
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 u m , 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 x . exp y is, in fact, the equation of
indices, a x+v = a" . a" ; exp a; 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=lH --- h , + ,- ,, + &> that is, e = 2'7l28... But as well theoretically as
1 1.2 1 . 2 . 3
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 ( a;)},
sin ix = {exp x exp ( x)},
viz. these values are
1 .
X s
of
x*
1 . ji . o . 4?
+ . . .
17273 > "
each of them real when x is real. The functions in question 1 +
and a; + T -+..., regarded as functions of x, are termed the hyperbolic cosine and
L . Z . V
sine, and are represented by the notations cosh x and sinh x respectively ; and similarly
we have the hyperbolic tangent tanh#, &c. : although it is easy to remember that
cos ix, -. sin ix, are, in fact, real functions of x, and to understand accordingly the formulae
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 log M,
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 s ; in fact, if log u be any one of its values,
then log u + 2sTri is also a value, for we have exp (log u + 2siri) = exp log u exp Zsiri, or
since exp Zsiri is =1, this is =w; that is, Iogw + 2s7n is a value of the logarithm of w.
We have
uv exp (log uv) = e'xp log u . exp log v,
787] FUNCTION. 527
and hence the equation which is commonly written
log ivo = log M + log v,
but which requires the addition on one side of a term Zsvi. And reverting to the
equation x + iy = r (cos 6 + i sin 0), or as it is convenient to write it, x + iy = r exp 16,
we hence have
log (x + iy) = \ogr + i(0 + Isir),
where logr may be taken to denote the real logarithm of the real positive quantity r,
and 6 the completely determinate angle denned as already mentioned.
Reverting to the function U M , we have u = exp log u, and thence u m = exp (m log u),
which, on account of the infinity of values of log u, has in general (as before remarked)
an infinity of values ; if = e, then e m , = exp (m log e), has in general in like manner
an infinity of values, but in regarding e m as identical with the one-valued function
exp TO, 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 = + 1 v 7 ! - n*, or we have cos" 1 M = - log( + i Vf- it 2 ), and from the same
equation, writing secondly sin x = u, we have sin" 1 u = - log (Vl u" + iu). But the
formula for tan" 1 u is a more elegant one, as not involving the radical Vl u- ; we have
exp ix exp ( ix) exp 2ix 1
i tan x = *. ; ; , = ^-jp q- ,
exp ix + exp (ix) expzw: + l
and thence
] + 1 tan x
exp Zix = ; r- ,
1 - 1 tan x
that is,
1 1 4- i tan x
or, if tan x = u, then
1 1 + iu
tan" 1 u = ^-. log = .
The logarithm (or inverse exponential function) and the inverse circular functions
present themselves as the integrals of algebraic functions
[dx
= log x,
J x
whence also
= ^-. log , - = tan" 1 x,
[ + a? 2* & 1 - w
and
dx
= sin" 1 x.
Vi-o 2
528 FUNCTION.
[787
6. Each of the functions exp u, sin u, COSM, tan u, &c., as a one-valued function
in this respect analogous to a rational function of u; and there are further
analogies of exp u, sin u, cosu, to a rational and integral function; and of tanw, sec &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,...
be the roots, p, q, ,-,... their multiplicities, then the form is A (l - -Y (l - -Y ..
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 expw has no finite roots, but an infinity of roots each = _; this
appears from the equation exp=(l+)", where n is indefinitely large and positive.
The function sinM has the roots r where s is any positive or negative integer zero
included; or, what is the same thing, its roots are and S7r , s now denoting any
posmve integer from 1 to oo ; each of these is a simple root, and we in fact have
/ It? \
1 ( l -^J Similarly the roots of COSM are (s + $, s denoting any positive
or negative integer, zero included, or, what is the same thing, they are + ( S + A).
s now denoting any positive integer from to oo ; each root is simple, and we have
(A2 \
l + (s+^ J- Obvi ously tanw, as the quotient smu + cosu, has both roots
and infinities, its roots being the roots of sin u, its infinities the roots of COSM; sec*
the reciprocal of COSM has infinities only, these being the roots of COSM, &c.
In the foregoing expression sin = M O (l - -) , the product must be understood
to mean the limit of IT," (l - ) for an indefinitely large positive integer value of ,
viz. the product is first to be formed for the values ,= 1, 2, 3, ... up to a determinate
then n is to be taken indefinitely large. If, separating the positive
and the negative values of s, we consider the product MlWl + ^-) iWl - -), (where
in the first product s has all the positive integer values from 1 to n, lid 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 sin*
be a ratio of equality*. And similarly as regards COSM, the product
"""i 1 + (T+^)7rj II ((7T^r)' m and B indefinitely large, does not approximate to
COSM, unless m : n be a ratio of equality.
* 
