MATHEMATICAL PAPEKS.
JLonHon: 0. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE,
ffilasgofo: 263, ARGYLE STREET.
F. A. BROCKHAUS.
got*: THE MACMILLAN CO.
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 |
II
v^x -'+-
i i i. 9
] S B i
+ 2- *N -^- S
M OD O 00 W 00
-S. * o
i S i i 111 i
**
m
3 ^
CJ
x* v
M | g
flfi fc,
N 1 8
2
1
p(
<; i-H
2 S
O
$ x Nig ^.
O
2^ |8 N
N 1 8 "' "18 +
>" S
!* 1
+ rt 2s
1 f-t
c
^- * r ^
^"^ ^^ ^^ ^^s- ? i
1 J
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
.2
+
O O
S
^
rH i t
o "5
Cfl
= 2
Cl
JT ** 7^
* s
M| 8 "* e"
S S
.2 2
W5 CO O
* ^
^ + i 1
~ a- J 5 !
Ofl 00 _j_ to 1C o
It:
o X
7 i 7 g
i? s- ^ 3 i ||
ftf g
II
o-
< T>_ 1 8 S
X + -? ? ! +
HI! Iii r
ft
o
iii ii
1 1 1 1 III 1
JL
1
9
*S c
1
g -
^.
3 iS
*^ .5
"N
1
t J
en *a
1 >>
J-o
5-^ **
+ 1
S
r* s
^-* % *?
*M "
00 ^ O
O *y
2-
+ f*- CO
- ^J
"n7 ^ ^1
H 8 ^ + o? t-
3 $
II
1 -^ ^ M
i i- N 4! "
o
"^ 5 S ^*
"e |
f ^* ii' *il "ii-
"V i -2- 5 tit f
1>
&
*^ CQ CO ^ *^
^F H M B 9 S ^
BH q x x x
M
M
p *
><
745]
AND THE POLYHEDRAL FUNCTIONS.
159
a
13
a
i
, 3
, <a
I I !!!
1 1 11 Ii
S s .s J r S
O rg rg
f -a a 21
flcaHgcc- - ~ * *
g H 8
** - J3 " hH
! ..
O 3 **" O M
d, O H O
HOP
o
S" "N s*~~**\ - - /" "X s" X S -x S X ^ v s* ^
H
iH ^H ^^ IH|W rH
^H
c_:
r-< a H *
a
* ">-.
8 Q 8 8 ^c 8
^H
^H +
rH OO
*[* t^
^^ f^ 3 OV^
14 N
**
sj \^*J s^~^ \fj. ^~^ y^V S^i^ V~^ \^--J
f--^\ f^3*> /^^S ^ l /^^S /^^ r^ /^^ /^^S
><X >o<
O CO 100 * l J5 |S SIS
. 00 ,* 1 *H |O to |0 iH |O GO ^ Ofl U5 O
OpH "5|,_( ciliO ^ W (Nlws W CS <N3 ^H(N *io
s g "a
S CO 100 CO 100 05 00 CO IX
1
** 'O 12 I?? ^ ;O5 SIS 2 <N KkH S IO * S S |S
co a S
r f
1 1
j j i- Iff) IG^ _H ~ <N |*O .us -, _4< ifls ^H (O _u r (N 1ft
* IS5 r4lo5 ^ l05 i-l IcN w la> ^ l05 tN IS * IC2 rt
^ O * los
a. a.
-fn -C1
i
SIS og
Nil
o
I
II
3 -S
II
M
O
-c
r-< |O m !-H IO tN W3
CM I*O ^ Icffl CN ! O iH CM
|S
-i
II II II II II
ii ii
3
<N CO CN Ig i-t
O O3
I
o o o o o
... X ... 100 i-l |O .-I* 1 CO l5 CM |5 rH O (N |>O <N I'O
10 -! * IrH CM l>0 "5 IrH S Wfl r-ilcN tN W rH ItN i-H IcN
S 10 S l>0
'S
II II II II II
II II II II II II II II II
II II
1
rH <f> CO * "5
O
CMtNCQCMfHrHcMi-Hi-l
OOOOOCJOOO
^ ?
3
O
II
o
II
s
s
O
O O O O0500O O O O
05 100 O
II II II II II
II II II II II II II II II
II II
g IM e<i o o
CNCNectdrHeNeNeNCM
rH eg
.0 ^a .Q & &
O
II
_. IA tyt _. _, CO IIO i^ 100
* ' S loj * I0) -H ItN "'I.H
II II II II II II
II II
iH iH rH iH CO <N
C8 05 A C8 a C3
* s
O
^t 1O9 J2 IcO "* '^ S IS O ^ IOS O "* 'OS ^
* IO9 * |e
II II II II II
II II II II II II II II II
II II
8 8 05 05 CO
rHrH!-4rHCOtHO5fHCO
aaaaaaaai
OS IX -* 10 3|g 3 IS
,< ,< sis -*i | . - = =
- =
f
*
7 OS |X 05 100 05 100 05 100
co ico co IQO co ico co ico co ix x r r:
r :
"o*
i 1
4
Ha
.
7 7
as
| | ^ 1C5 ^ ICft ^d* ICft
^f r*
^*ic> SIS ^ics 2IS ^i = = = -
^H ICO ^ 15*1
* "
i
I
- . CO ,0
1 ( HH
l-t . HH b-d . j 1 * HH
B r^StX-^H-
5 s
13
S
bo
tu a
t 3
% ^
v
QJ >O O
? I ^
W5 1O3 rt O
^H |co r *^
1 fl 1!
S 2
s g 1 a
3
^
a
a
ta
'%
a 1 1
*3 a>
ts I
to
a
t-H
5 1
, a
1
co g
h
i g
8" -
fc
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,
a 
