Full text of "The collected mathematical papers of Arthur Cayley"

MATHEMATICAL PAPEES. 



ILonDon: 0. J. CLAY AND SONS, 
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MARIA LANE. 
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$t&lt;as got*: THE MACMILLAN CO. 



THE COLLECTED 



MATHEMATICAL PAPERS 



OF 



AETHUE CAYLEY, Sc.D., F.E.S., 

LATE SADLEEIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE. 



VOL. XI. 




CAMBRIDGE : 
AT THE UNIVERSITY PRESS. 

1896 

[All Rights reserved.] 




CAMBRIDGE : 

PRINTED BY J. AND C. F. CLAY, 
AT THE UNIVERSITY PRESS. 



ADVEETISEMENT. 



rilHE present volume contains 93 papers numbered 706 to 798, published, 
with the exception of one series, for the most part in the years 1878 
to 1883. This series is constituted by the articles which Professor Cayley 
wrote for the Encyclopaedia Britannica between the years 1878 and 1888 ; 
it seemed desirable to place these together in the same volume, in spite 
of the departure from the chronological arrangement which governs the 
sequence of the papers in the volumes generally. The Syndics of the 
University Press desire to acknowledge their obligation to Messrs Adam 
and Charles Black, Publishers of the ninth Edition of the Encyclopedia 
Britannica, for their courteous consent in allowing these articles to be 
included in the Collected Mathematical Papers. Exact references to the 
volumes, from which the articles are extracted, will be found in the Table 
of Contents. 

The frontispiece to the present volume is a reproduction by Mr 
A. G. Dew-Smith, of Trinity College, of a photograph of Professor Cayley 
which he made in the year 1885. The Syndics of the Press desire to 
acknowledge their obligation to Mr Dew-Smith. 



ADVERTISEMENT. 

The Table for the eleven volumes is 

Vol. I. Numbers 1 to 100, 

II. 101 158, 

III. 159 222, 

IV. 223 299, 

V. 300 383, 

VI. 384 416, 

VII. ;, 417 485, 

VIII. 486 555, 

IX. 556 629, 

X. 630 705, 

XL 706 798. 

A. K FORSYTH. 

21 November, 1896. 



Vll 



CONTENTS. 

[An Asterisk means that the paper is not printed in full.] 



PAGE 



706. On the distribution of electricity on two spherical surfaces . 1 

Phil. Mag., Ser. 5, t. v. (1878), pp. 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. Fran9., Compt. Rend., t. ix. (1880), pp. 135139 

709. On the number of constants in the equation of the surface 



. ...... . . . 14 

Tidsskrift for Mathematik, Ser. 4, t. iv. (1880), pp. 145148 

710. On a differential equation ..... ... 17 

Collectanea Mathematica, in memoriam Dominici Chelini, (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. Rowes "Memoir on Abel s theorem" . 29 

Phil. Trans., t. CLXXII. (1881), pp. 751758 



CONTENTS. 

PAGE 

37 

714. Various notes ...- 

Messenger of Mathematics, t. vn. (1878), pp. 69: 115: 124: 125 

715. Note on a system of algebraical equations 

Messenger of Mathematics, t. vn. (1878), pp. 17, 18 

716. An illustration of the theory of the ^-functions . 41 

Messenger of Mathematics, t. vn. (1878), pp. 2732 

717. On the triple theta-functions . . ^ 

Messenger of Mathematics, t. vn. (1878), pp. 4850 

718. Addition to Mr. Geneses paper " On the theory of envelopes" . 50 

Messenger of Mathematics, t. vn. (1878), pp. 62, 63 

719. Suggestion of a mechanical integrator for the calculation of 

(Xdx+Ydy) along an arbitrary path . . 52 

Messenger of Mathematics, t. vn. (1878), pp. 9295; British Asso 
ciation Report, 1877, pp. 18 20 

720. Note on Arbogast s method of derivations . 55 

Messenger of Mathematics, t. vn. (1878), p. 158 

721. Formula involving the seventh roots of unity . .56 

Messenger of Mathematics, t. vn. (1878), pp. 177182 

722. A problem in partitions ..... . . 61 

Messenger of Mathematics, t. vn. (1878), pp. 187, 188 

723. Various notes .....- .63 

Messenger of Mathematics, t. vin. (1879), pp. 45, 46: 126: 127 

724. On the deformation of the model of a hyperboloid . . . 66 

Messenger of Mathematics, t. vin. (1879), pp. 51, 52 

(i it* r/yy 

725. New formula for the integration of 7^ + Tv~ ^ 

\l ** v * 
Messenger of Mathematics, t. vin. (1879), pp. 60 62 

726. A formula by Gauss for the calculation of log 2 and certain 

other logarithms . . . . . 70 

Messenger of Mathematics, t. vm. (1879), pp. 125, 126 



CONTENTS. ix 



PAGE 



727. Equation of the wave-surface in elliptic coordinates . . . 71 

Messenger of Mathematics, t. vm. (1879), pp. 190, 191 

728. A theorem in elliptic functions . . . . . . . 73 

Proc. Lond. Math. Soc., t. x. (1879), pp. 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. XL (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 Abel s relating to a quintic equation . . 132 

Camb. 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 *- II (w) up to m = n = 20 . . . . _ 144 

Camb. Phil. Trans., t. xin. (1883), pp. 14 

745. On the Schwarzian derivative, and the polyhedral functions . 148 

Camb. Phil. Trans., t. xin. (1883), pp. 568 

*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 " Versucli einer allgemeinen Auffass- 

ung der Integration und Differentiation," Werke, pp. 331 

344 . . . 235 

Mathematische Annalen, t. xvi. (1880), pp. 81, 82 

752. On the finite groups of linear transformations of a variable ; 

with a correction ......... 237 

Mathematische Annalen, t. xvi. (1880), pp. 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 formulce in elliptic functions . 250 

Messenger of Mathematics, t. ix. (1880), pp. 23 25 

755. On the matrix (a, b ), and in connection therewith the function 

I 

U 4 

ax + b 

c^+d . 252 

Messenger of Mathematics, t. ix. (1880), pp. 104 109 

756. A geometrical construction relating to imaginary quantities . 258 

Messenger of Mathematics, t. x. (1881), pp. 1 3 

757. On a Smith s Prize question, relating to potentials . . . 261 

Messenger of Mathematics, t. XL (1882), pp. 15 18 

758. Solution of a Senate-House problem . . . . . 265 

Messenger of Mathematics, t. XL (1882), pp. 23 25 

759. Illustration of a theorem in the theory of equations . . 268 

Messenger of Mathematics, t. xi. (1882), pp. Ill 113 

r ax 

760. Reduction of i , _ .T-J to elliptic integrals . . . . 270 

Messenger of Mathematics, t. xi. (1882), pp. 142, 143 

761. On the theorem of the finite number of the covariants of a 

binary quantic . . . . . . . . . 272 

Quart. Math. Journ., t. xvn. (1881), pp. 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 

b 2 



CONTEXTS. 



PAGE 



767. On the Gaussian theory of surfaces . 

Proc. Lond. Math. Soc., t xii. (1881), pp. 187192 

768. JVofe OH Landen s theorem 337 

Proc, Lond. Math. Soc,, t xm. (1882), pp. 47, 48 

769. Oi o formula relating to elliptic integrals of the third kind . 340 

Proc, Lond. Math. Soc., t xm. (1882), pp. 175, 176 

770. On the 34 concomitants of the ternary cubic . 342 

American Journal of Mathematics, t, IT. (1881), pp. 1 15 

771. Spi dnien of a literal table for binary qualities, otherwise a 

partition table . . . 357 

American Journal of Mathematics, t. iv. (1881), pp. 248 255 

772. On the analytical forms called trees 365 

American Journal of Mathematics, t. iv. (1881), pp. 266 268 

773. On the S-square imaainaries . 

American Journal of Mathematics, t. IT. (1881), pp. 293296 

774. Tables for the binary sextic . - 

Americau Journal of Mathematics, t. iv. (188 IX pp. 379 o&gt;4- 

775. Tables of corariante of the binary sejctic. 377 

Written in 1894 : now first published. 

776. On the Jacobian sejctic equation ... 3M) 

Quart. Math. Journ., t mil. (1882), pp. 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. Joxirn., t. mil. (1882), pp. 226 lit 

[Xote on Mr. Jefrcry s paper " On certain quartic curves, 

ichich haw a cusp at infinity 9 whereat the line at injinity 

is a tangent"] . ... .. , . . 408 

Proc, Lond. Math. Soc., t. XIY. (1883X P* & 



CONTEXTS. x iii 

PiOK 

780. [Addition to J/r. Hammond s paper " Xote on an exceptional 

case in which the fundamental postulate of Professor 
Sylvesters theoiy of tamisage fails " | . . 409 

Proc. Loud. Math. Soc., t. xiv. (1883), pp. 8891 

781. On the automorphic transformation of the binary cubic 

fimction .. .". .*. . . . . 411 
Proa Lond. Math. Soc., t xiv. (1883), pp. 103 108 

782. On Monge s " Memoire sur la theorie des deblais et des 

remblais" . . . . . . . . 4^7 

Proc. Lond. Math. Soc., t. xiv. (1883), pp. 139 142 

783. On Jfr. Wilkinson s rectangular transformation . . 421 

Proc. Lond. Math. Soc., t xiv. (1883), pp, 222229 

784. Presidential Address to the British Association, Southport, 

September 1888 ........ 429 

British Association Report, 1883, pp. 3 37 

785. Cun-e ......... 460 

Encyclopaedia Britannica, 9th ed., t, vi. (1878), pp. 716 728 

786. Equation . . . . . . . . 490 

Encyclopaedia Brit&nnica, 9th ed, t viu. (1878), pp. 497 509 

787. Function 522 

Encyclopjedia Britannica, 9th ed., t. ix. (1879), pp, 818824 

788. Galois . . . . 543 

Encyclopaedia Britannica, 9th ed., t. x. (1879), p. 48 

789. Grauss 544 

Encyclopaedia Britannica, 9th ed., t. x. (1879), p. 116 

790. Geometry (analytical) 546 

Encyclopaedia Britannica, 9th ed., t. x. (1879), pp. 408420 

791. Landen 5S.S 

Encyclopaedia Britannica, 9th ed., t xiv. (1882), p. 271 

792. Locus 585 

Encyclopaedia Britannica, 9th ed., t. xiv. (1882), pp. 764, 765 



CONTENTS. 

PAGE 



793. Monge . . . ; . . . 586 

Encyclopaedia Britannica, 9th ed., t. xvi. (1883), pp. 738, 739 

794. Numbers (partition of) . . . . 589 

Encyclopaedia Britannica, 9th ed., t. xvn. (1884), p. 614 

795. Numbers (theory of) ....... 592 

Encyclopaedia Britannica, 9th ed., t. xvn. (1884), pp. 614624 

796. Series ......... -. 617 

Encyclopaedia Britannica, 9th ed., t. xxi. (1886), pp. 677682 

797. Surface ....... " .- . . . : 628 

Encyclopaedia Britannica, 9th ed., t. xxn. (1887), pp. 668 672 

798. Wallis (John] .......... 640 

Encyclopaedia Britannica, 9th ed., t. xxiv. (1888), pp. 331, 332 

Portrait ......... To face Title. 



XV 



CLASSIFICATION. 



ANALYSIS. 



Calculation of log 2 ; 726. 

Series, 796. 

Prime roots of unity, 721. 

8-square imaginaries, 773. 

Squares, theorems of 2, 4, 8, 16; 763. 

Difference-table for A m O" -4- H (m) ; 744. 

Equations, theory of, 736, 741, 743, 759, 776, 777, 778, 786. 

Numbers, theory of, 731, 764, 795. 

Partitions, 722, 771, 794. 

Trees, 772. 

Matrices, 755. 

Elimination, 733. 

Transformation of cubic function, 781. 

Covariantive forms and tables, 737, 761, 770, 774, 775, 780. 

Fractional differentiation, 751. 

Mechanical integrator, 719. 

Differential Equations, 725. 

Schwarzian derivative, 745. 

Hypergeometric series, 710, 738. 

Finite groups, 752. 

Polyhedral functions, 739, 745. 

Elliptic functions, 728, 740, 754, 760, 768, 769. 

Transformation of elliptic functions, 711. 

Abel s theorem, 712, 713. 

Theta Functions, 716, 717, 753. 

Function, 787. 

Conformal representation, 729. 



XVI CLASSIFICATION. 

GEOMETRY. 

Analytical geometry in general, 790, 792, 797. 

Plane Curves, 746*, 785. 

Degenerate forms of curves, 747. 

Quartic Curves, 748, 779. 

Deblais et remblais, 782. 

Tortuous curves, 708, 785. 

Geodesic curvature, 766. 

Theory of surfaces, general, 709, 749*, 767. 

Transformation of coordinates, 742, 783. 

Reciprocal surfaces, 750. 

Wave-surface, 727. 

Apsidal surfaces, 735. 

Deformation and flexure of surfaces, 724, 765. 

Hypergeometry, 730. 

Schubert s numerative geometry, 762. 

VARIOUS SUBJECTS. 

Spherical Trigonometry, 732. 
Kinematics of a plane, 734. 
Maps, colouring of, 707. 

Electricity, distribution of, on spherical surfaces, 706. 
Potential, 757. 

Presidential Address to the British Association, 784. 
Biographical articles ; Galois, 788. 
Gauss, 789. 
Landen, 791. 
Monge, 793. 
Wallis (John), 798. 
MISCELLANEOUS 

714, 715, 718, 720, 723, 756, 758. 



706] 




706. 



ON THE DISTRIBUTION OF ELECTRICITY ON TWO SPHERICAL 

SURFACES. 

[From the Philosophical Magazine, vol. v. (1878), pp. 54 60.] 

IN the two memoirs " Sur la distribution de I e lectricite a la surface des corps 
conducteurs," Mem. de I Inst. 1811, Poisson considers the question of the distribution 
of electricity upon two spheres : viz. if the radii be a, b, and the distance of the 
centres be c (where c &gt; 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 &lt;(#), and for his F [j] writing &lt;(;) the question depends 
on the solution of the functional equations 



c x 



(x) = g, 



c x \c xi 
where of course the x of either equation may be replaced by a different variable. 

It is proper to consider the meaning of these equations : for a point on the axis, 
at the distance x from the centre of the first sphere, or say from the point A, the 

a 2 (o?\ 
potential of the electricity on this spherical surface is a$x or &lt;p f ) , according as 

the point is interior or exterior; and, similarly, if x now denote the distance from 
the centre of the second sphere (or, say, from the point B), then the potential of 

TO /7i2\ 

the electricity on this spherical surface is b&lt;3?x or -&lt;( ), according as the point is 

x \x J 

interior or exterior; &lt;/&gt;(#) is thus the same function of (ss, a, b) that &lt;(#) is of 
C. XI. 1 



2 ON THE DISTRIBUTION OF ELECTRICITY [706 

(x, b, a). Hence, first, for a point interior to the sphere A, if a: denote the distance 
from A, and therefore c x the distance of the same point from B, the potential of 
the point in question is 

6 2 , / 6 2 \ 
= ad&gt;x + - 3&gt; -- ; 
c x \c-xj 

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 



cx \c-x 



The two equations thus express that the potentials of a point interior to A and of 
a point interior to B are =h and g respectively. 

It is to be added that the potential of an exterior point, distances from the points 
A and B = x and c x respectively, is 



b- 



, 

* 



I-J 



c x \c 

and that, by the known properties of Legendre s coefficients, when the potential upon 
an axial point is given, it is possible to pass at once to the expression for the potential 
of a point not on the axis, and also to the expression for the electrical density at a 
point on the two spherical surfaces respectively. The determination of the functions 
(J&gt;(x) and &lt;&(x) gives thus the complete solution of the question. 

I obtain Poisson s solution by a different process as follows: Consider the two 
functions 

a 2 (c x) ax + b 

T^jsZ: = T7[ suppose, 

and 

b- (c x) OLX + 8 

- ~ -, = -%, suppose; 
c 2 a 2 ex &lt;yx+ o 

and let the ?ith functions be 

3^ and ^ 

c n x + d n &lt;y n x + b n 

respectively. 

Observing that the values of the coefficients are 

(a, b ) = ( -a 2 , a-c ), and (a, 8 ) = ( -b 2 , b*c ), 

c, d c , c 2 b- 7, 8 c , 

so that we have 

and consequently that the two equations 

(\+l) 2 _(a 



X ad -be 



706] ON TWO SPHERICAL SURFACES. 

are in fact one and the same equation 

(X + l ) 2 (c 2 -a 2 -6 2 ) 2 



for the determination of X, then (by a theorem which [686, 687] I have recently 
obtained) we have the following equations for the coefficients 

( , b n ), ( On, n ) 



of the nth functions ; viz. these are 



and similarly 







b) + (X" - X) (- d* + b)}, 



ca?-a)}; 



- a)}. 



Observe that these equations give, as they ought to do, 

a # + bo = #, c # + d =l, a 1 cc+ bj 
and similarly 



= yx + 8. 



a? 



Substituting in the first two equations - - in place of x, and in the second two 

~~ 



~~ (X) 



equations - - in place of x, we obtain the following results which will be useful : 

C "" 3C 

2 + b n (c #) = a 2 (y n x + S n ), 



c n a 2 + d n (c - a?) = (a 



n (c - a?) = 



the las.t two of which are obtained from the first two by a mere interchange of 
letters ; it will therefore be sufficient to prove the first and second equations. 



For the first equation we have 



a n a 2 + b n (c - a?) = 



A, 



{(\ n+1 - 1) [aa 2 + b (c - a?)] + (X" - X) [- da 2 + b (c - a)]}, 

12 



4 ON THE DISTRIBUTION OF ELECTRICITY [706 

where the term in { } is 

= (X l+1 - 1) [- a 4 4- a 2 c (c - x)] + (\ n - X) [a 2 (6 2 - c 2 ) 4- a 2 c (c - x)] ; 

viz. this is 

= a 2 {(\ n+1 - 1) (c 2 - a- - ex) + (X n - X) (6 2 - ca?)} ; 

or it is 

= a 2 {(X n+1 - 1) (yx +8) + (\ n - X) (yx - a)}, 

whence the relation in question. 

The proof of the second equation is a little more complicated. We have 

1 /a 4- d\ n-1 
c n a 2 4- d (c - x) = f - ^ J {(X n+1 - 1) [ca 2 4- d (c - a?)] + (X n - X) [ca 2 - a (c - a;)]}, 



where the term in { } is 

= (X n+I - 1) [- ca 2 + (c 2 - 6 2 ) (c - x)] + (X n - X) [- ca 2 + a 2 (c - x)]. 
Comparing this with 

" /3) + (X - X) (- 



where the term in { } is 

= (X+ 2 - 1) [6 2 (c - x)] + (\ n+1 - X) [- c (c 2 - a 2 - 6 2 ) + (c 2 - a 2 ) (c - a?)], 

it is to be observed that the quotient of the two terms in { } is in fact a constant : 
this is most easily verified as follows. Dividing the first of them by the second, we 
have a quotient which when x = c is 



- 1) (- ca 2 ) + (\ n - X) (-ca 2 ) a 2 (X" +1 - 1 +X-X) a 2 (X+l) 



(X n+1 - X) {- c (c 2 - a 2 - 6 2 )} (\ n + l - X) (c 2 - a 2 - 6 2 ) ~ (c 2 - a 2 - 6 2 ) X 

and when # = is 

(X+ 1 -l)c(c 2 -a 2 -6 2 ) 



2 -6 2 ) = (X M+1 -l)(c 2 -a 2 -6 2 ) _ c- - a 2 - 6 2 
- X) 6 2 c ~(X n+2 -l+X w + 1 -X)6 2 = 6 2 (X + 1) : 



these two values are equal by virtue of the equation which defines X ; and hence the 
quotient of the two linear functions having equal values for x = c and x = 0, has 

2 _ fj2 _ J2 

always the same value ; say it is = , 2 . . Hence, observing that a + d = a + 8, 

= c 2 -a 2 -6 2 , the quotient, c n a~ + d n (c - cc) divided by a n+1 # + /3 w+1 , is 



X + l c 2 - a 2 - 6 2 _]. 
-a 2 -6 2 6 2 (X + D ~6" 2 



or we have the required equation 



n (c - x) = 



706] ON TWO SPHERICAL SURFACES. 5 

Considering now the functional equations, suppose for the moment that g is = ; 
the two equations may be satisfied by assuming 

klr 



r 

c x + d das + 



We in fact, from the foregoing relations, at once obtain 



a 2 a 3 , [ w &) 2 ) o?L 



x cx 
62 



cx c x 



To satisfy the first equation we must have M = aL ; viz. this being so, the equation 
becomes 



- -- = =- 

c x \c-xj c x + d 

or, since C # + d =l, the equation will be satisfied if only aL = l, whence also M=\. 

9 7 2 f 

And the second equation will be satisfied if only - = bM ; viz. substituting for L, M 
their value, we find &&amp;gt; = ab. 

Supposing, in like manner, that h = 0, g retaining its proper value, we find a like 
solution for the two equations ; and by simply adding the solutions thus obtained, we 
have a solution of the original two equations 



c x \c X; 

o? . f a? \ ,, . , 

&lt;z&gt; I + 6&lt;P (x) = q : 

x r \c-xj 



viz. the solution is 



JL(\_ h | 1 ab } _ f ab (ab) 2 

* ^ rt ] si M i s3 rti&gt;-l " t/lrt&gt;-vilV* h HM _l_ n 



We have a general solution containing an arbitrary constant P by adding to the 
foregoing values for &lt;/&gt;# a term 

Pb(a-b) 

Va 2 (c - 0) - a? (c 2 - 6 2 - c#) 
and for &lt;!&gt;# a term 

= _____ Pa (b - a) ___ 

V6 2 (c x) x (c 2 a 2 ex) 



6 THE DISTRIBUTION OF ELECTRICITY ON TWO SPHERICAL SURFACES. [706 

as may be easily verified if we observe that the function 

a 2 (c x) x (c 2 b" ex), 

writing therein for x, becomes 

c-oc 

{b 2 (c-x)-x(c*-a 2 -cx)}: 



(c-x)* 
and similarly that 

/j2 // __ sr&gt;\ _ /r ( V*- __ /^2 ^_ /*/y^ 
t/ \ L* **/ / tv i v vv v*/ /j 

J2 

writing therein for x, becomes 

c - x 

= rr~, ^ l a * ( c ~ *) ~ x ( G ~ b " ~ cx )}- 



More generally, the terms to be added are for tj&gt;x a term as above, where P denotes 

Q% iff _ ,\ 

a function of x which remains unaltered when x is changed into , and for 

c- b 2 ex 

&lt;&gt;x a term as above with P instead of P, where P denotes what P becomes when 
x is changed into - . But these additional terms vanish for the electrical problem, 

C ~~ $/ 

and the correct values of (f&gt;x, &lt;&x are the particular values given above. 
It is to be remarked that the function 

a 2 (c x) 



c- - b 2 - ex 



is = 



c x 



viz. considering x as the distance of a point X from A, then taking the image of X 
in regard to the sphere B, and again the image of this image in regard to the 
sphere A, the function in question is the distance of this second image from A. And 
similarly the function 

6 2 (c - x) b- 

is = ; 



ex 

c 



viz. considering here x as the distance of the point X from B, then taking the image 
of X in regard to the sphere A, and again the image of this image in regard to 
the sphere B, the function in question is the distance of this second image from B. 
It thus appears that Poisson s solution depends upon the successive images of X in 
regard to the spheres B and A alternately, and also on the successive images of X 
in regard to the spheres A and B alternately. This method of images is in fact 
employed in Sir W. Thomson s paper " On the Mutual Attraction or Repulsion between 
two Electrified Spherical Conductors," Phil. Mag., April and August, 1853. 



707] 



707. 



ON THE COLOURING OF MAPS. 



[From the Proceedings of the Royal Geographical Society, vol. I., no. 4 (1879), 

pp. 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 is one or more than one 
interior boundary (that is, if there is within the exterior boundary one or more than one enclave not 
belonging to the area), then the area is "multiply connected." The theorem extends to multiply connected 
areas, but there is no real loss of generality in taking, and we may for convenience take the areas of the 
theorem to be each of them a simply connected area. 



s 



ON THE COLOURING OF MAPS. [707 



be sufficient, and taking a third colour for the surrounding area, three colours only 
would be wanted; and so in general according as the number of sectors is even or 
odd, three colours or four colours are wanted. And in any tolerably simple case it can 
be seen that four colours are sufficient. But I have not succeeded in obtaining a 
general proof: and it is worth while to explain wherein the difficulty consists. 
Supposing a system of n areas coloured according to the theorem with four colours 
only, if we add an (w+l)th area, it by no means follows that we can without 
altering the original colouring colour this with one of the four colours. For instance, 
if the original colouring be such that the four colours all present themselves in the 
exterior boundary of the n areas, and if the new area be an area enclosing the n 
areas, then there is not any one of the four colours available for the new area. 

The theorem, if it is true at all, is true under more stringent conditions. For 
instance, if in any case the figure includes four or more areas meeting in a point 
(such as the sectors of a circle), then if (introducing a new area) we place at the 
point a small circular area, cut out from and attaching itself to each of the original 
sectorial areas, it must according to the theorem be possible with four colours only 
to colour the new figure; and this implies that it must be possible to colour the 
original figure so that only three colours (or it may be two) are used for the 
sectorial areas. And in precisely the same way (the theorem is in fact really the 
same) it must be possible to colour the original figure in such wise that only 
three colours (or it may be two) present themselves in the exterior boundary of the 
figure. 

But now suppose that the theorem under these more stringent conditions is true 
for n areas: say that it is possible with four colours only, to colour the n areas 
in such wise that not more than three colours present themselves in the external 
boundary : then it might be easy to prove that the n + l areas could be coloured 
with four colours only: but this would be insufficient for the purpose of a general 
proof; it would be necessary to show further that the n + l areas could be with the 
four colours only coloured in accordance with the foregoing boundary condition; for 
without this we cannot from the case of the n + l areas pass to the next case of 
n + 2 areas. And so in general, whatever more stringent conditions we import into 
the theorem as regards the n areas, it is necessary to show not only that the n + l 
areas can be coloured with four colours only, but that they can be coloured in 
accordance with the more stringent conditions. As already mentioned, I have failed 
to obtain a proof. 



708] 



708. 

NOTE SUR LA THEORIE DES COURSES DE L ESPACE. 

[From the Compte Rendu de I Association Frangaise pour I Avancement des Sciences (1880), 

pp. 135139.] 

EN considerant dans 1 espace ime courbe d espece donne e, determinee au moyen 
d un nombre suffisant de points, la courbe n est pas determinee uniquement; mais on 
a par les points un certain nombre de telles courbes. Par exemple, la courbe unicursale 
d ordre^ 2p depend, comme on voit sans peine, de 8p constantes et sera ainsi 
determinee par 4p points (le cas p = l est urie exception): on ne connait pas, je 
pense, le nombre des courbes par les 4&gt;p points; mais pour le cas particulier p = 2 
(c est-a-dire pour une courbe quartique de seconde espece, ou autrement dit, une 
courbe excubo-quartique) ce nombre est =4: theoreme de montre par moi depuis 
longtemps par des considerations geome triques. (Voir Salmon, Geometry of three 
dimensions, 3 e ed. 1874, p. 319.) Ce n est que dernierement que j ai conside re la 
question analytique, de trouver les equations d une courbe excubo-quartique qui passe 
par 8 points donne s ; et meme j ai pris pour les 8 points une disposition qui n est 
pas tout a fait generale : 1 investigation elle-meme, et la forme du resultat, m ont 
paru assez interessantes pour que je les soumette a 1 Associatiou. 

En considerant sur une courbe excubo-quartique 4 points donne s, le plan passant 
par 3 quelconques de ces points rencontre la courbe dans un seul point ; et 1 on 
obtient ainsi encore 4 points sur la courbe: voila mon systeme de 8 points donnes, 
savoir en partant de 4 points quelconques, je prends un point quelconque dans chacun 
des plans qui passent par 3 de ces points, et j obtiens ainsi les autres 4 points. Et 
par un tel systeme de 8 points, je cherche a faire passer une courbe de 1 espece dont 
il s agit. 

En prenant cc=Q, y = Q, z = , w = 0, pour les equations des plans du tetraedre 
forme par les 4 premiers points, les coordonndes de ces points seront (1, 0, 0, 0), 
(0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1): et pour les coordonnees des 4 autres points, 
je prends (0, &, z lt Wl ), (x,, 0, * 8 , w 2 ), ( y 3 , (), w ,\ (x., y ti z 4 , 0). 

c. xi. 



10 NOTE SUR LA THEORIE DES COURBES DE L ESPACE. [708 

Les equations de la courbe sont x : y : z : w = P : Q : R : S, ou P, Q, R, S 
sont des fonctions (*)(#, I) 4 d un parametre variable ; il s agit de faire passer une 
telle courbe par les 8 points. 

Je prends a, ft, 7, B, a, b, c, d pour les valeurs du parametre qui correspondent 
aux 8 points respectivement. 

Pour que la courbe passe par les premiers 4 points, il faut et il suffit que les 
Equations soient de la forme 

. 0-a r&gt;0-b n O-c ^0-d_ 
: w = A 7: : B ,= : O ., 



0=~a "0--0 "0-y "0-8&gt; 
les conditions pour les autres 4 points seront alors 

2/i : z i w i = 



: 2/3 



= 





E a ~ 


b 


r a 


c 


a 


-d 


a 


ft 


a 


-7 


-B 


i b 


a 







c b 


c 


X 


-d 


- A b 


a 


L b 


-7 


-B 


1 


a 


7? C ~ 


b 






D c 


-d 


c 


a. 


c 


ft 






c 


-B 


d 


a 


B d ~ 


b 


c d 


c 






d 


a. 


d- 


ft 


d 


~7 







Evidemment il y a deux Equations qui donnent la valeur de B : C, et qui servent 
ainsi pour eliminer cette quantite. De cette maniere on obtient six equations que 
j ecris comme voici : 

a b.d c a .d @ 



_ 



a c . d b a 7? .d 



_w t y 3 _a d.c b a /3 . c B 



_ ZjWz a c .b d a 8 .b 






Z.M! a d.b c a 7.6 8 

z 2 x 4 b c .d a b a.d 7 

z x 2 b a.d c b y.d a 

x 2 w 3 b a . c d b B . c a. 

x 3 w 2 b d.c a b a.c 8 



^2/4 _c a.d b c {3 .d a. 
^2/3 c b.d a c a.d /3 

II Z 

savoir X, p., v, CT, K, p denotent ici les quantites donnees X = 4 , etc. Le nombre 



des equations independantes est 5, car Ton a identiquement Xfwi&Kp = 1. Je remar- 
que que 1 on peut faire sur le parametre une transformation lineaire quelconque 
(h0 + i) : (j0 + k), et introduire ainsi 3 constantes arbitrages ; on peut done prendre a 



708] NOTE SUR LA THEORIE DES COURBES DE I/ESPACE. 11 

volonte 3 valenrs du parametre 6, c est-a-dire les valeurs de 3 quelconques des quantite s 
a, /3, 7, 8, a, b, c, d ; et cela dtant les 5 equations donneront les valeurs des autres 
5 quantites. Si au moyen des equations on elimine a, /3, 7, 8, on obtient entre 
a, b, c, d une equation qui sera, comme on va voir, de 1 ordre 4 par rapport a 
chacune de ces quantite s : en prenant comme donnees a, b, c il y aura done 4 valeurs 
de d; et pour 1 une quelconque de ces valeurs, celles de a, /3, 7, 8 seront de termine es 
uniquement : il y aura ainsi 4 courbes qui passent chacune par les 8 points ; ce qui 
est le the oreme dont il s agit. 

J introduis, pour abreger, la notation 

a d, b d, c d, b c, c a, a b, 
= f, g; h, a, b, c: 

on a done identiquement 

a, b, c = g-h, h-f, f-g, 

a + b + c = 0, 
fa -f gb + he = 0. 

Les Equations prennent ainsi la forme 

he a y.d ft 

A* ~~ T~ 7^ -j " 9 6tC. . 

gb a p . d 7 
ou, en introduisant pour plus de commodite, les symboles 

L, M, N, P, Q, R, 

pour designer respectivement 

gb % he fa he gb fa 

"~ C~ ^&gt; ~ ~~ -- ~ ~~ ~ -- - 

he 
les equations seront 



C~ &gt; ~ ~c~ -- U v &gt; ~ ~c~ &gt; ~, -- L - 

he fa ^ gb fa he gb 



T 
- 



ii*-_a /3 . c 8 
~a- 8.C-/3 

AT a ~ ^ b ~ V 
- a -y.b-S 

p = ^r 7 Trf^a 

n _b 8 . c a 

r, _ c /3 .d a 
c a. rf /? 

avec la relation identique LMNPQR 1 ; il s agit entre ces 5 equations d eliminer 
a, , 7, S. 

22 



12 NOTE SUR LA THEORIE DES COURBES DE I/ESPACE. [708 

J e cris ot = a &lt;f), les facteurs b a, c a, d a de P, Q, R deviennent ainsi 
respectivement c + &lt;f&gt;, g + 4&gt;, f+&lt;/&gt;; cela dtant, les valeurs de P, Q, R servent a 
exprimer /3, 7, 8 en fonction de &lt; : substituant ces valeurs de /3, 7, B dans celles 
de L, M, N, on obtient sans peine 

h f(c-&lt;ft) + cP(-f+&lt;fr) 
gP b(- 

M - b ( ~ 






valeurs qui donnent, comme cela doit etre, LMNPQR = 1 : il faut entre ces equations 
e liminer ^&gt;. 

En retablissant \, p, v, CT, K, p au lieu de L, M, N, P, Q, R, ces equations 
deviennent 



a + 



a _ 
~g A = 






(evidemrnent i;r)%= 1), ou j ecris , ?;, ^ pour denoter les expressions -X-sr, etc., et ou 

O 

les valeurs des coefficients X, Y, etc., sont 

X = fc (fa + whc), 7 = - f 2 a - ^hc 2 , 



X 2 = be (he + *gb), F 2 = he 2 - gb-\ 
Les deux premieres equations donnent 



ou, ce qui est la meme chose, 



et Ton n a qu a substituer la valeur de ces coefficients. 

On a 
X,Y 2 - X,Y, = fb (gb + pfa) (he 2 - g b 2 ) - be (he + *gb) (- gb 2 + 

= fghb 2 c 2 - fg 2 bV + f 2 habc 2 p - f 2 gab 3 /cp + ghb 3 c 2 + g 2 b 4 c/e - f 2 habc-/j - f 2 gab 2 c*/j 
= ghb 2 c 2 (f + b) + g 2 b 4 (- f + c) p - f 2 gab 2 (b + c) K p 
= ghb 2 c 2 h + g 2 b 4 (- g) p + f 2 gab 2 a*p 
= gb 2 (h 2 c 2 - g 



708] NOTE SUR LA THEORIE DBS COURBES DE I/ESPACE. 13 

et de meme 



Done 



2 - X,Y= he 2 (f 2 a 2 - h 2 cV 
X F, - X l Y= f 2 a (g 2 b 2 - fVp 



- yit/j he 2 (f 2 a 2 - h 2 c 2 -sr 
a 

f 2 a (g 2 b 2 - f 2 a 2 /3 + h 2 c 2 CT/3) = 0, 



on enfin en multipliant par ai&gt;, et dans un terrae g 2 b 2 h 2 c 2 /Ayptzr/c, au lieu de 

ecrivant , 1 equation devient 
A 

(fa) 4 vp + (gb)* + (hc^ 1 - (gb) 2 (he) 2 (1 + i 

- (he) 2 (fa) 2 ^ ( + ^) - (fa) 2 (gb) 2 (i; + p) = 0, 
on, comme on peut 1 ecrire, 

2 , (gb) 2 , (hc) 2 ) 2 = 0. 

C est la deuxieme d un systems de trois equations e quivalentes ; savoir, en multipliant 
par -- et en reduisant par \PVGT &p = \, on obtient la premiere forme: et, en multipliant 
par \K et reduisant de me"me, on obtient la troisieme forme : le systeme est 



2j (gb) 2 , (hc) 2 ) 2 = 0, 



En ecrivant he = fa gb, on obtient une equation de la forme (*) (fa, gb) 4 = 0, 
savoir une Equation quartique pour avoir fa : gb, c est-a-dire, le rapport anharmonique 
(a d) (b c) : (b d) (c a) : en considerant a, b, c comme donnees, il y a done 4 
valeurs de d : et Ton a deja vu que les valeurs a, /3 ; 7, S sont donnees rationnelle- 
ment en fonctions de a, b, c, d: le theoreme est done demontre. 

Cambridge, juillet, 1880. 



14 [709 



709. 



ON THE NUMBER OF CONSTANTS IN THE EQUATION 
OF A SURFACE PS-QR = 0. 

[From the Tidsskrift for Mathematik, Ser. -t, t. iv. (1880), pp. 14-5148.] 

THE very important results contained in Mr H. Valentiuers paper "Xogle 
Ssetninger om fuldsta?ndige Skjaeringskurver mellem to Flader" may be considered 
from a somewhat different point of view, and established in a more simple manner, 
as follows*. 

Assuming throughout n &gt; p + q, p &gt; 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 

P,Q 
R, S 

so that the number of constants in the equation of a surface of the order n satisfying 
the condition in question is in fact the number of constants contained in an equation 
of the last-mentioned form. Writing for shortness 



the number of constants contained in a function of the order p is = a p + 1 ; or if 
we take one of the coefficients (for instance that of x p ) to be unity, then the number 

* Idet vi med stor Glsede optage Prof. Cayley s simple Forklaririg af den Reduktion af Konstanttallet i 
Ligningen PS-QR=Q, som Hr. Valentiuer havde paavist (Tidsskr. f. Math. 1879, S. 22), skulle vi dog 
bemaerke, at Grunden til, at dennes Bevis er bleven saa vanskeligt, er den, at ban tillige bar villet bevise, 
at der ikke finder nogen ydtrligere Reduktion Sted. 



709] ON THE DUMBER OF CONSTANTS IN THE EQUATION OF A SURFACE. 15 

of the remaining constants is = a p ; viz. a p is the number of constants in the equation 
of a surface of tb order p. As regards the surface in question 



P, Q 

R, S 



= 0, 



we may it is ciar take P, Q, R each with a coefficient unity as above, but in the 
remaining functin S, the coefficient must remain arbitrary : the apparent number of 
constants is thus= Op + a g + a_p +a,t_ 9 + 1 ; but there is a deduction from this number. 



The equatioi may in fact be written in the form 
P + *Q, Q 



8+J3Q 



= 0, 



where a represens an arbitrary function of the order p q, and /3 an arbitrary function 
of the degree n-p-q: we thus introduce (a p _ q + 1) + (an-p- 9 4-1), = a p - q + (in- p -. q + 2, 
constants, and b means of these we can impose the like number of arbitrary relations 
upon the constats originally contained in the functions P, Q, R, S respectively (say 
we can reduce o zero this number a p _ 9 + (in- v -q + 2 of the original constants) : hence 
the real number f constants is 

ap + tt q + d n _ p + dn_ q + 1 (a p _ q + a-,^ p ^ q + 2), 

1 



= &lt;o suppose ; 
viz. this is the rquired number in the case n &gt; p + q, p&gt;q- 

If however =p + q, or p = q, or if these relations are both satisfied, then there is a 

P Q I 
further deduction of 1, 1, or 2 : in fact, calling the last-mentioned determinant jL , 

then the four caes are 



S 



+ IP + kS + klR 
S + IR 



p + q, 



p&gt;y&gt; 


p / , q 


= 


P , Q 

R, S . 


p&gt;v&gt; 


p, q f \ = 


p + kR, q 
R, 


*-* 


F, q 


F, Q + kP 

R, S + kR 


*K 


P cy 

R S 

J~V j U 


= 


F + kR, q 
R , 



where k, I dence arbitrary constants: these, like the constants of a and /3, may be 
used to impose arbitrary relations upon the original constants of P, Q, R, S; and 
hence the numbr of constants is = &&amp;gt;, to - 1, &&amp;gt;-!, eo - 2 in the four cases respectively; 
where as above 



a f 



16 ON THE NUMBER OF CONSTANTS IN THE EQUATION OF A SURFACE. [709 

If n = 4, there is in each of the four cases one system of values of p, q ; viz. the 
cases are 

P&gt; ( 1 = 

21 No. = a s + a, + a, + a. - a, - a, - 1 = 9 + 3 + 9 + 19 - 3 - 3 - 1, = 33, 

31 a 3 + a, + a, + a, - a z - a - 2 = 19 + 3 -!- 3 + 19 - 9 - - 2, = 33, 
11 a 1 + a 1 + o s + a 3 -a -a 2 -2= 3 + 3 + 19 + 19 - - 9 - 2, = 33, 

22 a 2 +a, + a 2 + a 2 -a -a -3= 9 + 9 + 9 + 9 - - - 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 341, =33 as above: the reasoning is, in fact, the 
same as for the case of a plane passing through a line ; the line contains 4 constants, 
hence the plane, qua arbitrary plane through the line, would contain 1+4, =5 constants ; 
but the line being one of a doubly infinite system of lines on the plane the number is 
really 5 2, = 3, as it should be. 



Cambridge, 2nd Sept., 1880. 



710] 



17 



710. 

ON A DIFFEBENTIAL EQUATION. 

[From Collectanea Mathematical in memoriam Dominici Chelini, (Milan, Hoepli 1881) 

pp. 1726.] 

IN the Memoir on hypergeometric series, Crelle, t. xv. (1836), Kummer in effect 
considers a differential equation 

(a z* + 2b z + c) dz* = (ay? + 2bx + c) da? 
z*(z-iy a?(x-l)* . 

viz. he seeks for solutions of an equation of this form which also satisfy a certain 

differential equation of the third order. The coefficients a, b, c are either all arbitrary 

they are two or one of them, arbitrary; but this last case (or say the case 

where the function of x is the completely determinate function a? + 2bx + c) is scarcely 

considered: a , b , c are regarded as determinate in terms of a, b, c; and z is to 

found as a function of x independent of a, b, c: so that when these coefficients 

re arbitrary, the equation breaks up into three equations, and when two of the 

coefficients are arbitrary, it breaks up into two equations, satisfied in each case by 

he same value of ,; and the value of z is thus determined without any integration- 

these cases will be considered in the sequel, but they are of course included in the 

a b &gt; C are re S arded as havin S a y g^en values 



Writing for shortness X = ax* + 2bx + c, in general the integral 

f Ndx 
) D^/X 

where D is the product of any number n of distinct linear factors x-p and N is 
a rational and integral function of * of the order n at most, and therefore also the 
integral 



D 
C. XL 



[NXdx 
~J D^X 

3 



18 ON A DIFFERENTIAL EQUATION. [710 



a 



where N is now of the order n 2 at most, is expressible as the logarithm of 
quasi-algebraical function, that is, a function containing powers the exponents of which 
are incommensurable (for instance, x^ z is a quasi-algebraical function) : in fact, the integral 
is of the form 

//.. A B \ dx 

(M+- - + + 

J\ xp x q 

where each term is separately integrable, 

[dx _ J^ lo f b 

JVJ~ \/a gl 

dx I {(ap + b)x + (bp + c) + *JP- 

(x-p)*fit~ ~VP g l *-P 

where P is written to denote ap- + 2bp + c: the integral is thus = log H, where 
is a product of factors 



ax + b + ^a.^X, ^ ^^ - etc -&gt; 

x p 

raised to powers ^ , ^-= , etc. : hence, if we have a differential equation 
va vP 

N dz Ndx N ^Zdz _ N\/Xdx 

Ol "r^i ~~~ ~r\ 5 



where (= aV + 26 ^ + c ), and ^V, D are functions of z such as X, N, D are of 
# ; then, taking log C for the constant of integration, the general integral is 

log n = log c + log n : 

viz. we have the quasi-algebraical integral n (7O = 0. 

The constants a, b, c, p, q, ... etc. may be such that the exponents are rational, 
and the integral is then algebraical: in particular, for the differential equation 



+ I4&gt;z + I dz VarM-14r+ Idx 



z(z-\ x(x-\) 

the general integral is in the first instance obtained in the form 

(z + I + &lt;/Z) (z - 1 ) 2 _ (x + 1 + VZ) (a; - 1 ) 2 

" 



which, observing that (2#+ 2) 2 - X = 3 (aj - I) 2 , may also be written 

(* + 1) (^-34^ + 1) + ^^ = c (a?+l)(^-34a?+l) + 
V^ - I) 2 Va? (a- - 1)" 



710] ON A DIFFERENTIAL EQUATION. 19 

I had previously obtained the solution 






and I wish to show that this is, in fact, the particular integral belonging to the value 
C = 1 of the constant of integration : for this purpose I proceed to rationalise the general 
integral as regards z. 

Writing for a moment 

P = (z+l)(z"- 34* + 1), 



Q = (z- + Uz + 1) \/z + 14z + 1, 

R 
where 



O + 1) (& - 34a? + 1) + (a? 



the integral is P + Q + R = 0; or rationalising, it is 

(p, _ QJ _ 2 #2 ( p, + Q 2) + R = Q . 

we have 

P- = (l, -66, 1023, 2180, 1023, -66, l\z, I) 6 , 

Q- J = (1, 42, 591, 2828, 591, 42, l\z, 1), 
and thence 

P^-Q^ = (0, -108, 432, -648, 432, -108, 0\z, I) 6 , 



P 2 +Q 2 = 2(1, -12, 807, 2504, 807, - 12, l\z, If. 
Writing the equation in the form 



it thus becomes 

(1, -12, 807, 2504, 807, -12, l\z, 1) -z(z- I) 4 \M* + v -^(- = 0, 



where Jf has its above-mentioned value; and if we now assume (7=1, then 
(a? + 1) (a 8 - 34# + 1) + (a; 2 



108 = (as + 1) (a; 2 - 34a; + 1) - (^ + 14a; + l)Vfe"+ 14,-g +"l 

M V#O-1) 2 

and thence 

M , (108? = f^_10Sy _(^ + 1)^-34^+ I) 2 

2 : H * 



= ^ iy .(l, -12, 807, 2504, 807, -12, l"$x, I) 6 : 

32 



20 ON A DIFFERENTIAL EQUATION. [710 



and the rationalised equation is 

(1, -12, 807, 2504, 807, -12, \\z, I) 6 



#(# I) 
This is a sextic equation in z, of the form 



n _12, 807, 2504, 807, -12, ijx, 1) 6 = 0. 

4 



where 

X, /*, i/ = -12-fl, 807 + 4Q, 2504 -6O, 

if H denote the function of x which enters into the equation ; and writing z + - = 0, this 

2 

becomes 



But the equation in z is satisfied by the value z = x t and therefore the equation in 9 by 
the value 6 = x + - = a suppose, we have therefore 

10 

a 3 - 3a + X (a 2 - 2) + /* + v = 0, 
and thence subtracting, and throwing out the factor 6 a, 

0- 2 + 0a + a 2 -3 
viz. writing for X, p, a their values, this is 



x 
or, what is the same thing, 



where 

n= l -(1, -12, 807, 2504, 807, -12, Ija, I) 6 . 

t27 (A? ^~ J. J 

Hence in the quadric equation, the coefficients, each multiplied by (x- I) 4 , are 

12+i (1, -12, 807, 2504, 807, -12, l\x, I) 6 , 



00 

and 

12 1 



/ 12 1\ 

(x - I) 4 1 a? - 1 2 + 806 -- + ) 
\ x a?/ 

.^ + lVl, _12, 807, 2504, 807, -12, l$a?, I) 6 , 



which are respectively rational and integral quartic functions of x ; and, writing for its 
value, the equation finally is 

1, 188, 646, 188, \\x, I) 4 . (1, -644, 3334, -644, \\x, I) 4 

e**v. -=o 



710] ON A DIFFERENTIAL EQUATION. 21 

Writing 

4/- 1 - D * + rr 1 ~ * r 1 + * / --- 

f-v^ 4.-f+f. jB= r^r C= ir4 jD= r-^| (*=^-i as usual), 

this is 

^-J.OO-^O 

or, what is the same thing, 



that is, 

for we have 



i (A* _(!&gt; 28, 70, 28, 



(-!) 

4 , rwv (1. -28, 70, -28, 



And substituting these values, the coefficients will be rational functions of 4 , that is, of 
x, and it is easy to verify that they have in fact their foregoing values. 

It thus appears that for C=l, besides the values x and -, we have for z onlv the 

x J 

values 



viz. that the only solution is 



The example shows that although the differential equation 



V + 26^ + c dg Voa;-- 3 + Zbx + c dx 



can be integrated generally in a quasi-algebraical or algebraical form as above, yet 
we cannot from the general solution deduce, at once or easily, the various particular 
integrals comprised therein: nor can we find for what values of the constants a, b, c 
and a , b , c the differential equation admits of a simple solution, or say of a solution 
where z is expressed as an explicit (irrational) function of x. 

In the cases considered by Kummer there is a second (or it may be also a 
third) differential equation of the like form, the equations being each of them satisfied 
by the same value of z : hence eliminating the differentials dan, dz, the relation between 
x and z is of the form 

P _P 

Q ~Q 



22 ON A DIFFERENTIAL EQUATION. [7LO 

where P, Q are quadric functions of x\ P , Q quadric functions of z. But P and 
Q may contain a common factor, and the integral is then expressible in the form 

P 

x= r r, , the quotient of two quadric functions of z ; or P arid Q may have a common 

HJ 

p 

factor, and the integral is then expressible in the form z = ^ , the quotient of two 
quadric functions of x; or there may be a common factor of P, Q, and also a common 
factor of P and Q , and the integral is then of the form * = j^, the quotient of two 
linear functions of x. 

In the general case the differential equation is 

\(aP + bQ )dz* 

~~~ 



where a, b are arbitrary constants, X is a constant the value of which can in each 

P 

particular case be at once determined ; so when the integral is Z -Q&gt; tne differential 

equation is 

X (az + b) dz 2 _ (aP + bQ) dx z 
z*(z-I)* a?(x-].)- 

where a, b are arbitrary constants, but A, is now a linear function of z the value 
of which can in each particular case be at once determined. When the integral 

is z=^, the differential equation is 

X (az* + 26z + c) dz* _ (aL- + 2bLM + cM-} dx- 
z* (z-\f a- 2 (#-l) 2 

containing the three arbitrary constants a, b, c; X is a constant the value of which can 
be at once determined. 

There are in all 6 integrals of the form z = ^-,, for which the differential equation 

P I 
contains three arbitrary constants : 18 integrals of the form z = ^ f and of course the 

P \ P P 

same number of integrals of the form # = 7y)&gt; an d 9 integrals of the form -Q =/y&gt; f r a ^ 

of which the differential equation contains two arbitrary constants. It is to be remarked 
that Kummer, considering the values of z as a function of x, obtains the 72 rational and 
irrational values mentioned in his equations (31), (35), (36), (37), (38), and (39) : but the 
72 values are made up as follows, viz. the 18 values of z as a rational function of x, the 
36 irrational values obtained from the 18 expressions of x as a rational function of z, and 
the 18 irrational values of z obtained from the 9 integrals in which neither of the 
variables is a rational function of the other: 18 + 36 + 18 = 72. 



710] 



ON A DIFFERENTIAL EQUATION. 



23 



The several integrals together with the expressions of the functions 

a z 2 + Ib z + c and ax 1 + 2bx + c 
which enter into the differential equation are as follows : 

a z 2 + 2b z + c - ax 2 + 2bx + c = 



1. 



3. 



3 = 



X 

l-x 

1 

X 

1 


az 2 + 2bz + c 
j&gt; 

5) 

)) 
)) 


aa 2 + 26a? + c 
(^--l) 2 -26(-l) + c 

a + 2bx + ex 2 
a-2b(x-l) + c(x-iy 


l-x 

X 


x-l 
x-l 


a(xl) 2 + 2bx (x 1 ) + ex 2 


X 


(x+ ly 


az 2 + 6s 
&gt;&gt; 

5) 



n 

D 


a(x+ l) 2 + b(x-lf 
a (2x - I) 2 + 6 
a (x - 2) 2 + bx* 

a (x+ 1) 2 + 46ai 
(2x I) 2 + 4:bx (x - 1) 
a (Y *&gt;\3 4.A /-&gt;. I \ 


U- J 

(2.x- I) 2 

( x - 2 \ 

( x ) 

(x+ir 


4:X 

(2x-l 

402 (X 1) 

(x-Vf 


4(*-l) 




/*-iy 


6s + c 



M 

)) 
&gt;i 


b(x-iy + c(x + if 

b + c(2x-l) 2 


U+ 1) 
( l Y 


\2x-l) 

( x Y 


U-2; 

4:X 


4:bx + c(x + I) 2 
46a; (a; 1 ) + c (2aj - 1 ) 2 
-4b(x-l)+c(x-2f 


(x+iy 

4x(x-l) 

~(2x-iy 

4 (aj-1) 


(* -2) 2 



24 



z = 



ON A DIFFERENTIAL EQUATION. 

a z* + 2b z + c ay? 4- 2bx + c 



4. 



4a3 


as 2 - (a + c) z + c 


a (x I) 2 + 4ca: 


-4*(*-l) 





4aa; (oj 1 ) + c 


4(-l) 





4a (a; - 1 ) + ca; 2 


ar 2 


4x 


?&gt; 


4aw + c (a; - 1 ) 2 


(a;-!) 2 


-1 


5) 


a + 4raj (a; 1) 


4as(*-l) 


x 2 


!? 


aar 2 4c (a; 1) 


4(C- 1) 



5. 

6. [ same as 2, 3, 4 interchanging cc and s. 

7. 



9. 



10. 



2:2 



a (z I) 2 + 46z 
as 2 + 46 (s 1) 



4 (-!) = 



t ^_ 1 )^- 4a? ( a; -l) 
( 2 -l) 2 _ 4(a-l) 



-46(*-l) 



-!) 



r 2 



j - , 

4 (as- 1) 



4 - 



4az(z 



- 4a (a - 1) + 



c = 



4ax + 6 (a;- I) 2 

4a (a; 1) - 6ar 

+ 46a?(a;- 1) 



4aa; (a? 1) + b 
4ax (x 1 ) + b 
4a (x 1) + bx 2 



a(x- 1 ) 2 + 46a3 
aa; 2 46 (x 1) 



10] ON A DIFFERENTIAL EQUATION. 25 

The six functions of the set (1), that is, 

., 1 1 x x 1 

&gt; *^&gt; ~ &gt; 1 - &gt; - T~ ~~ 1 

x I as x 1 # 

form a group; and by operating with the substitutions of this group, and of the like 
group 

I 1 2 z-1 

~ 2 &gt; z &gt; i- 2 &gt; i^i&gt; - 2 ~&gt; 

upon any value of z in the sets (2), (3), (4), for instance upon z = ( ~V, we form all 

\*E 1 / 

the 18 functions of these sets. 

In any one of these sets (2), (3), and (4), comparing two forms (the same or 
different), for instance in the set (2), writing y for z and then in one form z for x, 

/a? + iy , /2 + i\" , 

= U-f) and -\f=V whence 

or 

) 2 , 

- whence = 



we obtain either the equations of the set (1) or those of the sets (8), (9) and (10); and 
whether we use the set (2), (3) or (4), the only new equations obtained are thus the 9 
equations of the sets (8), (9) and (10). These several equations present themselves 
however in different forms: for instance, instead of the equation 

(z 1) 2 _ 4&gt;ae 

~4z~ ~^l a 
we may obtain 



If, to get rid of this variety of form, we multiply out the denominators, the 9 
equations are 

0= x-z-- Zx-z- 2x2-+ x n --l2x2+ 2 2 - 2x- 22+ 1, 

= x *z- - Ifaz + IQx + IGz - 16, 

= 16#V - IQx-z - IQxz- + IGxz - 1, 

= a?z- - 2x 2 2 + x n -+ 16x2 - 162 

= 16x-2 - 16x2 - z- +2.0-1, 

0= lQx-2 -I6x*-16xz+ z + lQx 

0= x-z" 2x2- +16^+ 2--16x 

0= 16x2-- x 2 -16xz + 2x + 1, 

0= 16xz* + x 2 - 16x2 -162- +162 

These 9 equations are derivable all from any one of them by the changes of the set (1) 
upon x and z. 

Cambridge, 3rd June, 1879. 
C. XL A 



[711 



711. 



ON A DIAGRAM CONNECTED WITH THE TRANSFORMATION OF 

ELLIPTIC FUNCTIONS. 

[From the Report of the British Association for the Advancement of Science, (1881), p. 534.] 

THE diagram relates to a known theorem, and is constructed as follows. Consider 
the infinite half-plane y= + ; draw in it, centre the origin and radius unity, a 
semicircle; and draw the infinite half-lines # = -, and =J; then we have a 
region included between the lines, but exterior to the semicircle. The region in 
question may be regarded as a curvilinear triangle, with the angles 60, 60, and 0. 
The region may be moved parallel to itself in the direction of the axis of x, through 
the distance 1; say this is a "displacement"; or we may take the "image" of the 
region in regard to the semicircle. Performing any number of times, and in any 
order, these two operations of making the displacement and of taking the image, we 
obtain a new region, which is always a curvilinear triangle (bounded by circular 
arcs) and having the angles 60, 60, 0; and the theorem is that the whole series 
of the new regions thus obtained completely covers, without interstices or over 
lapping, the infinite half-plane. The number of regions is infinite, and the size of 
the successive regions diminishes very rapidly. The diagram was a coloured one, 
exhibiting the regions obtained by a few of the successive operations. 

ao&gt; + /3 
The analytical theorem is that the whole series of transformations, &lt;o into -$ , 

where a, /3, y, 8 are integers .such that a&-j3y&lt;=I, can be obtained by combination 
of the transformations to into o&gt; + 1 and o&gt; into . 



712] 27 



A PARTIAL DIFFERENTIAL EQUATION CONNECTED WITH THE 
SIMPLEST CASE OF ABEL S THEOREM. 



[From the Report of the British Association for the Advancement of Science, (1881). 

pp. 534, 535.] 

CONSIDER a given cubic curve cut by a line in the points (a?,, yj t ( Xn , y s ), 
03&gt; 2/3); taking the first and second points at pleasure, these determine uniquely the 
third point. Analytically, the equation of the curve determines y l as a function of 
x i, and 7/2 as a function of x. 2 : writing in the equation 

a? 3 = Xo?, + (1 - X) #o, y 3 = \y t + (1 - X) y st 

we have X by a simple equation, and thence x.,- viz. x, is found as a function of 
aii, a? 2 , and of the nine constants of the equation. Hence forming the derived equations 
(m regard to ac lt ar a ) of the first, second, and third orders, we have (1 + 2 +3 + 4=) 10 
equations from which to eliminate the 9 constants; x s , considered as a function of 
x, and a? s , thus satisfies a partial differential equation of the third order, independent 
of the particular cubic curve. 

To obtain this equation it is only necessary to observe that we have, by Abel s 
theorem, 

dXi dx 2 ,dx s _ 

y -T y T ^F~ v, 
1 ^2 -^3 

where X, is a given function of x, and y lt that is, of #, ; X 2 and X 3 are the like 
functions of # 2 and x s respectively. Hence, considering x, as a function of ^ and x. r 
we have 

^=_ dx^__X^ 
dx, X, dx,~ Z 2 

42 



2 g A PARTIAL DIFFERENTIAL EQUATION. [712 

and consequently 

dx 3 ^_^2. 
dx^ dx., Xi 

where X,, X, are functions of x,, x, respectively: hence taking the logarithm and 
differentiating successively with regard to x, and x,, we have 



which is the required partial differential equation of the third order. 

This differential equation has a simple geometrical signification. Consider three 
consecutive positions of the line meeting the cubic curve in the points 1, 2, 3; 
1 2 , 3 ; 1", 2", 3" respectively: qua equation of the third order, the equation 
should in effect determine 3" by means of the other points. And, in fact, the three 
positions of the line constitute a cubic curve; the nine points are thus the inter 
sections of two cubic curves, or, say, they are an "ennead" of points; any eight of 
the points thus determine uniquely the ninth point. 



713] 29 



713. 

ADDITION TO MR HOWE S MEMOIR ON ABEL S THEOREM. 

[From the Philosophical Transactions of the Royal Society of London, vol. 172, Part in. 
(1881), pp. 751758. Received May 27, Read June 10, 1880.] 

IN Abel s general theorem y is an irrational function of x determined by an 
equation ^ (y) 0, or say ^ (x, y} = 0, of the order n as regards y : and it was shown 
by him that the sum of any number of the integrals considered may be reduced to 
a sum of 7 integrals; where 7 is a determinate number depending only on the form 
of the equation x (*. y) = 0, and given in his equation (62), [(Euvres Completes, (1881), 
t. I. p. 168] : viz. if, solving the equation so as to obtain from it developments of y 
in descending series of powers of x, we have* 

a 

%!/*! series each of the form y = &lt;*?* + ..., 



* The several powers of x have coefficients: the form really is y = A l x^ + ..., which is regarded as 

J^ 

representing the MI different values of y obtained by giving to the radical x^ each of its MI values, and 
the corresponding values to the radicals which enter into the coefficients of the series: and (so understanding 
it) the meaning is that there are MJ such series each representing MI values of y. It is assumed that the 

_1 

series contains only the radical x 1 * 1 , that is, the indices after the leading index ^ are TOI ~ 1 l ~ 2 a 

4 a Mi Mi Mi 

series such as y = A 1 x* + B 1 x + ... , depending on the two radicals x^, x~ represents 15 different values, and 
would be written y-A-^x^ + ..., or the values of Wj and MI would be 20 and 15 respectively: in a case like 

7/1 

this where -- is not in its least terms, the number of values of the leading coefficient A l is equal, not to 
Mi, but to a submultiple of n l . But the case is excluded by Abel s assumption that ^, ^...., are fractions 

Ml /*o 

each of them in its least terms. 



30 ADDITION TO MR ROWE s [713 

(so that w = n,^ 1 + n a /i g +...+wt/A*), then 7 is a determinate function of n lt wi,, //,; 

Mr Rowe has expressed Abel s 7 in the following form, viz. assuming 
then this expression is 



7 = 

*&gt; 



or, what is the same thing, for n writing its value 

where in the first sum r, s have each of them the values 1, 2, ...,&, subject to the 
condition 6- &gt; r ; in each of the other sums n, m, and /z, are considered as having the 
suffix r, which has the values 1, 2, ..., k. 

It is a leading result in Riemann s theory of the Abelian integrals that 7 is the 
deficiency (Geschlecht) of the curve represented by the equation %(%, y) = 0: and it 
must consequently be demonstrable a posteriori that the foregoing expression for 7 is 
in fact = deficiency of curve % (x, y} = 0. I propose to verify this by means of the 
formulae given in my paper "On the Higher Singularities of a Plane Curve," Quart. 
Math. Jour., vol. vii., (1866), pp. 212223, [374]. 

It is necessary to distinguish between the values of which are &gt;, =, and &lt; 1 ; 
and to fix the ideas I assume k = 7, and 

?HI 7H 2 m 3 , i 
, , , eacn &gt; I, 

= 1 ; say m 4 = /u, 4 = X, and ?i 4 = 6 ; 



but it will be easily seen that the reasoning is quite general. I use 2 to denote 
a sum in regard to the first set of suffixes 1, 2, 3, and 2" to denote a sum in 
regard to the second set of suffixes 5, 6, 7. The foregoing value of n is thus 

n = 2V + \6 + 2 V. 

Introducing a third coordinate z for homogeneity, the equation %(#, y) = of 
the curve will be 



where it is to be observed that ( )"^ is written to denote the product of n^ 

W, MI 

different series each of the form yz^~ l - A^ - ... ; these divide themselves into n, 



713] MEMOIR ON ABEL S THEOREM. 31 

groups, each a product of /^ series; and in each such product the ^ coefficients A t 

!_ 

are in general the ^ values of a function containing a radical a^ and are thus 
different from each other: it is in what follows in effect assumed not only that this 
is so, but that all the n^^ coefficients A } are different from each other* : the like 

/ A\A 

remarks apply to the other factors. It applies in particular to the term [y #M 

A 

viz. it is assumed that the coefficients A in the \6 series y = Ax* + ... are all of 
them different from each other. These assumptions as to the leading coefficients 

really imply Abel s assumption that , ..., are all of them fractions in their least 

LL, ILl. 



\ 



terms, and in particular that - is a fraction in its least terms, viz. that \ = 1 : I 

A, 

retain hoAvever for convenience the general value X, putting it ultimately =1. 

In the product of the several infinite series, the terms containing negative powers 
all disappear of themselves ; and the product is a rational and integral function 
F(a, y, z) of the coordinates, which on putting therein z=l becomes = x(, y). 
The equation of the curve thus is F(ac, y, z) = ; and the order is 



viz. if K is the order of the curve x ( A&gt; &gt; 2/) = &gt; tnen K % nm + \0 + ^"np. 

The curve has singularities (singular points) at infinity, that is, on the line z = 



viz.- 



First, a singularity at (z = 0, x = 0), where the tangent is x 0, and which, 
writing for convenience y = 1, is denoted by the function 



/ *i y^i-M, 

where observe that the expressed factor indicates n l branches [jr "-*l , or 

_ w i^ 
say iHCmi-ft) partial branches z - a?" -*, that is, n, (m^ - ^} partial branches 

! 

z = A 1 x m i~^ + ..., with in all 11^(11^-^} distinct values of A^: and the like as regards 
the unexpressed factors with the suffixes 2 and 3. 

Secondly, a singularity at 0=0, 2/ = 0), where the tangent is y = 0, and which, 
writing for convenience xI, is denoted by the function 



* This assumption is virtually made by Abel, (I. c.) p. 162, in the expression "alors on aura en general, 
excepte quelques cas particuliers que je me dispense de considerer : h(y -y") = hy , &c.": viz. the meaning is 
that the degree of y being greater than or equal to that of y", then the degree of y -y" is equal to that 
of y" : of course when the degrees are equal, this implies that the coefficients of the two leading terms must 
be unequal. 



32 ADDITION TO MR HOWE S [713 



where observe that the expressed factor indicates branches \z - y** " 5 J , or 

_*5 

say n 5 (/4 B -m 6 ) partial branches z - y**~ m * , that is, *,(/!, -,) partial branches 

IN 

2 = ^5^-^ + ..., with in all n 5 (/* B -wi 5 ) distinct values of ^4 5 : and the like as 
regards the unexpressed factors with the suffixes 6 and 7. 

Thirdly, singularities at the points (z = 0, y-Ax = 0), A having here distinct 
values, at any one of which the tangent is y-Ax = 0, and which are denoted by 
the function 



but in the case ultimately considered X is = 1 ; and these are then the ordinary 
points at infinity, (z = 0, y Ax = 0). 

According to the theory explained in my paper above referred to, these several 
singularities are together equivalent to a certain number B + K of nodes and cusps; 
viz. we have 



= S(a-l), 

hence 



Assuming that there are no other singularities, the deficiency 






s&gt;r 



This should be equal to the before-mentioned value of 7 ; viz. we ought to have 

(K - 1) (K - 2) - M + 2 (a - 1) = ^n r m r n sf ji s + 2n-mfji - 2wm - 2w/4 - S?i + 2, 
or, as it will be convenient to write it, 
M = K-- %K + 2 (a - 1) - 

which is the equation which ought to be satisfied by the values of M and 
calculated, according to the method of my paper, for the foregoing singularities of 
the curve. 

We have as before 



s&gt;r 



The term 2n r m r n 8 p s , written at length, is 



+ 0\ ( n^^ 

+ n-,m 5 ( n 6 n 6 + n 7 /u 7 ) 



713] MEMOIR ON ABEL S THEOREM. 33 

which is 

= 2 / w r w r n g /A s + 6\ (S nw + 2 V) + 2 ?i?&gt;i . 2 V + "Z" 

s&gt;r s 

We have moreover 



2wm = 2 wra + 6\ + 2"nm, 

2V = 2V + 0X + 2 V&gt; 

2?i = 2 n + + 2"w. 

We next calculate 2 (a 1). 
For the singularity 



_ 

each branch U x m ^-^j gives a = TOJ ^, and the value of 2 (a 1) for this 

singularity is 

% (TO! - /A : - 1) + w 2 (m 2 - yu 2 - 1) + ?? 3 (m s - ^ - 1), 
which is 

= S wi?i 2 w/i S /?. 
For the singularity 



/ Mg \H5-ms 

each branch I j 7/^-n 5 J gives a = ^5 - ??i 5 , and the value of S (a 1) for this 

singularity is 

n s (p 5 - m s - 1) + w 6 (&gt; 6 - ??z g - 1 ) + w 7 (yu 7 - m 7 -I), 
which is 

= 2 V - 2"nm - 2 V. 
For each of the ^ singularities 



we have a = X and the value of 2 (a 1) is =0(X 1) : this is = for the value 
X = l, which is ultimately attributed to X. 

The complete value of 2 (a 1) is thus 

= S wm - 2"nm - 2V + 2 V ~ % n - 2"?i + 0\-0. 
Substituting all these values, we have 

M = (2 nm + 2 V) 2 + 26&gt;X (2W + 2 V) + (^) 2 

- 3 (2 nm + 2 V) ~ 30X 

+ 2 nm 2"rm 2 w/t + 2"7iyu. 2 w 2"w + ^X ^ 

2^ n r m r n s fjL s W\ (Z nrn + 2 V) ~ 22 nm . 2"w/A 22"?i r w r ?? s yu, s 

. &gt; r &gt; r 

- 2 tfm/j, - (6&gt;X) 2 - 2"wm/A 
+ 2 wrn + 0X + 2"wm 

+ 2V + 0* + 2 V 
+ 2 n + 6&gt; + 2"w, 
c. xi. 5 



34 ADDITION TO MR ROWE s [713 

or, reducing, 

M= (2nm - ^!nm Z tfm/j, 2^ n r m r n s fi s 



and it is to be shown that the two lines of this expression are in fact the values 
of M belonging to the singularities 



... f and * Jf" 

/ 

respectively. We assume \ = 1, and there is thus no singularity (y 

I recall that, considering the several partial branches which meet at a singular 
point, M denotes the sum of the number of the intersections of each partial branch 
by every other partial branch : so that for each pair of partial branches the inter 
sections are to be counted twice. Supposing that the tangent is x 0, and that for 
any two branches we have z 1 = A^x^, z z = A^pc^ (where p 1} p 2 are each equal to or 
greater than 1), then if p 2 = pi, and z 1 z 2 = (A l A^x^ where A A,, not =0 (an 
assumption which has been already made as regards the cases about to be considered), 
then the number of intersections is taken to be =p ; and if p 1 and p 2 are unequal, 
then taking p 2 to be the greater of them, the leading term of ^ z. 2 is = A^x^, and 
the number of intersections is taken to be =p l ; viz. in the case of unequal ex 
ponents, it is equal to the smaller exponent. 



/ _J2i_YiG"i-W 

Consider now the singularity (z - x m ^^j ...; and first the intersections of 

a partial branch z x m ^~^ by each of the remaining n l (m 1 fa) I partial branches 

777 

of the same set : the number of intersections with any one of these is = - ; 



A?7 

and consequently the number with all of them is = - - [MI (m^ /ij) 1]. But we 

mi /MI 

obtain this same number from each of the n 1 (m^ /^) partial branches, and thus the 
whole number is 

"?7? 

n, (m, - /&!&gt; - - Dh (w a - /ij) - 1], = n, m, [n, (Wj - ^) - 1]. 
m l fa 

Taking account of the other sets, each with itself, the whole number of such 
intersections is 

^1^1 [MJ (m l - /AI) 1] + w 2 m 2 [?? 2 ( 2 ^2) - 1] + n 3 m 3 [n 3 (m 3 fi 3 ) 1], 
which is 



713] MEMOIR ON ABEL S THEOREM. 35 

Observe now that &gt; , that is, D &lt; c2 1 an( j that, these being each &lt; 1 we 

/^i A^2 iYl&gt;\ TLi) 

thence have 1 &gt; 1 - Ma , that is, -3 ^ &gt; ^ 2 : and we thus have 



Considering now the intersections of partial branches of the two sets 



m l \ni (M|^t|} / 7/3 

TO 1 J I 

3C l J 3/HCl I .S ^/ 2 

?^ } 

respectively, a partial branch s - a;^i-^. gives with each partial branch of the other 
set a number = -_- ; and in this way taking each partial branch of each set, 
the number is 

??! (m l /AJ) . n* (m 2 fj, 2 ) . , = n^rt^n^ (m. 2 /i 2 ) : 

and thus for all the sets the number is 
which is 



where in the first sum the 2 refers to each pair of values of the suffixes. But the 
intersections are to be taken twice ; the number thus is 



Adding the foregoing number 

2 n 2 rn 2 S / n*0t/u % 

the whole number for the singularity in question is 

= (2 nm) 2 - S rcra - S 



8&gt;r 



(&lt;* \%(M;-m 5 ) 
^_y^-m 5 J &lt;&gt;&gt;; taking each set with itself, the 

number of intersections is 



[n s (fi 8 - r 6 ) - 1] + n,/^ [n 7 (& -m?}- I], 
which is 

KV/ o K 1 // o ^&lt;// 

= ft n-/j.- z wmfji 2 TI/A. 

52 



36 ADDITION TO MR ROWE s MEMOIR ON ABEL S THEOREM. [713 

We have here 5 &gt; 6 ; each of these being less than 1, we have 1 -&lt;!- ^ , 

fJ S A*6 ^ 5 ^ 6 

that is, Mg ~ Wg &lt;^ g ^ : , or ^ &gt; ^^ ; and so 



Hence considering the two sets 

.% (M 6 -W e ) 



yLlg 

a partial branch of the first set gives with a partial branch of the second set _ m 
intersections: and the number thus obtained is 



n 5 (/*, - ra 5 ) . n 6 (/*, - m 6 ) . , = w 5 w 6 /* 6 (p s - m 5 ). 
MB % 

For all the sets the number is 



or taking this twice, the number is 



where in the first sum the 2" refers to each pair of suffixes. Adding the foregoing 
value 

S"n&gt; 2 - 2 Vm/i - 2 V, 

the whole number for the singularity in question is 



s&gt;r 



and the proof is thus completed. 



Referring to the foot-note (ante, p. 31), I remark that the theorem 7 = deficiency, 
is absolute, and applies to a curve with any singularities whatever : in a curve which 
has singularities not taken account of in Abel s theory, the "quelques cas particuliers 
que je me dispense de considerer," the singularities not taken account of give rise 
to a diminution in the deficiency of the curve, and also to an equal diminution of 
the value of 7 as determined by Abel s formula; and the actual deficiency will be 
= Abel s 7 such diminution, that is, it will be = true value of 7. 



714] 37 



714. 

VARIOUS NOTES. 

[From the Messenger of Mathematics, vol. vn. (1878), pp. 69, 115, 124, 125.] 

An Identity. 

THE following remarkable identity is given under a slightly different form by 
Gauss, Werke, t. in., p. 424, 

3 



then 



On two related quadric functions. 

Assume 

fa = a 2 (c - a?) - ar (c 2 - 6 2 - car), 

^ra; = J 2 (c x) X (c 2 O? CX) . 



U~ 

In the first of these for oc write - ; then 

c x 



a?(c-x)\ a*(c-x? _a 2 6_ 

- ^ - * 2 9 1 



38 VARIOUS NOTES. [714 

A Trigonometrical Identity. 

cos (b - c) cos (b + c + d) + cos a cos (a + d) 

i 

= cos (c a) cos (c + a + d) + cos b cos (6 + d) 
= cos (a - b) cos (a + b + d) + cos c cos (c + d) 
= cos a cos (a + d) + cos b cos (b + d) + cos c cos (c + d) cos d. 



Extract from a Letter. 
" I wish to construct a correspondence such as 

(x + iy) x + O + iy) = X + iY, 
or, say, for greater convenience 

4 (a? + iy) s - 3 (x + iy) = X + iY; 
viz. if 

x + iy = cos u, 
then 



Suppose 3 is a value of 3u corresponding to a given value of X + i Y, then the 

three values of x + iy are of course cos^ , cos(w -^-); but I am afraid that the cal- 

V o / 

culation of , even with cosh and sinh tables, would be very laborious. Writing 

X + iY = R (cos + i sin ), 

the intervals for might be 5, 10 or even 15, those of R, say O l from to 2, 
and then 5 up to 4 or 5 ; and 2 places of decimals would be quite sufficient ; but 
even this would probably involve a great mass of calculation. 

It has occurred to me that perhaps a geometrical solution might be found for 
the equation X + iY= cos 3u." 

October 31, 1877. 



715] 39 



715. 

NOTE ON A SYSTEM OF ALGEBEAICAL EQUATIONS. 

[From the Messenger of Mathematics, vol. vn. (1878), pp. 17, 18.] 

ASSUME 

x + y +z =P, 

yz + zx + xy = Q , 

xyz R, 

A=x (nyz + Q)-w 2 (mx + P\ 
B =y (nzx + Q) W 2 (my + P\ 
C =z (nxy + Q) up (mz + P), 

= - mnR + PQ. 
Then 

(mz + P)B-(my + P)C 

= (myz + Py) (nzx + Q) (myz + Pz) (nxy + Q) 

= myz (nzx + Q nxy Q) + Pnxyz + PQy Pnxyz PQz 

= mnxyz (z y) PQ (z y) 

= (z-y} {mnxyz - PQ} =(y-z)@; 

whence, identically, 

(mz +P)B - (my + P)C = (y-z)&, 
(mx + P)C - (mz +P)A=(z-ae)&, 
(my + P)A- (mx + P)B =(x-y)S. 

Hence any two of the equations .4 = 0, B = 0, (7 = imply the third equation. 



40 NOTE ON A SYSTEM OF ALGEBRAICAL EQUATIONS. [715 

We have 

A=x {(n + l)yz + zx + ocy}-w z {(m + 1) x + (y + 2)} 

= (x- - w 2 ) (y + z) - x [(m + 1) w z - (n + 1) yz\ 
and similarly for B and G. The three equations therefore are 

x 



_ _ 
x- w z ~ (m + 1) w* (n + 1) yz 



x + y 



z^- uf- (m + 1) w*-(n 
and any two of these equations imply the third equation. 



716] 41 



716. 

AN ILLUSTKATION OF THE THEORY OF THE ^-FUNCTIONS. 

[From the Messenger of Mathematics, vol. vii. (1878), pp. 27 32.] 

IF X be a given quartic function of x, and if u, or for convenience a constant 

multiple au, be the value of the integral I -^ taken from a given inferior limit to 

J v \-A- ) 

the superior limit as; then, conversely, x is expressible as a function of u, viz. it is 
expressible in terms of ^-functions of u, where ^u, or say %(u, g) (g a parameter 
upon which the function depends), is given by definition as the sum of a series of 

exponentials of u; and it is possible from the assumed equation * f-J^r, and the 

J vv&) 

definition of ^n, to obtain by general theory the actual formulas for the determination 
of x as such a function of u. 

I propose here to obtain these formulae, in the case where X is a product of 
real factors, in a less scientific manner, by connecting the function &lt;bu (as given by 

such definition) with Jacobi s function @, and by reducing the integral t-L- by a 

^ ^ 

linear substitution to the form of an elliptic integral; the object being merely to 

obtain for the case in question the actual formulae for the expression of x in terms 
of ^-functions of u. 

The definition of *bu or, when the parameter is expressed, S- (u, $) is 



where s has all positive or negative integer values, zero included, from - oc to + oo 

(that is, from - S to + 8, S = oo ) ; the parameter g, or (if imaginary) its real part, 
must be positive. 

C. XL 6 



OF THE 

UNIVERSITY 



42 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. [716 

Evidently ^w is an even function: *r(-u*) = ^u. Moreover, it is at once seen that 
we have 



S- ( u + TT) = &lt;$u, *(u + t ft) = - e%~ 2iu STM, 

whence also 

S- (u + m-rr + raft), 

where m and w are any positive or negative integers, is the product of &lt;bu by an 
exponential factor, or say simply that it is a multiple of *bu. 

Writing u = -$ift, we have ^ (- i8) - ^ (i*8), that is &gt; 
and therefore also 



The above properties are general, but if ft be real, then k, K, K , q being as in 
Jacobi (consequently k being real, positive, and less than 1, and K and K real and 



positive), and assuming ft = -^- , or, what is the same thing, 



i%Ku\ 
the function S- is given in terms of Jacobi s by the equation !h~8f - V; or, 



what is the same thing, w = ^ ^-^ } . 

\Z/L/ 



We hence at once obtain expressions of the elliptic functions sn u, en u, dn u in 
terms of ^, viz. these are 



cn 



//^ \ 

- V U) 



Consider now the integral 

dx 



f dx f 

-77^^ -^r~ ji = 

J a v{( ) CL.X b.x c.x a} J a 



su PP ose . 



where a, b, c, d are taken to be real, and in the order of increasing magnitude, viz. 
it is assumed that b -a, c -a, d - a, c - b, d - b, d - c are all positive; x considered 
as the variable under the integral sign is always real; when it is between a and b 
or between c and d, X is positive, and we assume that */(X) denotes the positive 
value of the radical; but if x is between b and c, X is negative, and we assume 



716] AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. 43 

that the sign of \/(X) is taken so that -77-^ is equal to a positive multiple of i, 

TV* / 

and this being so the integral is taken from the inferior limit a to the superior 
limit x, which is real. 

Take x a linear function of y, such that for 

x = a, b, c, d, 

y = 0, 1, ri , x , respectively, 

so that, x increasing continuously from a to d, y will increase continuously from to x . 
We have 

b a . d c 
~d b . c a 

b d x a 
b a x d 

_d a x b 
b ax d 

, 2 _d ax c _ 
c a x d 
and, thence, 



y * = - A - r rr &gt; 

c - a V Vc - a/ (x - d? 

where A /( ------ ) is taken to be positive, and the sign of \f(X) is fixed as above. Then 

V Vc - a/ 

for y between and 1 or &gt; , 2 , y . 1 y . 1 k 2 y will be positive, and *J(y . 1 y . I k 2 y) 
will also be positive ; but y being between 1 and y^ , y.ly.I k*y will be negative, 

K 

and the sign of the radical is such that -r, = - = - r^ is a positive multiple of i. 

- y - 1 - 



We have moreover 

7 d a , , ,. dx 
CM/ = 7 (d b)-, -- = 
y 6 - a v (fB^-dy 

and therefore 

7 
- 6 . c - a) 



where *j(d b.c a) is positive ; or, say, 

dy ., , . [ dx 
= &gt;J(d b.c ft) I 



62 



44 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. [716 

Hence, writing y = z z = sn 2 u, we have 

\ r ^ 

2w = V( b.c a) 

i a 

and it is to be further noticed that to 

x = a, b, c, d, 
correspond 

sn u= 0, 1, T , oo , 

or we may say 

u = 0, K, K+iK , 

Writing for shortness 

2 

= &gt; 



b. c a) 
we have 

dx 



and moreover 



or if for a moment we write 

r dx 

~WY\ = A &gt; &c " 
J v(^) 

then these equations are 

aK = B -A, 



Hence B+ C-2A = D-A, that is, A -B- C + D = 0, or B-A=D-C, that is, 



where observe as before that x= a to x = b, or as = c to x= d, X is positive, and the 
radical &gt;J(X) is taken to be positive. 



We have also 

fb 

oK = B- A = 



716] AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. 45 

where, as before, from b to c, X is negative, and the sign of the radical is such that 
_ is a positive multiple of i; the last formula may be more conveniently written 



, 

dx 



where, from b to c, X is positive, and \J(X} is also taken to be positive. 
Collecting the results, we have 

dx 2 7o b - a.d-c 



b . c ct) d b . c a 

and also 



d b . c a 
and then conversely 



a (d b) + d (b a) sn 2 u _ 

& 7~5 T~\ . " / 7 \ rt 1 



or, what is the same thing, 

b d . x a 

sn 2 u = , , , 

b a . x d 

d a . x b 
b a . x d 

d a . x c 
c a . x d 



en- 5 u = 



dn 2 w = 



where, in place of the elliptic functions we are to substitute their ^--values; it will 
be recollected that $, the parameter of the ^--functions, has the value 



dx f b dx 

\ K J j b V(~~ X) 
and, as before, 



Hence, finally, a, ^, k , K, denoting given functions of a, &, c, d, if as above 



we have conversely 

b d. x a 
b d . x d 



c-a.x-d 
which are the formulae in question. 



_ , 

K 



46 AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. [716 

The problem is to obtain them (and that in the more general case where a, b, c, d 
have any given imaginary values) directly from the assumed equation 



dx 

= au, 



and from the foregoing definition of the function S-. 

It may be recalled that the function *bu is a doubly infinite product 

u 

TTiTT + (n + ) 

m and n positive or negative integers from oo to +00; I purposely omit all further 
explanations as to limits; or, what is the same thing, 



iK \ 



and consequently that, disregarding constant and exponential factors, the foregoing 
expressions of 

b d.ac a d a.x b d a.x c 

b a.x d ba.x d ca.x d 

X Y Z 

are the squares of the expressions ^., ., -, where X, Y, Z, W are respectively of 

the form 

( i 

M nn + 7 -- , nn 



(m, n)} ( (m, n}\ 



\ I ) -i.AJ.j.\-i. i t . i j 

m, n)j { (m, n)\ 

where (m, n) = 2mK + 2niK , and the stroke over the m or the n denotes that the 
2m or the 2n (as the case may be) is to be changed into 2m + 1 or 2n -f 1. But 
this is a transformation which has apparently no application to the ^--functions of 
more than one variable. 



717] 47 



717. 



ON THE TEIPLE THETA-FUNCTIONS. 



[From the Messenger of Mathematics, vol. vn. (1878), pp. 48 50.] 

As a specimen of mathematical notation, viz. of the notation which appears to 
me the easiest to read and also to print, I give the definition and demonstration of 
the fundamental properties of the triple theta-functions. 

Definition. 

*(U, V, F) = Sexp. @, 
where 

= (A, B, C, F, G, H) (I, m, n)* + 2(U, V, W)(l, m, n), 

2 denoting the sum in regard to all positive and negative integer values from 
ao to +00 (zero included) of I, m, n respectively. 

^- ( U, V, W) is considered as a function of the arguments ( U, V, W), and it 
depends also on the parameters (A, B, C, F, G, H). 

First Property. ^(U, V, F) = 0, for 

U = %{x7ri + (A, H, G)(, ft, 7)}, 
V = ^{yTri + (H, B,F)(*,0, 7)}, 
W=${z7ri + (G, F, G)(a, 0, 7)}, 

x, y, z, a, /3, 7 being any positive or negative integer numbers, such that ax + fty + yz 
= odd number. 

Demonstration. It is only necessary to show that to each term of S- there corre 
sponds a second term, such that the indices of the two exponentials differ by an odd 
multiple of iri. 



48 ON THE TRIPLE THETA-FUNCTIONS. [71 7 

Taking I, m, n as the integers which belong to the one term, those belonging to 
the other term are 



(where observe that one at least of the numbers a, /3, 7 being odd, this system of 
values is not in any case identical with I, m, n). The two exponents then are 

@, =(A, B, C, F, G, H)(l, m, n)*+2(U, V, W)(l, m, n), 
and 

, = (4, B, C, F, G, H)(l + a, m + & n + y}*-2(U, V, W)(l + a, ml/3, w + 7 ); 



viz. the value of is 

= (A, B, C, F, G, H}(1, m, iif + (A, B, C, F, G, #)(, /3, 7 ) 2 
+ 2(4, B, C, F, G, H)(l, m, n)(a, 0, 7) 
- 2 (U, V, W)(l + a, m+/3, n + 7), 

and we then have 

-0 = 2(4, B, C, F, G, H)(l, m, ?i) (a, /?, 7) 

+ (4, B, C, F, G, H}(CL, /3, 7 ) 2 
-2(U, V, W)(2l+a, 2m + @, 2?i+ 7 ). 

Substituting herein for U, V, W their values, the last term is 

= - {(21 + a) x + (2m + (3)y + (2n + 7) z] 
-2(4, B, C, F, G, H)(l, m, n)(, & 7) 
- (4, B, C, F, G, H}(a, 0, 7 ) 2 &gt; 
and thence 

_ = _ {(ft + a ) x + (2m + /3) y + (2w + 7) ^} TTI, 

which proves the theorem. 

As to the notation, remark that, after (4, B, C, F, G, H) has been once written 
out in full, we may instead of 

(4, B, C, F, G, H}(1, m, n^, &c., write (4, ...)(, m, r&lt;) 2 , &c., 
and that we may use the like abbreviations 

(4, ...)(l, m, n), to denote (4, H, G)(l, m, n) respectively, 
(H, ...)(l, m, n), (H, B, F) (I, m, n) 
(G, ...)(l, m, n), (G, F, C) (I, m, n) 

These are not only abbreviations, but they make the formulas actually clearer, as 
bringing them into a smaller compass; and I accordingly use them in the demonstration 
which follows. 



ON THE TRIPLE THETA-FUNCTIONS. 49 

Second Property. If U lt F 1; W^ denote 

xiri+(A, H, )(, 13, 7), 
(H, B, F)(a, /3, 7), 
(Q, F, C) (a, /3, 7), 

respectively, where x, y, z, a, 0, 7 are any positive or negative integers (zero values 
admissible), then 



.* f (U, V, W), 



or say 



= exp. {-(A, ...)(, 13, 7) 2 }-exp. {- 2( 



, F, F). 



Demonstration. Writing *(#;, F lf TTO = 2 . exp. 1; then in the expression of t 
we may in place of I, m, n write I -a, m-0, n-y; we thus obtain 



- a 



which is 



= (A, ...)(l, m, nf 



y W t + ( J ff, ...)(, 7)] 
+ (n -7) [W + m+(G, ...)(, A 7)]}. 



/, m, n) (a, & 7 ) 
;, m, n)(a, /3, 7) 



+ (^ .)(, A 7) 2 &gt; 
2(lU+mV+nW 



which is 



-(A, ...)(, A 7) 2 - 

+ 2 [(^ - a) a; + ( m -/3)y+ (n - 7) 0] m. 

Hence, rejecting the last line, which (as an even multiple of vn) leaves the exponential 
unaltered, we see that *(U lt F lf W,) is =^(JT, F, TF) multiplied by the factor 



. {-(A, ...)(, /3, 7) 2 }.exp. {- 2 (^+ /9F+ 7 F)}, 
which is the theorem in question. 

In many cases a formula, which belongs to an indefinite number s of letters is 
most easily intelligible when written out for three letters, but it is sometimes con 
venient to speak of the s letters l,m,..., n, or even the s letters l,..., n , and to write 
out the formulae accordingly. 



C. XI. 



50 [718 



718. 



ADDITION TO MR GENESE S NOTE ON THE THEORY 

OF ENVELOPES. 

[From the Messenger of Mathematics, vol. vii. (1878), pp. 62, 63.] 

THE example, although simple, is an instructive one. Introducing z, p for 
homogeneity, the equation is 



\ 2 y (y bz) + 2\/*xy + fj?x (x az) = 0, 

giving the envelope 

xy [(x - az} (y - bz) - xy] = ; 

that is, 

xy (bx + ay abz) z = ; 

viz. we have thus the four lines 



Writing these values successively in the equation of the curve, we find respectively 

\-y (y - bz) = 0, 
fj^ac (as az) = 0, 

(&X- a^l-0, 

(\y + fjuK) 2 = ; 



viz. in each case the equation in \, /j, has (as it should have) two equal roots; but 
in the first three cases the values are constant ; viz. we find \ = 0, /u, = 0, 6\ a/u, = 0, 

dC \l 

respectively ; and the curves x = 0, y = 0, - + r z = 0, are for this reason not proper 
envelopes. 



718] ADDITION TO MR GENESE s NOTE ON THE THEORY OF ENVELOPES. 51 

It is to be remarked that writing in the equation of the parabola these values 
X = 0, /u = 0, b\-ap = successively, we find respectively 

x(x az) = 0, 

y(y- fa) = o, 

(bx + ay) (&. ,- + 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 X, /j, is (\y + /j,x)~ = 0, giving a variable 
value \^-fju = x-^y; hence 2 = 0, the line at infinity is a proper envelope. 

The true geometrical result is that the envelope consists of the three points A, B, C, 
and the line at infinity; a point qua curve of the order 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 j(Xdx+Ydy) ALONG AN ARBITRARY 
PATH ". 

[From the Messenger of Mathematics, vol. vn. (1878), pp. 9295 ; British Association 

Report, 1877, pp. 1820.] 

I CONSIDER an integral f(Xdx+Ydy), where X, Y are each of them a given 

function of the variables (as, y)- Xdx+Ydy is thus not in general an exact differential; 
but assuming a relation between (x, y\ that is, a path of the integral, there is in 
effect one variable only, and the integral becomes calculable. I wish to show how 
for any given values of the functions X, Y, but for an arbitrary path, it is possible 
to construct a mechanism for the calculation of the integral : viz. a mechanism such 
that, a point D thereof being moved in a plane along a path chosen at pleasure, the 
corresponding value of the integral shall be exhibited on a dial. 

The mechanism (for convenience I speak of it as actually existing) consists of a 
square block or inverted box, the upper horizontal face whereof is taken as the plane 
of xy, the equations of its edges being y = 0, y=l, x =0 &gt; x = i respectively. In the 
wall faces represented by these equations, we have the endless bands A, A , B, E 
respectively; and in the plane of xy, a driving point D, the coordinates of which are 
(x, y\ and a regulating point R, mechanically connected with D, in suchwise that 
the coordinates of R are always the given functions X, Y of the coordinates of f ; 
the nature of the mechanical connexion will of course depend upon the particular 
functions X, Y. 

This being so, D drives the bands A and B in such manner that, to the given 
motions dx, dy of D, correspond a motion dx of the band A and a motion dy of 

* Bead at the British Association Meeting at Plymouth, August 20, 1877. 

t It might be convenient to have as the coordinates of R, not X, Y but f, ,, determinate functions of 
X, Y respectively. 



719] SUGGESTION OF A MECHANICAL INTEGRATOR. 53 

the band B; A drives A with a velocity-ratio depending on the position of the 
regulator R in suchwise that, the coordinates of R being X, Y, then to the motion 
da; of A corresponds a motion Xdx of A ] and, similarly, B drives B with a 
velocity-ratio depending on the position of R, in suchwise that to the motion dy of 
B corresponds a motion Ydy of B . Hence, to the motions dx, dy of the driver D, 
there correspond the motions Xdx and Ydy of the bands A and B respectively ; 
the band A drives a hand or index, and the band B drives in the contrary sense 
a graduated dial, the hand and dial rotating independently of each other about a 
common centre ; the increased reading of the hand on the dial is thus = Xdx + Ydy ; 
and supposing the original reading to be zero, and the driver D to be moved from its 
original position along an arbitrary path to any other position whatever, the reading on 

the dial will be the corresponding value of the integral I (Xdx + Ydy). 

It is obvious that we might, by means of a combination of two such mechanisms, 
calculate the value of an integral If (if) du along an arbitrary path of the complex 
variable u, = x + iy ; in fact, writing f(x + iy) = P + iQ, the differential is 

(P -(- iQ) (dx + idy), = Pdx - Qdy + i (Qdx + Pdy) ; 
and we thus require the calculation of the two integrals 

f(Pdx-Qdij) and ^(Qdx + Pdy), 

each of which is an integral of the above form. Taking for the path a closed curve, 
it would be very curious to see the machine giving a value zero or a value different 
from zero, according as the path did not include or included within it a critical 
point; it seems to me that this discontinuity would really exhibit itself without the 
necessity of any change in the setting of the machine. 

The ordinary modes of establishing a continuously-variable velocity-ratio between two 
parts of a machine depend upon friction ; and, in particular, this is the case in Prof. 
James Thomson s mechanical integrator there is thus of course a limitation of the 
driving power. It seems to me that a variable velocity-ratio, the variation of which is 
practically although not strictly continuous, might be established by means of toothed 
wheels (and so with unlimited driving power) in the following manner. 

Consider a revolving wheel A, which by means of a link BC, pivoted to a point B 
of the wheel A and a point C of a toothed wheel or arc D, communicates a reciprocating 
motion to D; the extent of this reciprocating motion depending on the distance of B 
from the centre of A, which distance, or say the half-throw, is assumed to be variable. 
Here during a half-revolution of A, D moves in one direction, say upwards; and 
during the other half-revolution of A, D moves in the other direction, say downwards ; 
the extent of these equal and opposite motions varying with the throw. Suppose 
then that D works a pinion E, the centre of which is not absolutely fixed but is so 
connected with A that during the first half-revolution of A (or while D is moving 
upwards), E is in gear with D, and during the second half-revolution of A, or while 



54 SUGGESTION OF A MECHANICAL INTEGRATOR. [719 

D is moving downwards, E is out of gear with D ; the continuous rotation of A 
will communicate an intermittent rotation to E, in such manner nevertheless that, to 
each entire revolution of A or rotation through the angle 2-rr, there will (the throw 
remaining constant) correspond a rotation of E through the angle n.Zir, where the 
coefficient n depends upon the throw*. And evidently if A be driven by a wheel 

A , the angular velocity of which is - times that of A, then to a rotation of A 



2-7T 



through each angle , there will correspond an entire revolution of A, and therefore, 

as before, a rotation of E through the determinate angle n . 2?r ; hence, X. being 
sufficiently large to each increment of rotation of A , there corresponds in E an 
increment of rotation which is ?i\ times the first-mentioned increment; viz. E moves 
(intermittently and possibly also with some "loss of time" on E coming successively 
in gear and out of gear with D, or in beats as explained) with an angular velocity 
which is = n\ times the angular velocity of A . And thus the throw (and therefore n) 
being variable, the velocity-ratio n\ is also variable. 

We may imagine the wheel A as carrying upon it a piece L sliding between guides, 
which piece L carries the pivot B of the link EG, and works by a rack on a toothed 
wheel a concentric with A, but capable of rotating independently thereof. Then if a 
rotates along with A, as if forming one piece therewith, it will act as a clamp upon L, 
keeping the distance of B from the centre of A, that is, the half-throw, constant ; whereas, 
if a has given to it an angular velocity different from that of A, the effect will be to 
vary the distance in question; that is, to vary the half-throw, and consequently the 
velocity-ratio of A and E. And, in some such manner, substituting for A and E the 
bands A and A of the foregoing description, it might be possible to establish between 
these bands the required variable velocity-ratio. 

* If instead of the wheel or arc D with a reciprocating circular motion, we have a double rack D with a 
reciprocating rectilinear motion, such that the wheel E is placed between the two racks, and is in gear on the 
one side with one of them when the rack is moving upwards, and on the other side with the other of them 
when the rack is moving downwards; then the continuous circular motion of A will communicate to a 
continuous circular motion, not of course uniform, but such that to each entire revolution of A or rotation 
through the angle 2ir, there will correspond a rotation of E through an angle n.lir as before. This is in 
fact a mechanical arrangement made use of in a mangle, the double rack being there the follower instead of 
the driver. 



720] 55 



720. 

NOTE ON ARBOGAST S METHOD OF DERIVATIONS. 

[From the Messenger of Mathematics, vol. vn. (1878), p. 158.] 

IT is an injustice to Arbogast to speak of his first method, as Arbogast s method*. 
There is really nothing in this, it is the straightforward process of expanding 



7 7Q 7 

by the differentiation of &lt;f&gt;u, writing a, b, c, d, ... in place of u, ^- , ^- , ^ , &c. or 
say in place of u, u , u", u ", &c. respectively; thus 

cfjft, (f&gt; a . b, ^ {fy u c + &lt;f)"a . b 2 }, ^ (4* ^ ~^~ ( f ) " ^ 

( +&lt;f&gt;"a.2bc + (f&gt; "a.b 3 

= {fia . d + &lt;f&gt;" a . 3bc + &lt;f&gt; "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 &lt;f&gt; (a + bx + cx~ + da? +...), 
are obtained by the rule of the last and the last but one ; thus we have 



&lt;j&gt;a, &lt;f) a .b, &lt;f&gt;a.c + fi a . ^b 2 , &lt;j&gt;a.d + &lt;f&gt;"a . be + &lt;j&gt; "a. ^b 3 , &c., 
where each numerical coefficient is found directly, without an addition in any case. 

* See Messenger of Mathematics, vol. vn. (1878), pp. 142, 143. 



56 



[721 



721. 

FORMULA INVOLVING THE SEVENTH ROOTS OF UNITY. 

[From the Messenger of Mathematics, vol. vn. (1878), pp. 177 182.] 

LET tw be an imaginary cube root of unity, &&amp;gt; 2 + w + I = 0, or say w = |- {- 1 +i V(3)} ; 
a 3 = -7(l+3&&amp;gt;), /3 3 = - 7 (1 + 3&lt;o 2 ), values giving a 3 /3 3 =343, and the cube roots a, ft 

being such that a/3 = 7 ; then a + /3, = a + -, is a three-valued function (since changing 

7\ 
the root &&amp;gt; we merely interchange a and -J ; and if r be an imaginary seventh root 

of unity, then 

3 (r + r 6 ) = a + p-1, 



3 (r 4 + r 3 ) = a&gt; 2 a + a&gt;/3 - 1. 

Any one of these formulae gives the other two; for observe that we have 3 = -a/3(l + 3&lt;w), 
/3 3 = - a/3 (1 + 3&lt;y 2 ), that is, 2 = -/3(l +3w), /3- = -a(l + 3a&gt; 2 ); hence, starting for instance 
with the first formula, we deduce 



= - /3 (1 + 3(w) + 14 - a (1 + 3&lt;y 2 ) - 2a - 2/3 -f 1, 
= - a (3 + 3a&gt; 2 )-/3(3 + 3a&gt;) + 15, 
15, 



that is, 

3 (r- + r 5 ) = too. + w 2 /3 - 1 ; 

and in like manner by squaring each side of this we have the third formula 

3 (r 4 + r 3 ) = &) 2 + w/3 - 1. 



721] FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. 57 

The foregoing formulae apply to the combinations r + r s , r- + r\ r* + r 3 of the seventh 
roots of unity, but we may investigate the theory for the roots themselves r, r 2 , r 3 , r*, r 5 , r 8 . 
These depend on the new radical V(- 7) or i V(7) ; introducing instead hereof X, Y, 
where 



then if 

A 3 = G + So) Z + (1 + 3&) 2 ) F, 

B* = 6 + 3&&amp;gt; 2 Z + (1 + 3o&gt; ) Y, 
where 



we have (Lagrange, Equations Numeriques, p. 294), 

3r = Z + 4 + 5. 

I found that, in order to bring this into connexion with the foregoing formula, 
3 (r + ?) = a + - 1, where as before a 3 = - 7 (1 + 3a&gt;), /3 3 = - 7 (1 + Sty 2 ), a/3 = 7, it is 
necessary that B, A should be linear multiples of a, j3 respectively, the coefficients 
being rational functions of co, X ; and that the actual relations are 



A = [a + o&gt; + X (3 + 2&lt;w)} ; 

in verification of which, it may be remarked that these equations give 
= {(20- *&gt; - o&gt; 2 ) + Z (17 - 4o&gt; - 4) + Z 2 (3 - 4o&gt; - 



viz. in virtue of the equation w 2 + &&amp;gt; + 1 = 0, the term in { } is =21 
= 7(Z 2 + 3Z + 3), or since Z 2 + X + 2 = 0, this is =7(2Z+1), =7iV(7); the equation 
thus is 1AB = a/3 . t VCO, which is true in virtue of AB = i^/( t f) and a/3 = 7. The same 
relations may also be written 

- a = 5 (w 2 + X), 

- /3 = A (ai + Z). 
I found in the first instance 

3r = X + A + B, 

8r--l_X+ A(a&gt;"-- X) + B(a&gt;- X), 

3r- = X + co-A + w B, 

3?- r = - 1 - X + A (o&gt; - &) 2 Z) + B (w 2 - w Z), 

3r = Z + wA + tfB, 

3?- 3 = -l-Z+ ^1(1 - o&gt;Z) + B (1 - ft, 2 Z), 
C. XL 



58 FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. [721 

which in fact gave the foregoing formulae 

3 ( r + r 6 ) = - 1 + a + j3, 
3 ( r 2 + ^) = _ l + toa + 0,2/3, 
3 (r 4 + r 3 ) = - 1 + w"a + o)/3. 

But there is a want of symmetry in these expressions for r, r", &c., inasmuch as the 
values of r, r 2 , r 4 are of a different form from those of r 6 , r 5 , r 3 ; to obtain the proper 
forms, we must for A, B substitute their values in terms of a, /3, and we thus obtain 

3r = X + ^{ 4- a&gt; + X( l-2)}+{ 5+ o&gt; + X ( 3 + 

o&gt; + X - 3 - 



1 {- 3 - 2o&gt; + 

3r 4 = X + y {- 5 - 4ft) + X (- 3 - to)} + ^ {- 1 + 4o, + X (- 2 + &&amp;gt;)}, 

3r^ = -l-Z + |{-2-3a,4-X( 3+ )} + f { l + 3w + Z( 2- )} ; 

viz. each of the imaginary seventh roots is thus expressed as a linear function of the 
cubic radicals a, /3 (involving &&amp;gt; under the radical signs) with coefficients which are 
functions of &&amp;gt;, -X". 

Recollecting the equations a 2 = - (1 + 3a&gt;), /3 2 = - a (1 + 3&&amp;gt; 2 ), a/3 = 7 ; &&amp;gt; 2 + a&gt; + 1 = 0, 
X 2 + X + 2 = 0; it is clear that, starting for instance from the equation for 3r, and 
squaring each side of the equation, we should, after proper reductions, obtain for 9r 2 
an expression of the like form ; viz. we thus in fact obtain the expression for 3r 2 ; 
then from the expressions of 3r and 3r 2 , multiplying together and reducing, we should 
obtain the expression for Sr 3 ; 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 

yjp _ \ 

starting-point is the following theorem : if x be a root of the equation - =- = 0, 

CC ~~ J. 

p a prime number, and if g is a prime root of p, and 

F (a) = x + axv + aW* + . . . + a^- 1 x"~~, 



a _ 
where a is any root of y- = 0, we have 

F(a m ) F(a n ) = ^ (a) 



721] FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. 59 

where ^ (a) is a rational and integral function of a with integral coefficients ; or, what 
is the same thing, if a and ft be any two roots of the above-mentioned equation, then 

F(a)F(/3) = + (*, ft) F (aft), 

where ^(, ft) is a rational and integral function of a, ft with integral coefficients. 
As regards the proof of this, it may be remarked that, writing a? for x, F (a), F(ft), 
and F(aft) become respectively ar l F(a), ft^F(ft), (aft)~ l F (aft) hence, F(a)F(ft) + F(aft) 
remains unaltered, and it thus appears that the function in question is expressible 
rationally in terms of the adjoint quantities a and ft. With this explanation the 
following extract will be easily intelligible : 

" The true form (never yet given) of the roots of the equation x? 1 = 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 pl and a A = 1, then it is further known that 
{J T (a)} A is a mere function of a. But it is only necessary to know those values of 
F(a) for which \ is the power of a prime number. For suppose XX X"... is a factor 
of p l , further let X, X , X", ... be powers of different prime numbers, and a, a , a", . . . 
prime Xth, X th, X"th, ... roots of unity, then 



.. 

^(a, a , a , ...) 

where ^(a, a , a",...) denotes a rational and integral function of a, a , a",... with 
integral coefficients. Hence, considering always the (p l)th roots of unity as given, 
there are contained in the expression for x only radicals, the exponents of which are 
powers of prime numbers, and products of such radicals. But if X is a power of a 
prime number, = (M n , suppose, the corresponding function F(a.) can be found as follows : 
Assume 



then 

* = &lt;X{*i() *,() ...+*-( 

. . . ^ (a^) F (a*)}, 



and so on, up to 

P-I 



a* 1 "" 1 ) +* ( M "" 1 ) *M-I (O (-) " Pi" 

so that the formnlffi contain ultimately /ith roots only. It is remarked in a foot 
note that, when n=l, the p 1 functions can always be reduced to one-sixth part in 
number, and that by an induction continued as far as /m 31, Jacobi had found that 
all the functions -v|r could be expressed by means of the values of a single one of 
these functions. 

" The p I functions determine, not only the values of all the magnitudes under 
the radical signs, but also the mutual dependence of the radicals themselves. For 
replacing a. by the different powers of a, one can by means of the values so obtained 
for these functions rationally express all the fi n l functions F (a 1 ) by means of the 
powers of F(a); since all the /j, n I magnitudes [F(a)} i -4- F(o. { ) are each of them 

82 



60 FORMULA INVOLVING THE SEVENTH ROOTS OF UNITY. [721 

equal to a product of several of the functions ty (a). Herein consists one of the great 
advantages of the method over that of Gauss, since in this the discovery of the 
mutual dependency of the different radicals requires a special investigation, which, on 
account of its laboriousness, is scarcely practicable for even small primes ; whereas the 
introduction of the functions ty gives simultaneously the quantities under the radical 
signs, and the mutual dependency of the radicals. The formation of the functions ty 
is obtained by a very simple algorithm, which requires only that one should, from the 
table for the residues of g m , form another table giving g m = 1 + 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 x p 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, x 4 + x s + x 2 +x + \ =0. 
According to Gauss, we have x 4- .* 4 and x 2 + a 3 , the roots of the equation u? + u 1 = ; 
say x + x* = % { 1 + V(5)}, x 2 + a? ^ { 1 V(5)}- The first of these, combined with 
x.x* = l, gives x-x* = V[- | {5 + \/(5)j] ; and thence 4# = - 1 + V(5) + V[- 2 {5 + V(5)}] ; 
if from the second of them, combined with o?.x 3 = l, we were in like manner to obtain 
the values of a? and a?, it would be necessary to investigate the signs to be given 
to the radicals, in order that the values so obtained for x 2 and a? might be consistent 
with the value just found for x. For the Jacobian process, observing that a prime 
fourth root of unity is a = i, and writing for shortness F 1} F 2 , F 3 , F 4 to denote F(a), 

), F(a. 3 ), F(a.*) respectively, these functions are 

F l = x - x* + i (# 2 - x 3 ), 



viz. we have F, = -l, F/=5, or say #, = V(5), Fi&gt; = - (I + 2i) F,, = - (1 + 2i) V(5) ; and 
similarly F&lt;?= - (1 - 2i) F 2&gt; = - (1 - 2*) V(5) ; but also F 1 F 3 = o, so that the values 
F 1 = ^/{-(l + 2t)V(5)}, F 3 = V{-(1- 2t) V(5)}, must be taken consistently with this last 
equation F t F s = V(5). The values of F lf F 2 , F 3 , F t being thus known, the four equations 
then give simultaneously x, x 4 , x 2 , x 3 , these values being of course consistent with each 
other. It may be remarked that the form in which x presents itself is 



with the before-mentioned condition as to the last two radicals; with this condition 
we, in fact, have 

V{- (1 + 2t) V(5)} + VI- (1 - 2i) V(5 
as is at once verified by squaring the two sides. 



722] 



61 




722. 



A PEOBLEM IN PARTITIONS. 



[From the Messenger of Mathematics, vol. vn. (1878), pp. 187, 188.] 



TAKE for instance 6 letters ; a partition into 3 s, such as abc . def contains the 6 
duads ab, ac, be, de, df, ef. A partition into 2 s such as ab.cd .ef contains the 3 
duads ab, cd, ef. Hence if there are a partitions into 3 s, and /3 partitions into 2 s, 
and these contain all the duads each once and only once, 6a + 3/3 = 15, or 2a + /3=5. 
The solutions of this last equation are (a = 0, /3=5), (a = l, /3 = 3), (a = 2, /3 = 1), and 
it is at once seen that the first two sets give solutions of the partition problem, but that 
the third set gives no solution; thus we have 



x = 0, /3 = 5 


a = 1, /3 = 3 


ab . cd . ef 


abc . def 


ac .be . df 


ad .be .cf 


ad . bf. ce 


ae . bf. cd 


ae .bd .cf 


af. bd . ce. 


af. be . de 





Similarly for any other number of letters, for instance 15 ; if we have a partitions 
into 5 s and yS partitions into 3 s, then, if these contain all the duads, 4a + 2/3 = 14, 
or what is the same 2a + (3 = 7 ; if a = 0, j3 = 7, the partition problem can be solved (this 
is in fact the problem of the 15 school-girls) : but can it be solved for any other values 
(and if so which values) of a, /3 ? Or again for 30 letters ; if we have a partitions into 
5 s, /3 partitions into 3 s and 7 partitions into 2 s ; then, if these contain all the duads, 
4a + 2/3 + 7 = 29 ; and the question is for what values of a, /3, 7, does the partition- 
problem admit of solution. 



62 A PROBLEM IN PARTITIONS. [722 

The question is important from its connexion with the theory of groups, but it 
seems to be a very difficult one. 



I take the opportunity of mentioning the following theorem : two non-commutative 
symbols a, /3, which are such that fia = a?/3 2 cannot give rise to a group made up of 
symbols of the form a*/S. In fact, the assumed relation gives /Sa 2 = a 2 a 2 /3 2 ; and 
hence, if /3a 2 be of the form in question, =a?fty suppose, we have 



= a- . 



that is, l= 2 /3 2 , and thence /3a=l, that is, p = or l , viz. the symbols are commutative, 
and the only group is that made up of the powers of a. 



723] 63 



723. 

VARIOUS NOTES. 

[From the Messenger of Mathematics, vol. viu. (1879), pp. 4546, 126, 127.] 

An Algebraical Identity: p. 45. 
Let a, b, c, f, g, h be the differences of four quantities a, /3, y, 8, say 

a, b, c, f, g, h = 0-y, y-a, a-/3, a - 8, /3 - 8, y - 8 ; 
then 

h g + a = 0, 

-h 



a b c . = 0. 
Now Cauchy s identity 

(a + 6) 7 - a 7 - 6 7 = 7a6 (a + b) (a 2 + ai + ft 2 ) 3 , 
putting therein a + 6 = c, becomes 

a 7 + b 7 + c 7 = lobe ( be + ca + ab)~ ; 
hence we have 

A 7 - g 7 + a 7 = - 7agh (- ga + ah - hg) z , 

+ b 7 = - nhf (- hb + bf -fh)\ 

+ C 7 = _ 7c fg (_/ C + C g - gff, 



whence, adding, 

(- ga + ah- hg)* + bhf(- hb + bf-fhf + cfg (-/c + cg- gff + abc (be + ca + ab)- = 0, 



G4 VARIOUS NOTES. [723 

or, as this may also be written, 

agh (g* + A 2 + a- ) 2 + bhf(h- +/ 2 + 6 2 ) 2 + cfg (/ 2 + g* + c 2 ) 2 + abc (a 2 + 6 2 + c 2 ) 2 = 0, 
an identity if a, b, c, f, g, h denote their values in terms of a, ft, 7, 8. 



Note on a Definite Integral : p. 126. 
The integral 



i 

* frafdx 



used by Weierstrass, is at once seen to be =K E ; but the proof that the other integral 

_ 

~ 

is = " is not so immediate. 
We have 

d 
and thence 



viz. replacing the numerator by 

-+|a -**&gt;. 

this becomes 

_^ 2 n rfy i 

^ 2 Jo (l-2/ 2 )*(l-% 2 ) A? 
that is, 

f 1 

Jo (l- 
or, writing k for /;, 



T- -^- = ^ 

Jo (1-/)*(1-^V) ? k ~ 



The integral J writing therein x = -77^ -- ^7 - becomes 



J = k- i 

o 
viz. its value is thus =E . 



1 dy 



723] VARIOUS NOTES. 65 

On a Formula in Elliptic Functions: p. 127. 

cn ?/ 
Writing enM=B ( j nM tnen the formulae p. 63 of my Elliptic Functions give 

sn(u , V \ = T ~ T&gt; n(u =& + &_. 

and, substituting for T 7 , T , B, B , and (7, C" their values, we obtain 

, sn u en ?; + sn v en M 
sn(w + w) = ,-,-rr 

1 + A;- sn w en it sn v en v 

, . en w en v sn w sn v 

en (w + v) = TT- 

1 & sn w en u sn v en v 

formulae which, as regards their numerators, correspond precisely with the formula?, 

sin (u- + v} = sin u cos v + sin v cos u 
and 

cos (u + v) = cos w cos v sin w sin v, 
of the circular functions, and which in fact reduce themselves to these on putting k = 0. 

The foregoing formula?, putting therein & 2 = -l, are the formula} given by Gauss, 
Werke, t. in., p. 404, for the lemniscate functions sin lemn (a b) and cos lemn (a 6) ; 
where it is to be observed that these notations do not represent a sine and a cosine, 
but they are related as the sn and en, viz. that 

cos lemn a = V(l sin lemn 2 a) -4- \/(l + sin lemn 2 a). 



C. XI. 



66 [724 



724 



[From the Messenger of Mathematics, vol. vin. (1879), pp. 51, 52.] 

THE following is a solution of Mr Greenhill s problem set in the Senate-House 
Examination, January 14, 1878. 

"Prove that, if a model of a hyperboloid of one sheet be constructed of rods 
representing the generating lines, jointed at the points of crossing; then if the model 
be deformed it will assume the form of a confocal hyperboloid, and prove that the 
trajectory of a point on the model will be orthogonal to the system of confocal 
hyperboloids." 

Let (#j, y 1} zj, (x. 2 , ?/ 2 , z 3 ) be points on the generating line of 




4-2- - -1 

a 2 T 6 2 c 2 
then 

22 

a 2 6- c 2 

/ - / / ~ " ~ 

a 2 + T 2 " "c 2 = 



4. 2 _ - 1 

a 2 " 6 2 "~ c 2 " 
or, what is the same thing, if 

^ ^ ?}- ^ 2/ 2 ^ 

a b c ~^ 5l&gt; a 6 c 
then 

Pi + qi 2 - r-c = 1, 



724] ON THE DEFORMATION OF A MODEL OF A HYPERBOLOID. 67 

Similarly, if (ft, r^ l , ^), (&gt;, rj.,, 2 ) be points on generating line of 

e+-f.i 

**& 7 2 ~ 

and if 

& ^ &-D a r- ? 2 ^ ?-D o r- 
a 7~ P " q " a 7 ~ P2 q2 r2 

then 

Pi 2 + qi 2 - I? = 1, 

p 2 2 + q 2 2 - r 2 2 = 1, 
Pip 2 + qiq a - ^r,, = 1. 

Hence if (ss 1 , y l , Zj), (ft, ^ l , ) be corresponding points on the two surfaces, that 
is, if 

*i yi *i_ft *?i. Si 

i T f F 7P-7 1 ~^ 3l&gt; n&gt; 

and similarly, if (? 2 , 7/ 2 , ^ 2 ), (^, %, ^ 2 ) are corresponding points, that is, if 

5? y !_& 5? ??_ a r . 
a b j c~ /3 y" 1 ?2 2 

then we have, as before, the system of three equations 

Pi" + qi - n 2 = 1, 
p** + qs - ^ = 1, 

Pipa + qiqi-rfa = 1. 

Then if the two surfaces are confocal, that is, if 

2 &gt; @~, 7 2 = a 2 + h, b- + k, c- + h, 
we shall have 

(a?, - ^ 2 ) 2 + (y, - ytf + (z, - ztf = (ft - ft) 8 + (ih - ^ + (?i ~ r.) 2 - 

For this equation is 

2 (Pi ~ Ptf + & (q, - q,)" + C- (r, - r.,)- = a 2 (p, -p,)- + fr (q, - q,)* + ^ ( TI - r 2 ) 2 , 

that is, 

(pi -ptf + (qi - ^ 2 ) 2 - (n - r 2 ) 2 = 0, 

an equation which is obviously true in virtue of the above system of three equations. 
Hence, if on confocal surfaces 



we take two points P 1} P 2 on the first, and Q lt Q 2 the corresponding points on the 
second ; then P 1? P, being on a generating line of the first surface, Qi, Q will be 
on a generating line of the second surface, and PjPa will be = Q^. The same 
is evidently true for the quadrilaterals P 1 P,P 3 P^ and Q^QsQ* where P^,, P,P 3&gt; 
, P*Pi are generating lines on the first surface : and therefore Q&, Q,Q 3 , Q 3 Q^, 
are generating lines on the second surface, which proves the theorem. 

92 



68 



725. 



NEW FORMULAE FOR THE INTEGRATION OF 



[From the Messenger of Mathematics, vol. vm. (1879), pp. 60 62.] 






I HAVE found in regard to the differential equation 
das dy 



*J(a-x.b-x.c-x.d-x) \f(a-y.b-y.c-y.d-y) 



= 0, 



a system of formulae analogous to those given, p. 63, of my Treatise on Elliptic 
Functions, for the values of sn (u + v), en (u + v), dn (u + v). Writing for shortness 

a, b, c, d = a x, b x, c x, d x, 
a,, b lt c,, d j = a-y, b-y, c-y, d-y, 
and (be, ad) to denote the determinant 

1 , + y, xy 

1, b + c, be 

1, a + d, ad 

and (cd, ab), (bd, ac) to denote the like determinants; then the formulae are 
fa z \ \/(a b.a c) { VCadbjCj) + 



/( a - z \- 
V\d-z) 



(be, ad) 

( b . a - c) (x y) 



(a - b . a - c) 



i) + 



(o - b . a - c) { V(acb 1 d 1 ) 



dx dv 

725] NEW FORMULAE FOR THE INTEGRATION OF -.^+ ^=0. 69 



1 ,1 * 

//" * \ _ V ^ 

V U - f) ~ 



J 1 

~ c) VCbdbA) + (b -d) V(aca A )} 



(a c) x/Cbdbidj) (b 

a - d 



(a 6) 

/ IB * 
/ / v * i 

V U - ,87 = 



) - ( 6 - c &gt; V(ada 1 d 1 )} 



(a c) VObdbjdi) (b d) ^( 

/(a-c\ n , 
A / ~ j (", ac) 

V \a-d ^ 



_ _ 
(a 6) V(cdcjdi) (c c?)V(aba 1 b 1 ) 

The twelve equations are equivalent to each other, each giving z as one and the 
same function of x, y ; and regarding z as a constant of integration, any one of the 
equations is a form of the integral of the proposed differential equation. 

Writing in the formulae x a, b, c, d successively, the formulae become 
x = a, x = b, x = c, x =d, 



a z &! 


c a I 


&gt;j 6 a Cj 


a 6 . a c QI 


d-z d x 


d b(. 


&gt;i d c b x 


d b.d c &i 


b z bj 


c b i 


i x 6 a . b c dj 


a b Cj 


d z dj 


rf d ( 


3j d a . d c bj 


Oy ^~ C BJJ 


C "~~ Z Cj 


c a . c b c 


1, 6 - c a, 


a c b : 



d z d/ da.d bc^ d a bj d 6 aj 

viz. in the first case we have 2 = y, and in each of the other cases z equal to a 
linear function - ^ of v. 

w + s 

Cambridge, July 3, 1878. 



70 [726 



726. 

A FORMULA BY GAUSS FOR THE CALCULATION OF LOG 2 
AND CERTAIN OTHER LOGARITHMS. 

[From the Messenger of Mathematics, vol. vm. (1879), pp. 125, 126.] 

GAUSS has given, Werke, t. IL, p. 501, a formula which is in effect as follows: 
/1025V /1048576V /6560\ 3 /15624V /9801V 

O196 1 A59 I II _ I I -- I I - ] I -- I 

V1024J U048575/ \Q5Ql) U5625/ \9800J 
viz. this is 

2*. 3 2 . 7.31V / 3 4 .11 2 V 



2 .3.11.31.4l 

where on the right-hand side the several prime factors have the indices following, viz. 

2, index is (59 + 160 + 15 + 24- 50- 12) = 196, 

3 (16+16-8-24 ) = 0, 

5 (59+10+3-16-48- 8) = 0, 

7 (8-8 ) = 0, 

11 (8-8 ) = 0, 

31 ( 8- 8 ) = 0, 

41 ( 5+ 3- 8 ) = 0, 

or the right-hand side is = 2 196 as it should be. The value of log 2 calculated from 
2 196 =10 39 is log 2 = ^- = 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 :96 = 10043 36277 66186 89222 13726 30771 

32266 26576 37687 11142 45522 06336.* 
Supposing log 2 calculated by the form, we then have 

41 - ({f) 2 12 ^ 10 2 , giving log 41, 
and 

3 8 =10. MM- 2 4 . 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. viu. (1879), pp. 190, 191.] 

THE equation of the wave-surface 

ax 2 _ by 2 cz 2 = 

o i . .o i .o 7, o i _ &gt; i ^ J 



a? -|- # 2 + z* -a a? + y 2 + z 2 -b x- + y- + z 
when transformed to coordinates p, q, r, such that 



of- V* z- 

i y. i 



a + p b+p c+p 

x- y 2 z 2 

-i r \- - = 1 

a + q b + q c + q 

x~ y- z 2 

| t7 I . J 

a + r b+r c+ r 

x 2 y 2 z~ 
(that is, to the elliptic coordinates belonging to the quadric surface - -\ r + =1), 

Q C/ "~ 

assumes the form 

(q+ r a b c)(r +p a b c}(p +q a b c) = 0, 
(Senate-House Problem, January 14, 1879). 

In fact, p, q, r are the roots of the equation 



-a + u b+u c + u 
we have therefore 

(u-p) (u-q}(u- r) = (M - a) (u - b) (u - c) 

- a;* ( - 6) (t&lt; -c)-if (u - c) (u - a) - 2 2 (jt - a) (u - 6) ; 



72 EQUATION OF THE WAVE-SURFACE IN ELLIPTIC COORDINATES. [727 

whence, writing for shortness 

A = a + b + c , P = p + q + r, 
B = be + ca + ab, Q = qr + rp +pq, 

C = abc , R = pqr, 

we have 

x- + y*+ z* = P-A, 

(b + c) x* + (c + a) y- + (a + b)z- = Q - B, 

bca? + cay- + abz~ R C, 
and thence also 

a(b+c)x n - + b(c+a)f + c(a+b)z 2 =B(P-A)-(R-C), 
aa? + bf + cz- = A(P-A}-(Q-B). 

The equation of the wave-surface is 

ale - {a (b+c)tf + b(c + a) if + c (a + b) z-\ + (x- + y- + z-} (ax 2 + by- + cz-) = 0. 
By the formulae just obtained, this is 

C-[B(P-A)-(R-C)] + (P-A)[A(P-A)-(Q-B)] = 0, 
that is, 

A 3 - 2A-P + A(P- + Q)-(PQ-R) = 0, 
that is, 

{A -(q + r)} [A -(r + p)} [A-(p + g) =0, 

or, substituting for A its value a + b + c, and reversing the sign of each factor, we 
have the formula in question. 

It is easy to see that, taking a, b, c to be each positive, (a&gt;b&gt; c), and assuming 
also p &gt; q &gt; r, we obtain the different real points of space by giving to these 
coordinates respectively the different real values from oo to a, a to b, and b to c 
respectively. Hence 

greatest, least value, is 

q + r, a + b, a + c, 

r +p, oc , a + c, 

p + q, x , a + b, 

so that r+p, p + q, may be either of them = a + b + c, but q+r cannot be = a + b + c, 
that is, q + r = a + b + c does not belong to any real point on the wave-surface. We 
can only have r + p and p + q each = a + b + c, if p = a + c, q = r = b, and these values 
belong as is easily shown to the nodes on the wave-surface ; hence, the equations 
r + p = a + b + c and p +q = a + b + c being satisfied simultaneously only at the nodes 
of the surface, must belong to the two sheets respectively. And it can be shown 
that p + r = a + b + c belongs to the external sheet, and p + q = a + b + c belongs to the 
internal sheet. In fact, for the point (0, 0, ^a), which is on the external sheet, we 
have p = a + c, q = a, r = b, and therefore p+r=a+b + c: for the point (0, 0, 
which is on* the internal sheet, either 

(p = b + c, q = a, r = b) or (p = a, q = b + c, r c), 
according as b + c&gt; a or b + c &lt; a : but in each case 



728] 73 



728. 

A THEOREM IN ELLIPTIC FUNCTIONS. 



[From ,the Proceedings of the London Mathematical Society, vol. x. (1879), pp. 48 _ 48. 

Read January 8, 1879.] 

THE theorem is as follows : 
If u + v + r + s = 0, then 

1 k 2 

k 2 sn u sn v sn r sn s + en u en v en r en s r dn u dn v dn r dn s = -- 

& 2 # 

It is easy to see that, if a linear relation exists between the three products, then 
it must be this relation: for the relation must be satisfied on writing therein 
v=u, s = r, and the only linear relation connecting sn 2 u sn 2 r, en 2 u en 2 r, dn 2 u dn 2 r 
is the relation in question 

k 2 sn 2 u sn 2 r + en 2 u en 2 r j- dn- w dn J r = . 

# &- 

A demonstration of the theorem was recently communicated to me by Mr Glaisher ; 
and this led me to the somewhat more general theorem 

- A/ 2 sn (a + /3) sn (a - /S) sn (7 + 8) sn (7 - 8) 
+ en (a + @) en (a - y9) en (7 + 8) en (7 - 8} 

- jp dri (a + /3) dn (a - ) dn (7 + S) do (7 - 8) 

_k^ ^k 2 (sn 2 a - sn 2 7) (sn 2 /3 - sn 2 8) 
~ 



A- 2 1 - k 2 sn 2 a sn 2 . 1 - & 2 sn 2 7 sn 2 8 
C. XT. 10 



74 A THEOREM IN ELLIPTIC FUNCTIONS. [728 



In fact, writing herein + 7 = 0, that is, y = ct, the right-hand side becomes =0; 
-and the arcs on the left-hand side are a. + @, a. /3, a + S, a 8, which represent 
any four arcs the sum of which is = 0. 

Writing in the last-mentioned equation x, y, z, w for the sn s of a, /3, 7, & 
respectively, also 



Q = 1 - x 2 - y 2 + k 2 x 2 y 2 , Q l = 1 - z* - w 2 + k 2 z 2 w 2 , 

R = l-k 2 x 2 - k 2 y 2 + k 2 x ~y 2 , R l = l- k 2 z 2 - k 2 w 2 + k 2 z 2 w 2 , 

D = l- k 2 x 2 y 2 , A = 1 ~ k 2 z 2 w 2 , 
the equation is 

% RR l _ k 2 2k *(x*-z*)(y*-w*) 



_ 
, DD, k 2 DD,~ k 2 DD, 

that is, 



- ^PP X + QQ l - RR^ + DA + 2^ 2 (a- - z*) (f - w 2 ) = 0. 

It is easy to verify that the terms of the orders 0, 1, 2, 3 and 4 in a; 2 , y 2 , z 2 , w- 
separately destroy each other; for instance, for the terms of the order 2, we have 

- k 2 (x 2 - y 2 } (z- - w-) + {(a? + y 2 ) (z- + w 2 ) + k 2 (x 2 y 2 + z*w 2 )} 





f 

+ ^ |_ fc (x 2 y 2 + zhu 2 )} + 2k 2 (x 2 - z 2 ) (y 2 - w 2 ) = 0, 

fC 

that is, 

- k 2 (x 2 - y 2 ) (z 2 -w 2 ) + (l- k 2 ) (x 2 + y 2 ) (z 2 + w 2 ) 

+ (k 2 -I- k 2 ) (x 2 if + z-w 2 ) + 2k 2 (x 2 - z 2 ) (y 2 - w 2 } = ; 

or, omitting the factor k 2 , this is 

- (x 2 - y 2 ) (z 2 - w 2 ) + (x 2 + y 2 ) (z 2 + w 2 ) - 2 (x 2 y 2 + z-w 2 } + 2 (x 2 - z 2 ) (y 2 - w 2 } = 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 

y 2 jL 

1 1 *- 



a +p b + p c +p 

x 2 y 2 z 2 

j_ 7 I J 

a + q b +q c + q 

x 2 y 2 z 2 

- + r i r +r ~ = 1 ; 

a+ r b + r c 4- r 



728] A THEOREM IN ELLIPTIC FUNCTIONS. 75 

or, what is the same thing, 

Pyx 2 = a+p.a+q.a + r, 

yay* == b -if-p .b + q .b +r, 

aftz- = c +p .c+q.c+r, 

where a, @, 7 denote b c, c a, a b respectively ; then, treating r as a constant, 
the coordinates x, y, z will belong to a point on the ellipsoid 

a 8 v 2 & 

h r 2 h = 1, 

a + r b +r c + r 

and the differential equation of the right lines upon this surface is 

dp dq 



va+p.b+p.c + 2) Va + q. b + q.c + q 

Take # , y , z the coordinates of a point on the surface, and p , q the corresponding 
values of p, q, so that 

ftyx - = a + p . a + q . a + r, 

72/o 2 = b + p . b + q Q .b +r, 

aftz,? = c + p . c + q . c + r, 

then the equation of the tangent plane at the point (ac , y , z ) is 

+ --^- \- 1 
a + r b + r c + r 

or, substituting for # 2 , ar 2 , &c., their values, we have 




. a + q , &c., 



CL ~r~ *?* 

and consequently the equation of the tangent plane is 

.a + q.a+p .a + q + @\/b+p.b + q.b+p 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 q ) is the integral 
of the proposed differential equation. 

Writing 

sn 2 u = A (a + p), crfu = B(b+p), dn?u = C(c+p), 



the values of A, B, C, k are determined ; and, assuming for q, p , q 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 i( = v v , while the foregoing integral equation, on reducing the constant 
coefficients contained therein, takes the form 

A; 2 sn u sn v sn u sn v 
+ en u en v en u en v 

j dn u dn v dn u dn v 



_ 
A; 2 

viz. this equation holds good if U U O = V V Q . And by a change of signs we have 
the theorem. 



If, as above, u + v + r+s = Q, the theorem gives a linear relation between the 
three products sn u sn v sn r sn s, cnucnvcnr en s, dn u dnv dn r dn s, and regarding at 
pleasure the sn s, the en s, or the dn s as rational, one of these products will be 
rational while the other two will be each of them a quadric radical; and hence, 
rationalising, we obtain an equation which contains the product in question linearly, 
and contains besides only the squares of the sn s, en s, or dn s; that is, we have 
three such equations containing the three products respectively. Bringing to one side 
the terms which contain the product, and again squaring, we obtain an equation 
involving only the squares of the sn s, en s, or dn s; but the three equations thus 
obtained represent, it is clear, one and the same rational equation, which may be 
expressed as an equation between the squares of the sn s, or of the en s, or of the 
dn s, at pleasure. This equation may be obtained, as I will show, from the ordinary 
addition-equations of the elliptic functions, but it is not obvious how to obtain from 
them the three equations involving the products respectively, and these last have the 
advantage of being of a degree which is the half of the equation which involves 
only the squared functions. 

Write x, y, z, w for sn u, sn v, sn r, sn s respectively ; then, writing 



A = x Vl - f. 1 - &y, a = z Ji 

A = y Vl - *Ti ~- k 2 a?, a = w V 1 - z 2 . 1 - fcz 2 , 

P = x- y-, TO = z- w 2 , 
D = 1 - 



we have 

sn (u + v) = - sn (r + s), 
that is, 

A+A P a + a 



D A-A ~ S a -a. 

and consequently 

CT = _ ( a _ a!) (A + A ), 



P8 = - (a + a ) (A - A ) 
whence 



728] A THEOREM IN ELLIPTIC FUNCTIONS. 77 

that is, (** - w 2 ) (1 - k 2 x 2 y 2 ) - (& - 2/ 2 ) (1 - tfzW) 

= 2 [xw Vl - y* . 1 - fcyTl - * 2 . 1 - ^ 2 - ys Vl - #* . 1 - &V . 1 - w 2 . 1 - Aftw*}. 

Rationalising, we obtain, as mentioned above, an equation containing only the squares 
x*, 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 (#*, y 2 , z 2 , w 2 ), but the 
calculation was lost, and the easier way of obtaining it is obviously by means of the 
equation involving xyzw. 

We have, by the theorem, 
k 2 xyzw 



K 

that is, 

k 2 (1 - k 2 xyzw) = k- Vl - a? . 1 - y 2 . 1 - ^ 2 . 1 - w 2 

- Vl - k 2 x? . l~-k 2 y 2 . 1 - k 2 z 2 . 1 - k 2 w 2 ; 
and then, writing 

71 ") i o i 

p = x- + y 2 + 

Z? rffinflyfi _L r, 

S\j J; U 6 T ** 



and using V^ to denote the rational function xyzw, we have 



- 2k 2 \(l- 

or, if for a moment the radical is called VA, then the factor k 2 divides out, and 
the equation becomes 

2 VA = 2 - (1 + & 2 ) P + 2k 2 Q - (k 2 + & 4 ) R 

whence 

4 ( i _ p + Q - R + S ) ( 1 - k 2 P + k* Q - kR 

_ (2 - (1 + fc 2 ) P + 2k 2 Q - (k 2 + k 4 )R+ 2k*S\ 2 - 4&gt;k s S 
= -2k*^S{2-(I+ k 2 ) P + 2k 2 Q - (k 2 + k*) R 

The factor k 4 divides out; omitting it, we have 

4 Q _ p* _ 4 (i + k 2 ) R + Wk*S + 2k 2 PR -4&lt;(k 2 + k 4 ) PS - k*R 2 

= - 2 V# {2 - (1 + k 2 ) P + 2k 2 Q - (k 2 
or, as this may also be written, 

k 2 (- 4&gt;R + 2PR + US - 4PS) -f ^ (- E 2 + 4QS - P&gt;Sf)} 



which is the required rational equation involving the product of the sn s. 



78 [729 



729. 

ON A THEOREM RELATING TO CONFORMABLE FIGURES. 



[From the Proceedings of the London Mathematical Society, vol. x. (1879), pp. 143 146. 

Read May 8, 1879.] 

CONSIDER two plane figures, say the figure of the points P referred to axes 
Ox, Oy, and that of the points P referred to axes Ox, Oy ; and let x, y be the 
coordinates of P, and x, y those of P . If the figures correspond to each other in 
any manner whatever, P and P being corresponding points, then we have x , y 
each of them a function of x, y\ and we may consider the second figure as derived 
from the first by altering the distance OP in the ratio VV 2 + y" 1 -4- v x* + y 1 , and by 

rotating it through the angle tan" 1 , tan" 1 - ; say by the Extension Va? 2 + y - -4- *la? + y z , 

x x 

11 ij 

and by the Rotation tan" 1 ^- tan" 1 -; where the Extension and the Rotation are each 

x x 

of them a determinate function of x, y, the coordinates of P. 

Passing from the point P to a consecutive point Q, the coordinates of which 
are x + dx, y + dy (the ratio dy -4- dx being arbitrary), then the coordinates of the 
corresponding point Q will be x + dx , y + dy , where 

7 . dx , dx , , . dif , dy , 



Writing -, and instead of dy -4- dx and dy -4- dx, the expressions 

CLvC \JLUU 

- 1 - - - 1 - 



dx 2 + dy 2 -4- d# 2 + dy\ and tan- 1 -, - tan- 1 - , 

ax ax 

will in general have values depending upon that of the arbitrary ratio dy : dx. But 
they may be independent of this ratio ; viz. this is the case when x , y are functions 
of x, y such that 

dx _ dy dy __ dx 

dy dx dy dx" 1 



729] ON A THEOREM RELATING TO CONFORMABLE FIGURES. 79 

and the two figures are then conformable (or conjugate) figures ; that is, figures similar 
as regards corresponding infinitesimal elements of area. We have, in this case, 

\/dx* + dy ~ 2 -T- Jdaf + dy 2 , and tan" 1 -/- t - tan" 1 ^ , 

doc ct^c 

each a determinate function of x, y, the coordinates of P; and we pass from the 
element PQ to the corresponding element P Q by altering the length in the ratio 



2 + dy 2 4- *Jdx- + dy 2 , and rotating the element through the angle tan" 1 -^- t tan" 1 -^- ; 

dx dx 

say, this ratio and this angle are the Auxesis and the Streblosis respectively, these 
being, as already mentioned, functions of x, y only. 

Considering now any two conformable figures, say the figure of the points P, 
and that of the points P ; we have the theorem that we can from the first figure 
obtain a third conformable figure by means of an Auxesis and a Streblosis which 
are respectively equal to the Extension and the Rotation by which the second figure 
is derived from the first. 

In fact, if in the three figures respectively we take x, y, x , y , and x", y", for 
the coordinates of the corresponding points P, P , P", the first and second figures 
are conformable : and we have therefore 

dx _ dy dy _ dx 
dy dx dy dx 



the third figure is to have the Auxesis V# 2 + y 2 H- V# 2 + r/ 2 , and the Streblosis 



tan- 1 y - - tan" 1 ^ ; 
x x 

viz. writing r for V&- 2 + y 2 , we ought to have 

1 xx + yy 7 xy x y , 
dx -- --?*- dx--^ -- ?-dy, 

IY& )- " 

, xy x y i xx + yy , 

d y = s r- 1 dx + f*~ d y ; 

and it is therefore to be shown that there exist x", y" functions of x, y satisfying 
these relations ; for, this being so, we have 

dx" = _dy^ dy"_ = drf^, 
dy dx dy dx 

and the third figure is thus conformable with the first. 

Writing, for shortness, 

_ xx + yy xy 1 - x y 





80 ON A THEOREM RELATING TO CONFORMABLE FIGURES. [729 

the equations are 

dx" = Ada; Bdy, 

dy" = Bdx + Ady; 
or the conditions for the existence of the functions x", y" are 



dA dB dA dB 

~T~ ~\~ ~f~~ == "&gt; ~~ 7 -- J 

dy dx dx dy 



We, in fact, have 



and similarly 



which proves the theorem. 

The theorem is closely connected with the theory of the function of an imaginary 
variable ; for, writing the conditions for the conformable figures in the form 



_ _ 

dx dy dy dx 

we have 

dx = Fdx Gdy, 

dy = Gdx - Fdy ; 

that is, 

dx + idy = (F + iG) (dx + idy) : 

whence F + iG is a function of x + iy, and then by integration x + iy is also a 
function of x + iy. In one point of view, any function such as $ (x, y) + ity (x, y) is 
a function of x -I- iy, for the quantity x + iy is only known by means of its real 
components x, y ; that is, knowing x + iy, we know x, y, and therefore also 

&lt;f&gt;(x, y) + i^(x, y); 

and Cauchy, adopting this definition, introduced the expression " fonction monogene " 
of x + iy, to denote that which is in the more restricted (and the ordinary) sense 
termed a function of x + iy. And MM. Briot and Bouquet, in their " Theorie des 
fonctions elliptiques" (Paris, 1875), although not using Cauchy s expression fonction 
monogene, but the simple term fonction, do this under the qualification stated p. 3 : 
" Dans tout ce qui suit, nous ne nous occuperons que des fonctions qui admetteut 
une derivee." Now, a function admitting of a derivative (that is, in the ordinary 



729] ON A THEOREM RELATING TO CONFORMABLE FIGURES. 81 

sense, a function) of the imaginary variable z, = x + iy, is a function such that, for a 
consecutive value z , = x + iy + dx + idy, we have 



z z 



= a quantity independent of the ratio of the real components dx, dy of the increment 
dx + idy of the imaginary variable. Or, what is the same thing, writing f(z) = x+iy , 
the condition in order that x + iy may be a function of x + iy is 

dx + idy = (F + iG) (dx + idy), 

where F and G are functions of x and y. It is not part of the condition that 
F + iG shall be a function of x + iy, and it is only a long way further on that the 
authors prove that this is the case (see the definition of a "function holomorphe," 
p. 14 ; and the proof, p. 137). The last-mentioned equation 

dx + idy = (F+ iG) (dx + idy), 

where F and G are only assumed to be functions of x and y, has, if we represent 
x + iy by means of the point P with coordinates (x, y), and in like manner x + iy by 
means of the point P with coordinates (x, y ), the geometrical interpretation that the 
figures of the points P and P are conformable figures, that is, figures similar as 
regards their infinitesimal elements. The foregoing theorem in regard to the Auxesis 
and the Streblosis is that we can, by means of F and G, construct a third conformable 

/nr 

figure, in fact, the Auxesis and the Streblosis are = VJ* 2 + G 2 and tan -1 -^ respectively ; 

and, using these as an Extension and a Rotation, we have the third conformable figure 
x" + iy" = (F + iG) (x+iy); that is, (F + iG) (x + iy), and therefore also F + iG, is a 
function of x + iy, and we have thus the derivative of a function of x + iy as itself 
a function of x + iy. 

It is to be remarked that, although the theorem of the Auxesis and the Streblosis, 
considered as a property of conformable figures, is not by any means geometrically 
self-evident, yet the foregoing analytical proof is only a proof conducted by means of 
real quantities, of what (admitting the theory of imaginary quantities) is in fact 
self-evident; viz. the analytical conclusion really is that, F, G denoting functions of 
x, y, then, if dx + idy = (F + iG) (dx + idy), that is, if (F + iG) (dx + idy) be a complete 
differential, then F + iG is a function of x + iy. 



C. XI. 11 



82 [730 



730. 



[ADDITION TO ME SPOTTISWOODE S PAPER "ON THE TWENTY- 
ONE COORDINATES OF A CONIC IN SPACE."] 

[From the Proceedings of the London Mathematical Society, vol. x. (1879), 

pp. 194196.] 

WRITE 

U=(a, b, c, d, f, g, h, I, m, n$x, y, z, t) 2 , 

^o = ( $fc 77, co) 2 , 

W=( fa, y, 

P = (a, , 7, 5$*, y, z, t), 

P O =(, & 7 , s$ 17, r, *&gt;) 



Then the equation of the cone, having for its vertex the arbitrary point (, 17, ^, &&amp;gt;), and 
passing through the conic U=0, P Q, is 



Or if, to put the coefficients , 77, ^, &lt;u in evidence, we write for a moment 

A = (a, A, 0, Z $&gt;, y, z, t), 
B=(h,b,f,m% ), 

G = (g, f&gt; o, w$ ), 

D = (/, m, n, d% ), 

and therefore 



then the equation is 

7(a + y87; 4- 7^+5a))- - 2P (a| -H ^17 

+ P 2 (a, 6, c, d, /, 0, A, I, m, $(?, 17, f, &lt;w) 2 = 0. 



730] 



ADDITION TO MR SPOTTISWOODE S PAPER. 



83 



And if we expand first in , 77, , &lt;w, and then in x, y, z, t, the final result is 

y? y* z 2 P yz zx xy xt yt zt 



e 

+ rf 

+ r 





C 


B 


F 


2A r 








2L 


2L 


c 




A 


G 




2B 




2M 




2M 





A 




H 






20 


2N 


2N 




F 


G 


H 




IF 


2G 


2ir 








2A 






2F 


-2A 


-20 


-2B 


2(Q-R) 


-2M 


-2N 




2B 




2G 


-20 


-2B 


-2A 


-2L 


2(R-P) 


-2N 






20 


211 


-IB 


-2A 


-20 


-2L 


-2M&gt; 


2(P-&lt;?) 




2M 


2N 




2(Q-JR) 


-2L 


-2L 


- 2F 


-2H 


-2G 


2L 




2jy 




-2M 


-2(R-P) 


-2M 


-2H 


-2G 


-2F 


2L 


2M 






- 2N 


-2N 


2(P-Q) 


- 2G 


-2F 


-2H 



= 0. 



In particular, if t] = 0, = 0, &&amp;gt; = 0, then we have the foregoing equation X = ; and the 
like for the equations F=0, Z=0, and TF=0 respectively. 

Take a, b, c, f, g, h for the six coordinates of the line through the points 



that is, write 



where, of course, 



x, y, z, t ; 



a = y% zrj, f = xw t%, 



h = zw 



af+bg + ch = 0. 
Then the foregoing equation of the cone is 

Atf + b 2 + Cc 2 + Ft* + Gg* + 
- ZA bc - 25 ca - 2(7 ab + 2^ gh + 2 hf + ZH fg 
+ 2Paf + 2Mag - 
2Qbg + 



= 0. 



And this may be regarded as the equation of the conic in terms of the twenty-one 
coordinates of the conic, and of the six coordinates of an arbitrary line meeting the 
conic. It is, in fact, the general form of the equation given in the paper Cayley, 
" On a new Analytical Representation of a Curve in Space," Quart. Math. Jour., 
vol. in. (1860), [284; this Collection, vol. iv. p. 453]. 

112 



84 [731 



731. 



ON THE BINOMIAL EQUATION a!"- 1=0; TRISECTION AND 

QUAKTISECTION. 

[From the Proceedings of the London Mathematical Society, vol. xi. (1880), pp. 4 17. 

Read November 13, 1879.] 

THE solution of the binomial equation x p 1 = 0, p a prime number, or, say rather, 
the equation 

X P~ l + cP- 2 + ...+# + 1 =0, 

depends upon the Jacobian function 

Fa. = x 1 + 0x9 + . . . + a?-*aca v ~ 2 , 

where g is a prime root of p, a. any root whatever of the equation u p ~ l 1 = 0. Taking 
e a factor of p 1, and / for the complementary factor (that is, p 1 = ef), then, if for 
a we write of, or, what is the same thing, taking a/, = /3, a root of u e 1 = 0, we have 

F/5 = X + /3Z, + . . . + /9- Z_ lf 

p 1 

where X , X 1} ..., X e ^ denote each of them a period or sum of f, = , roots, viz. 

e 

X =(1, g\ ...,&lt;7&lt; -n 



(read X = x 1 + a&* + . . . -f xP (f 1]e , and so for the other functions). 

We have, of course, F(l), = X + X l + ... +X e - lt the sum of all the roots = 1; 
and, further, the general property that any rational and integral function of these 
periods is expressible as a sum 

^o-^-o + QiXi + ... + Ct&gt;eiX e i 

with known coefficients 



731] ON THE BINOMIAL EQUATION X p 1=0. 85 

The several cases e = 2, 3, 4, ... may be termed those of the bisection, trisection, 
quartisection, &c., of the equation ; viz. 

e = 2, there are two periods, X, Y, and F( l) = X F; 

e = 3, three periods, X, Y, Z, and Fj = X + yY+y-Z, if 7 is a root of w 3 - 1 = 0; 
e = 4, four periods, X, Y, Z, W, and F8 = X + 8Y+8-Z + 8 3 W, if 8 be a root of w 4 -l=0. 
It is sufficient to attend to the prime roots 7 and 8 of the equations 

u 3 - 1 = 0, w 4 - 1 = 0, 

respectively; for, if 7 or 8 be = 1, we have simply -^(1), = 1; and if 8 be = 1, 
then the function is F(-I), = X + Z-(Y+W), where X+ Z and Y+ W are the 
periods for the bisection. The prime roots 8 are of course i and i, and we have 

F(i) = X + iY-Z-iW, 

F(-i} = X-iY-Z + iW, 
respectively. 

g-i 
As regards the bisection, it is known that (X F) 2 = ( ) 4 p, which is +p or p, 

according as p is = 1 or 3, mod. 4 ; and the values of X, Y are thus determined. 
In what follows, I consider the cases e = 3 and e = 4 of the trisection and the 
quartisection respectively. 

It is to be remembered that, not the division into periods, but the order of the 
periods, depends on the choice of g, a prime root at pleasure of p ; and, in what 
follows, I select the prime root used in Reuschle s Tafeln complexer Primzahlen 
welche cms Wurzeln der Einheit gebildet sind (4to, Berlin, 1875): viz. these are 

p = 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 

59, 61, 67, 71, 73, 79, 83, 89, 97, 

= 2,2,5, 2, 2, 3, 2, -2, 2, 3, 2, 6, 3,10, 2, 

2, 2, 2, 62, 5, 3, 2, 30, 10, 

where I quote the whole series, although I am here only concerned with the values 
of p which are = 1 (mod. 3), or = 1 (mod. 4). 

The periods are consequently those of Reuschle, viz. X, Y, Z are his T/ O , 771, ij z , and 
X, Y, Z, W his r) Q , i} lt 772, 7; 3 : they can of course, without referring to his work, be 
easily recalculated, but it is, I think, convenient to have for his values of g the 
series of residues such as are given (for differently selected values of g) in Jacobi s 
Canon Arithmetics (4to, Berlin, 1839); and I have accordingly taken out of Reuschle, 
and annex, such a table. 

For instance, p=13, the powers of g are 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7; 
and, by writing these down in order in columns of 3 or of 4, 

1 8 12 5 139 

2 3 11 10 265 
4697 4 12 10 

8 11 7 



86 ON THE BINOMIAL EQUATION X p 1=0. [731 

we have the periods X, Y, Z or X, Y, Z, W, belonging to the trisection and the 
quartisection of p = 13. 

I further remark that the equations which I am concerned with are all given in 
Reuschle, but in a somewhat different form ; thus, p = 13, quartisection (see p. 13), he has 

(where observe that here and in every case the value of 7/ 77 3 is at once obtained 
from that of 770771 by a mere cyclical interchange of the suffixes, so that the last 
equation is in fact superfluous) ; the other equations, using 770 + t] l + 772 + 773 = 1 to 
eliminate any constant term which occurs, give my values 

X 2 = ( 0, 1, 2, 0) (X, Y, Z, W), 
XY=( 1, 1, 0, 1)( ), 
XZ = (- 3, - 2, - 3, - 2) ( ). 

Similarly, in the case of a trisection, the equation for 770773 is superfluous, and the 
other equations give my values of X 2 and X Y. 

Reuschle gives also, and I take from him, the cubic and the quartic equations (such 
as &gt;=13, 77 s + 77 2 477+ 1 =0, 77" + T? 3 + 277 2 477 + 3 = 0), which determine the periods in 
the trisections and the quartisections respectively. 

Many of the results obtained accord with, and furnish exemplifications of general 
theorems contained in Jacobi s memoir, "Ueber die Kreistheilung und ihre Anwendung 
auf die Zahlentheorie," Crelle, t. xxx. (1846), pp. 166 189 ; [Ges. Werke, t. vi. pp. 

254274]. 

Trisection, e = 3; p = l (mod. 3). 
We have three periods X, Y, Z; and we thence obtain 

X 9 =(a, b, c)(X, Y, Z), 
XY=(f,g,h)( ), 

the coefficients a, b, c, f, g, h being determinate integers. And, by cyclical inter 
changes, we obtain equations which may be written 

X" = a, b, c, 
F 2 = c, a, b, 
Z- b, c, a, 
XY=f, g, h, 
YZ=h,f,g, 
ZX=g,h,f; 
viz. here and elsewhere the coefficients a, b, c are written to denote the sum 

aX + bY+cZ. 
It is easy to see that 



731] ON THE BINOMIAL EQUATION X p 1 = 0. 87 

in fact, a period contains %(p 1) terms, and in two consecutive periods X, Y, there 

are no terms the product of which is unity; hence XY contains ^(p I) 2 terms, 

each a power of x, and the sum XY+YZ+ZX contains ^(p I) 2 such terms, being 

in fact the sum X+Y+Z taken ^(p 1) times; whence the relation in question. 

Hence also 

YZ + ZX + XY=-(p-l\ 

From the equation X+Y+Z=-l, multiplying by X, and for X 2 , XY, XZ 
substituting their values, we obtain an expression 



which must identically vanish ; viz. the three coefficients must be each of them = ; or 

we must have 

a = -f-g-l, 

b = g h, 
--A-/; 

so that, taking /, g, h as known, the other coefficients a, b, c are given in terms of them. 
The equations give 



We have X.YZ=Y.ZX; that is, X (h, f, g)=Y(g, h,f): or, substituting for X 2 , 
XY, &c. their values, 

h (a, b, c) = g (f, g, h) 

,V +h(c, a, b) 

,/) +f(h,f, g); 

that is, 

ah+f* +0* = gf + ch+fh, 



ch+fh +fg=gh + bh +fg, 
equations which reduce themselves to the single equation 



and this is the only relation obtainable by consideration of the three equal values 

X.YZ, Y.ZX, Z.XY. 

Moreover, this equation being satisfied, the six functions in the three equations become 
each of them =fg h? ; or we have 



that is, 

XYZ-K-fy. 
We have 

Fy.Fy*= X+Y&gt; + Z--YZ-ZX-XY 

+ c-f-g-h)(X+ Y+Z) 



88 ON THE BINOMIAL EQUATION X p -l=0. [731 

that is, 

Fy.Fy-=p. 

We have, moreover, 

(Fy)"- = X*+2YZ + y(Z* 4- 2ZF) + 7 2 (F 2 

[(a, b, c) + 2 (A, / #)] 
+ 7 [(6, c, o) + 2(/ &gt;fl f, A)] 

+ 7 2 [(c, a, 6) + 2(flr, A./)], 
which is 

= {(a + 2A) + 7 (6 + 2/) + 7 2 (c + 20)} (X + 7 2 F + 

as is at once seen by comparing the coefficients of X, Y, Z respectively. 

Hence, writing 

a + 2h + 7 (b + 2/) + 7 2 (c + 20) 



we have 

4 = a + 2A - c - 20 = 3A - 3^r - 1, 

5 = b + 2/ - c - 2^r = 3/ - 3$r. 
We have 

and thence, writing 7 2 for 7, 

equations which give 

Fy.F&lt;f, =p, = 

or, say p = A- AB + B 2 ; viz. p has the complex factor 

A+By, =3/i-3&lt;7- 



Hence also 

(Fy)*=p(A+B 7 ) &gt; 

and, as before, 

Fy.Fy-=p; 

which equations determine Fy, Fy-, and from these and ^(1) = ! we obtain the 
periods X, Y, Z\ we have thus, in fact, the solution of the cubic equation which gives 
these periods. We have already found the coefficients of this cubic equation, viz. 

X+Y+Z=-l, YZ+ZX + XY=-^(p-l), XYZ = h 2 -fg; 
the equation thus is 

&lt;+tf-|&lt;p-i)*+0&-*)-ft 

As already remarked, the values of a, b, c ; /, g, h, and the equations in 77, are in effect 
given in Reuschle ; the complex factors of p, as given p. 1 (7 = 2y 3y 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 I = 0. 



89 



for the primes 7, 13, ..., to 97, the values from Reuschle of a, b, c; f, g, h, and of the 
coefficients of the ^-equation ; also the values of A and B derived from /, g, h by the 
foregoing formulae. It will be seen that all the values are consistent with the theory. 



TABLE FOR THE TRISECTION. 



f 


a, b, c 

/. g, h 


/ + % 


A B 


Page in 
Reuschle 


7 


2 1-2 
1 1 


- 2 1 


2 3 


p. 6 


13 


- 4 - 3 - 2 

1 2 1 


4 - 1 


4 -3 


P- 15 


19 


4 .5-4 
1 2 3 


6 - 7 


2 -3 


p. 26 


31 


7 - 6 - 8 
424 


- 10 - 8 


5 6 


P- 45 


37 


- 8 - 10 - 7 
543 


- 12 11 


- 4 3 


P- 54 


43 


- 11 8 -10 
644 


- 14 8 


1 6 


p. 69 


61 


- 14 - 13 - 15 

587 


-20 - 9 


4 -9 


P- 97 


67 


- 16 - 13 - 16 
967 


22 5 


2 9 


p. 105 


73 


- 16 - 18 - 15 
699 


- 24 - 27 


- 1 -9 


p. 128 


79 


-20 -17 -16 

9 10 7 


-26 41 


- 10 -3 


p. 138 


97 


- 20 - 23 - 22 
10 9 13 


- 32 - 79 


11 3 


p. 1 68 



C. XI. 



12 



90 ON THE BINOMIAL EQUATION 2^1 = 0. [731 

Quartisection, e = 4 ; p = 1 (worf. 4). 
We have four periods X, Y, Z, W ; and we obtain 

Z 2 =(a, 6, c, d)(Z, F, , F), 
XY=(f,g,h,k)( ), 

XZ=(l, m, l,m)( ), 

the coefficients being determinate integers. It can be shown that Z + m = ^(p-l) or 
i) according as /&gt; = 1 or 5 (mod. 8). And then, by cyclical interchanges, 



X" = a, b, c, d, 

F- = d, a, b, c, 

Z- = c, d, a, b, 

W 2 = b, c, d, a, 

X7=f, g, h, k, 

YZ =k,f,g, h t 

ZX =h, k,f, g, 

XW = g, h, k,f, 

XZ = I, m, I, m, 

YW = m, I, m, I. 

We have, in like manner as for the trisection, 



and so also the expression for 

2XY, =XY+XZ + XW+YZ+YW + ZW 

is 

h+k + l + m) = -i(p-l)-l-m; 



nnd, in virtue of the foregoing value of l + m, this is = f(p 1) or (_p + 3) according 
as p = 1 or 5 (mod. 8). 

Again, from the equation X+Y+Z+ W= l, multiplying by X and reducing, 

a = - 1 -/- g-l, 
b= g h m, 
c= h k l, 
d= kfm, 
and thence 

a + 6 + c + d = -l-2 (/+ g + h + k)-2(l+ m), 

and 

a-b + c-d = -l + 2 (m - 1). 



731] ON THE BINOMIAL EQUATION ^1=0. 91 

We have 

X.YZ=Y.ZX = Z.XY, 

that is, 

X(k,f, g, h}=Y(l, m, I, m) = Z(f, g, h, k), 
and thence 

k (a, b, c, d) = I (f, g, h, k) = / (I, m, I, m) 

+f(f&gt;9&gt; h k ) +m(d, a, b, c) +g(k,f, g, h) 
+ g(l, m, I, m) +1 (k, f, g, h) +h(c, d, a, b ) 

+ h(g, h, k,f) +m(m, I, m, I) + k(h, k, f, g), 
that is, 

ka+f 2 + gl + gh=lf + md + lk + m 2 = If +gk + ch + kh, 

kb+fg + gm + h 2 = Ig + am + If + ml =fm +fg + hd + k 2 , 
kc +fh + gl + hk = Ih + mb + Ig + m 2 =fl + g* + ah + kf, 
kd+fk+ gm + fh = kl + me + Ih + Im =fm +gh + bh + gk, 

in which equations a, b, c, d may be regarded as having their foregoing values. 
One of these equations is 

kc +fh +gl + hk = lf+g" + ah + kf, 
that is, 

-k(h + k + 1) +fh+gl + hk = lf+g* - h (/+ g + I + 1) + kf, 
or, reducing, 

l(ff + h ~f- k) = g- + k*- 2hf- hg -h + kf, 
which gives I. 

Again, another equation is 

^ +/9 +gm + h- =fm +fg + hd + k 2 , 
that is, 

-k(g + h + +fg + gm + A 2 =fm +fg - h (k +f+ m) + k\ 
or, reducing, 

m (g + h f k) = k- h- + gk hf, 
which gives m. 

And we have also 

md + Ik + m 2 = gk + ch -f kh, 
that is, 

- m (k+f+ m) + lk + m 2 = gk + kh -h(h + k + I), 
or, reducing, 

I (k + h) -m(f+ k) = gk - k\ 

Substituting herein for I, m their values, we have 

(A- + li) [g 2 + k 2 - 2hf- hg + kf- h] - (/+ k) [k 2 - h 2 + gk - hf] + (h 2 - gk) [g + h -f- k] = 0. 

122 



92 ON THE BINOMIAL EQUATION X p 1=0. [731 

In this equation the only terms of the second order are -h(h+ k), which contain the 
factor h; the terms of the third order contain this same factor h, and throwing it out, 
and reducing, the equation is found to be 



or, as it may also be written, 

gt + fc- 2hf- h + (h* +f*-2gk-k) = ; 

and the foregoing values of I, m are 

(0. + jfc. -2hf-h) - (gh-kf) 

g + h-k-f 

&-h* + gk- hf 

-. 

- 77 / ? 

g+h-k-f 

and by means of these three equations all the foregoing equations are satisfied. 

We have 

FiFi 3 = (X - Zy + (Y - Wy 

W--2(XZ+YW) 



or, substituting for a, b, c, d, this is 

= 1+2 (/+ g + h + k) + 4 (I + m), 
viz. it is 

l) + 4(1 *-); 



or, substituting for l + m its before- mentioned value, then, according as p = l or 5 (mod. 8), 
the value is =p or p ; that is, we have 

i-i 
FiFi 3 = (-) * p. 

Again, we have 

) 2 = (.Y + iY - Z - i W)* 

- YZ+ZW- WX) 
- W) 



where 

A = a b+c d + 2(m l\ = - 1 + 4 (m - I), 



or, since X Y+Z- W = F ( 1), this equation is 

(FiY = (A 
and similarly 



731] ON THE BINOMIAL EQUATION X p 1=0. 93 

Moreover 

[j-(-i)]=(-)V p&gt; = P - 

and we have therefore 

(py = (A* + &-)p, 
that is, 

A- + B 2 =p; 

or the expression A + Bi determined as above is a complex factor of p. 

We may investigate the quartic equation for the determination of the periods X, Y, 
Z, W. The values of X + Y+ Z + W and XY+ XZ+ XW + YZ + YW + ZW are already 
known: for the next coefficient XYZ + XYW + XZW + YZW, we have XYZ = (a, /3, 7, &), 
where each of the coefficients a, fi, 7, B is given under three different forms : the values 
of YZW, ZWX, WXY are (8, a, & 7), (7, 8, , ), (/?. 7, 8 } a); and the required sum 
therefore is 

W\ = -( a 



Taking the first expressions of these coefficients respectively, we have 



2g(l + m) 



We find XYZW most readily as the product of XZ and YW ; we thus obtain 

XYZW=lm(X*-+ Y* + Z*+ W* + 2XZ+2YW) + (l* + m*)(XY+ XW+ YZ+ZW), 
= lm(-a-b-c-d-2l- 2m) - (I 2 + m 2 ) (f+g + h + k\ 
= Im {1 + 2 (/+ g + h+k)}-(l* + m?)(f+g + h + k); 

or, substituting forf+g+h + k its value i (/)-!), this is 

Im -\(l- my (p - 1), = ^ {( + m) 3 -(I- rn^p}. 
Hence the required equation, having roots X, Y, Z, W, is 



where, for the sake of having a single formula, I have retained I + m in place of its 
value =-(p_l) or ^(p+ 3) according as p=l or 5 (mod. 8). 



94 ON THE BINOMIAL EQUATION 0^1=0. 

We thus have the following : 

TABLE FOR THE QUARTISECTION. 



[731 



p 


abed 

f 9 1&gt; * 
I in 


7? 4 + 77 ;! + 
7T T, 1 7,0 


A B 


Page in 
Reuschle 


5 


0100 
0001 

1 1 


1 1 1 


- 1 -2 


p. 2 


13 


0120 
1101 
3 2 


24 3 


3-2 


P- 13 


17 


4 2 3 4 
2011 
1 1 


6 1 1 


- 1 4 


p. 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. 6 1 


53 


2362 
4423 
- 11 9 


7-43 47 


7-2 


p. 80 


61 


4326 
3363 
-11 -12 


8 42 117 


-5 6 


p. 96 


73 


- 16 - 13 - 12 - 14 

6552 
4 5 


- 27 - 41 2 


3 8 


p. 126 


89 


- 19 - 18 - 16 - 14 

4855 
6 5 


- 33 39 8 


-5 -8 


p. 152 


97 


- 22 - 16 - 17 - 18 

8655 
7 5 


- 36 91 - 61 


-9 4 


p. 167 



731] ON THE BINOMIAL EQUATION X p l=Q. 95 

TABLE OF THE POWERS OF REUSCHLE S SELECTED PRIME ROOTS. 

3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 



























2252232 


21 


2 


3 2 


6 3 


10 2 


2 


2 2 


62 5 


3 2 


30 10 


1 


444494 


4 


4 


9 4 


36 9 


6 4 


4 


4 4 


10 25 


9 4 


10 3 


2 


... 3 6 8 8 10 8 


15 


8 


27 8 


11 27 


13 8 


8 


8 8 


52 52 


27 8 


33 30 


3 


2 5 3 13 16 


16 


16 


19 16 


25 38 


36 16 


16 


16 16 


29 41 


2 16 


11 9 


4 


3 10 6 5 13 


14 


3 


26 32 


27 28 


31 32 


32 


32 32 


23 59 


6 32 


63 90 


5 


9 12 15 7 


18 


6 


16 27 


39 41 


28 11 


5 


3 64 


6 3 


18 64 


21 27 


6 


7 11 11 14 


10 


12 


17 17 


29 37 


45 22 


10 


6 61 


17 15 


54 45 


7 76 


7 


3 9 16 9 


3 


24 


20 34 


10 25 


27 44 


20 


12 55 


60 2 


4 7 


32 81 


8 


6 5 14 18 


17 


19 


29 31 


19 32 


35 35 


40 


24 43 


28 10 


12 14 


70 34 


9 


10 8 17 


12 


9 


25 25 


32 10 


21 17 


21 


48 19 


32 50 


36 28 


53 49 


10 


... 7 7 15 


22 


18 


13 13 


28 30 


22 34 


42 


35 38 


67 31 


29 56 


77 5 


11 


4 11 


2 


7 


8 26 


4 4 


32 15 


25 


9 9 


36 9 


8 29 


85 50 


12 


12 3 


19 


14 


24 15 


24 12 


38 30 


50 


18 18 


31 45 


24 58 


58 15 


13 


2 6 


8 


28 


10 30 


21 36 


4 7 


41 


36 36 


5 6 


72 33 


49 53 


14 


6 12 


7 


27 


30 23 


3 22 


40 14 


23 


11 5 


26 30 


58 66 


46 45 


15 


5 


9 


25 


28 9 


18 23 


24 28 


46 


22 10 


50 4 


16 49 


45 62 


16 


10 


5 


21 


22 18 


26 26 


5 3 


33 


44 20 


47 20 


48 15 


15 38 


17 




13 


13 


4 36 


33 35 


3 6 


7 


27 40 


3 27 


65 30 


5 89 


18 




20 


26 


12 35 


34 19 


30 12 


14 


54 13 


44 62 


37 60 


61 17 


19 




6 


23 


5 33 


40 14 


18 24 


28 


47 26 


30 18 


32 37 


50 73 


20 




11 


17 


15 29 


35 42 


39 48 


56 


33 52 


14 17 


17 74 


76 51 


21 






5 


14 21 


5 40 


14 43 


53 


5 37 


16 12 


51 65 


55 25 


22 






10 


11 5 


30 34 


46 33 


47 


10 7 


69 60 


74 47 


48 56 


23 






20 


2 10 


16 16 


37 13 


35 


20 14 


18 8 


64 11 


16 75 


24 






11 


6 20 


14 5 


41 26 


11 


40 28 


51 40 


34 22 


35 71 


25 






22 


18 3 


2 15 


34 52 


22 


19 56 


38 54 


23 44 


71 31 


26 






15 


23 6 


12 2 


11 51 


44 


38 45 


13 51 


69 5 


83 19 


27 








7 12 


31 6 


16 49 


29 


15 23 


25 36 


48 10 


87 93 


28 








21 24 


22 18 


19 45 


58 


30 46 


59 34 


68 20 


29 57 


29 








11 


9 11 


2 37 


57 


60 25 


37 24 


46 40 


69 85 


30 








22 


13 33 


20 21 


55 


59 50 


22 47 


59 80 


23 74 


31 








7 


37 13 


12 42 


51 


57 33 


15 16 


19 77 


67 61 


32 








14 


17 39 


26 31 


43 


53 66 


7 7 


57 71 


52 28 


33 








28 


20 31 


25 9 


27 


45 65 


8 35 


13 59 


47 86 


34 








19 


38 7 


15 18 


54 


29 63 


70 29 


39 35 


75 84 


35 










23 21 


9 36 


49 


58 59 


9 72 


38 70 


25 64 


36 










15 20 


43 19 


39 


55 51 


61 68 


35 57 


38 58 


37 










8 17 


7 38 


19 


49 35 


19 48 


26 31 


72 95 


38 










7 8 


23 23 


38 


37 3 


42 21 


78 62 


24 77 


39 










24 


42 46 


17 


13 6 


48 32 


76 41 


8 91 


40 










29 


44 39 


34 


26 12 


65 14 


70 82 


62 37 


41 












11 25 


9 


52 24 


45 70 


52 81 


80 79 


42 












29 50 


18 


43 28 


11 58 


77 79 


86 14 


43 












8 47 


36 


25 29 


43 71 


73 75 


88 43 


44 












33 41 


13 


50 58 


39 63 


61 67 


59 42 


45 












29 


26 


39 49 


4 23 


25 51 


79 32 


46 












5 


52 


17 31 


35 42 


75 19 


56 29 


47 












10 


45 


34 62 


40 64 


67 38 


78 96 


48 












20 


31 


7 57 


66 28 


43 76 


26 87 


49 












40 


3 


14 47 


45 67 


50 69 


68 94 


50 



96 ON THE BINOMIAL EQUATION X p 1 = 0. [731 



TABLE (continued). 

53 59 61 67 71 73 79 83 89 97 



51 


27 6 28 27 


21 


43 


71 


55 


82 


67 


51 


52 


12 56 54 


24 


69 


55 


27 


57 


88 


52 


53 


24 51 41 


68 


53 


7 


54 


19 


7 


53 


54 


48 41 15 


27 


46 


21 


25 


36 


70 


54 


55 


37 21 30 


41 


11 


63 


50 


12 


21 


55 


56 


15 42 60 


57 


55 


31 


17 


4 


16 


56 


57 


30 23 53 


55 


56 


14 


34 


31 


63 


57 


58 


46 39 


2 


61 


42 


68 


40 


48 


58 


59 


31 11 


53 


13 


47 


53 


43 


92 


59 


60 


22 


20 


65 


62 


23 


44 


47 


60 


61 


44 


33 


33 


28 


46 


74 


82 


61 


62 


21 


58 


19 


5 


9 


84 


44 


62 


63 


42 


46 


22 


15 


18 


28 


52 


63 


64 


17 


12 


37 


45 


36 


39 


35 


64 


65 


34 


34 


39 


56 


72 


13 


59 


65 


66 




49 


49 


10 


61 


34 


8 


66 


67 




56 


26 


30 


39 


41 


80 


67 


68 




64 


57 


11 


78 


73 


24 


68 


69 




63 


66 


33 


73 


54 


46 


69 


70 






38 


20 


63 


18 


72 


70 


71 






44 


60 


43 


6 


41 


71 


72 








22 


3 


2 


22 


72 


73 








26 


6 


60 


26 


73 


74 








40 


12 


20 


66 


74 


75 








41 


24 


66 


78 


75 


76 








44 


48 


22 


4 


76 


77 








53 


13 


37 


40 


77 


78 










26 


42 


12 


78 


79 










52 


14 


23 


79 


80 










21 


64 


36 


80 


81 










42 


51 


69 


81 


82 












17 


11 


82 


83 












65 


13 


83 


84 












81 


33 


84 


85 












27 


39 


85 


86 












9 


2 


86 


87 












3 


20 


87 


88 














6 


88 


89 














60 


89 


90 














18 


90 


91 














83 


91 


92 














54 


92 


93 














55 


93 


94 














65 


94 


95 














68 


95 





















732] 97 



732. 

A THEOREM IN SPHERICAL TRIGONOMETRY. 

[From the Proceedings of the London Mathematical Society, vol. XL (1880), pp. 48 50. 

Read January 8, 1880.] 

IN a spherical triangle, where a, b, c are the sides, and A, B, C the opposite 
angles, we have 

tan \c tan \ a tan \~b sin (A B) = tan ^b sin A tan \a sin B, 
tan |c {1 tan^a tan |6 cos (/I B)} = tan ^b cos A + tan 

which are both included in the form 

. tan^c tani& (cosJ. +i sin . 

tan \a (cosB i sinB) = -= -. - = L n - ; -. 

1 + tanc tan 6 (cos A +tsin 

For the first of the two identities : from 

cos A + cos B cos C 



cos a 



cos 6 = 



sin B sin C 
cos B + cos A cos C 



sin -4 sin C 
we deduce 



cos a 



1 /cos J. cos 5\ cos G /cos 5 cos A 

A . -- _ I _ ---- 

y -y^l-T* 4 J /"Vl Tl A 

sm U \sm 5 sin A) sin (7 Vsm B sin .4 
1 ^ (sin 2^ -sin 2^) cos C sin (4 - B) 



sin C sin J. sin B sin C sin J. sin B 



^ r/Ar .. &lt;, 

,.-7, cos (A+B)+ cos C} 
l 



, 

smCsmA 

sin A 



c. xi. 



98 A THEOREM IN SPHERICAL TRIGONOMETRY. [732 

that is, 

f A r&gt;\ sin C , L x 

sin (A - B) = - (cos a cos 6) 
1 cos c 



sm G sine , 7 , 

= -s (cos a cos 6) ; 

sm c 1 cos c 



or, what is the same thing, 



tan ic sin ( A B} = (cos a cos b). 
sine v 

Here cos a cos & is =(1 + cosa) (1 + CGS&); substituting for successively 

sm c sm a 

and - y , the right-hand side is 
sm b 

1 + cos a . 1 + cos b . ,, 

= -: - sin A . = sm B, 

sm a sm b 

= cot ^a sin A cot ^b sin B ; 

whence, multiplying each side by tan \ a tan ^b, we have the relation in question. 
For the second identity which is 

tan \c {1 tan \a tan \ b cos (.4 J5)} = tan | b cos .4 + tan ^a cos 5 ; 
if on the right-hand side we substitute for cos A, cos B their values 

cos a cos b cos c , cos b cos a cos c 

: ; : and : : - , 

sm 6 sm c sm a sm c 

the right-hand side becomes 

1 (cos a cos b cos c cos 6 cos a cos c) 

J I I * 

sin c ( 1 + cos b 1 + cos a ) 

whence, multiplying the whole equation by sin c (1 + cos a) (1 + cos b), it becomes 

(1 - cos c) {(1 + cos a) (1 + cos b) sin a sin b cos ( A B)} 
= (1 + cos a) (cos a cos b cos c) + (1 + cos b) (cos b cos c cos a). 
We have here 

( * m , A D , i T&gt; ( cos a cos & cos c ) ( cos & C S C COS a) + D 

cos (J. B) = cos J. cos B + sm -d sm B = - : ~. : 

sm 2 c sm a sm b 

by substituting for cos .4, cos B their foregoing values, and for sin -4, sin B their values 

vn vn 

where 

sin 6 sin c sin a sin c 

D = 1 cos 2 a cos 2 6 cos 2 c + 2 cos a cos 6 cos c. 



732] A THEOREM IN SPHERICAL TRIGONOMETRY. 9&lt;&gt; 

The numerator is 

cos a cos b cos c (cos 2 a + cos 2 6) + cos a cos b cos 2 c 

+ 1 cos 2 c (cos 2 a + cos 2 6) + cos a cos 6 . 2 cos c ; 
viz. this is 

= cos a cos b (1 + cos c) 2 (cos 2 a + cos 2 b) (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, 

, A m _ cos a cos 6(1+ cos c) (cos 2 a + cos 2 6) + 1 cos c 

COS Lfl Jj) - -. ; 

(1 cos c) sin a sin b 
and the equation thus is 

(1 - cos c) (1 + cos a) (1 + cos b) - {cos a cos 6(1 + cos c) - (cos 2 a + cos 2 6) + 1 - cos c} 

= (1 + cos a) (cos a cos 6 cos c) + (1 + cos 6) (cos 6 cos c cos a),, 
where each side is in fact 

= cos a + cos 2 a + cos 6 + cos 2 6 cos c (cos a + cos 6) 2 cos a cos 6 cos c ; 
and the second identity is thus proved. 



132 



100 [733 



733. 

ON A FOKMULA OF ELIMINATION. 



[From the Proceedings of the London Mathematical Society, vol. XL (1880), pp. 139 141. 

Read June 10, 1880.] 

CONSIDER the equations 

(a, ...10, 1)=0, 

(A,... 10, i) = o, 

where a,..., A,... are functions of coordinates. To fix the ideas, suppose that each 
of these coefficients is a linear function of the four coordinates x, y, z, w. Then, 
eliminating 6, we obtain V = 0, the equation of a surface ; and (as is known) this 
surface has a nodal curve. 

It is easy to obtain the equations of the nodal curve in the case where one of 
the equations, say the second, is a quadric : the process is substantially the same 
whatever may be the order of the other equation, and I take it to be a cubic ; 

the two equations therefore are 

(a, b, c, d^e, 1) 3 = 0, 

(A, B, CIO, 1) 2 = 0; 
giving rise to an equation 

V, = (a, b, c, dy(A, B, C)\ =0. 

And it is required to perform the elimination so as to put in evidence the nodal 
line of this surface. 

Take 1} # 2 the roots of the second equation, or write 

(A, B, cie, i)*=A(0-0 1 )(0-0,)&gt;, 



that is, 

^ 



733] 
then, if 

we have 



ON A FORMULA OF ELIMINATION. 



101 



, = (, b, c, d^0 lt I) 3 , 
2 = (a, b, c, d$0. 2) I) 3 , 

V = A 3 ^ 



viz. on the right-hand side, replacing the symmetrical functions of # 1( 2 by their 
values in terms of A, B, C, we have the expression of V in its known form 



V = a * 



&c. 



Form now the expressions 



_ # (S) # 2 /? 3 d 3(S) _ d 3(S) 

l *^1^^2j - 2 vj/j ^~ l/i * 2 "2 ^-^1 "l ^-^2j 



each divided by Q^ 6. 2 . These are evidently symmetrical functions of 1} 0. 2 , the 
values being given by the successive lines of the expression 



( 



0, 

-i, 



1, 

o, 



l + 2 , e, 2 + 6 A + e.^d, Sc, 36, a) ; 

0,e,, 0&(0i + 
o, ^, 

Q?Q-?, o 



and, consequently, these same quantities, each multiplied by A 2 , are given by the 
successive lines of 



( 0, 



A*, -2AB, - 

-4 3 , 0, AC, -2BC 

2AB, -AC, 0, C 3 

-4&, 2BC, -C 2 , 



, 3c, 36, a). 



Calling these Jf, F, ^, W, that is, writing 



+ (- A C + 4B 2 ) a, &c., 



then X, F, ^, W are the values of 



each multiplied by J. 2 -=- (0! # 2 ) ; and the functions all four of them vanish if only 
1 = 0, 2 = 0; or, what is the same thing, the equations X = 0, F=0, Z=0, W=Q 
constitute only a twofold system. 



The functions 



( X, Y, Z ) 
F, Z, W 



102 ON A FORMULA OF ELIMINATION. [733 

contain each of them the factor ! 2 , that is, V ; they, in fact, each of them vanish 
if (S^ = o, and they also vanish if B 2 = ; or, by a direct substitution, we have 

XZ - Y* = A - (ft - ^) 2 i- - - A*!,, 



XW-YZ = -(tfi-Wfli + tfo)!*, = -A* l s (8 l + 3 ), 
YW-Z- = - (0i - 0^ 0Ai 3 , = - AW&S0&. 

Or, what is the same thing, these are = J.V, 25V, (7V, respectively; thus the 
first equation is 



- QABb + (- AC + 4 2 ) a} [2ABd - 3ACc + C*a] 

_ (_ A -d + 3ACb- 25(7a) 2 = -A (A 3 d 2 + &c.), = - A V ; 



and similarly for the other two equations. The nodal curve is thus given by the 
twofold system Z = 0, F=0, Z=Q, W=0. 

The method may be extended to the case where, instead of the quadric equation 
(A, B, C~6, 1) 2 = 0, we have an equation of any higher order, but the formulae are 
less simple. 



734] 



103 



734. 



ON THE KINEMATICS OF A PLANE. 



[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 

pp. 1-8.] 

IT seems desirable to bring together under this title various questions which 
have been, or may be, proposed or discussed. We consider two planes in relative 
motion one upon the other, but, for convenience, they may be distinguished as a 
moving plane and a fixed plane, the first moving upon the second. Any point of 
the moving plane traces out on the fixed plane a curve, and any line of the mo vino- 
plane envelopes on the fixed plane a curve; similarly, any point of the fixed plane 
traces out on the moving plane a curve, and any line of the fixed plane envelopes 
on the moving plane a curve. More generally, any curve of the moving plane envelopes 
on the fixed plane a curve, and any curve of the fixed plane envelopes on the 
moving plane a curve. There is, moreover, in the moving plane a curve which rolls 
upon a curve in the fixed plane, and these two curves (a single relative position 
being given) determine the motion. 

Fig. 1. 





MX 

The analytical theory presents no difficulty. Taking in the fixed plane the fixed 
axes Ox, Oy (fig. 1), and, fixed in the moveable plane so as to move with it, the 
axes O^j, 0,3^; then the position of the axes 0,0?^ may be determined, say by 



104 ON THE KINEMATICS OF A PLANE. [734 

a, 13, the coordinates of Oj in regard to Oxy; and by 6, the inclination of O l x l to 
Ox. And denoting by x, y, x l , y^ the coordinates of a point P in regard to the two 
sets of axes respectively, then 

x = a + x-i cos y l sin 0, 

y = ft + x l sin 6 + 3/3 cos 6 ; 
or, what is the same thing, 

a-j = (x at) cos 6 + (y /3) sin 0, 
y l = (x a) sin 6 + (y (3) cos ; 

or, as these last equations may be written, 

#j = otj -f # cos ( 0) y sin ( 0), 
7/j = & + ar sin (- 0) + y cos (- 0), 

where a 1} &, = a cos 6 /Ssin 0, a sin 6 ft cos 0, are the coordinates of referred to 
the axes OiX^, and 6 is the inclination of Ox to O^. 

When the motion is given, a, /3, are given functions of a single variable 
parameter, say of t* ; or, if we please, a, /3 are given functions of 6. 

The velocities of a given point (x, y) are determined by the equations 

x CL (%i sin + y l cos 6} 6 , 

y = j3 + (#! cos 6 y l sin 6} & : 
that is, 



or, as these equations may also be written, 

- (of - a!) sin 6 + (;/ - /3 ) cos 9 = x^ , 

- (x - a ) cos 6 - (y - j3 ) sin = y^ff. 

Hence if x =0, y = 0, we have 

#!# = a sin 6 /3 cos 0, or a = (y /3) , 
2A0 = a cos + sin (9, _ = (#_ a) , 

which equations determine in terms of t, x l and ^ the coordinates in regard to the 
axes 0^x^y l , and x and y the coordinates in regard to the axes Oxy, of /, the centre 
of instantaneous rotation. 

If from the expressions of x l} y l we eliminate t, we obtain an equation between 
(#!, 7/j), which is that of the rolling curve in the moveable plane ; and, similarly, if 

* t may be regarded as denoting the time, and then the derived functions of x, y in regard to t will 
denote velocities ; and, to simplify the expression of the theorems, it is convenient to do this. 



734] 



ON THE KINEMATICS OF A PLANE. 



105 



from the expressions of x, y we eliminate t, we obtain a relation between (x, y), 
which is that of the rolled-on curve in the fixed plane. 

The system may be written 

a . a /3 /3 

x l = sin 6 - , cos 0, X = CL- -, 



,= , cos 



or, if we take as the independent variable, 

X-L a sin 6 $ cos 0, x = a. /3 , 
y l = a. cos 6 + /3 sin 6, y = @ + . 

To find the variations of /, we have 

a?/ = a" sin - /3" cos 6 + a. cos + sin 0, = a" sin - fi" cos + y l , 
yi = a" cos + /3" sin - a sin + /3 cos 0, = a" cos ^ + /8" sin ^ - a; : , 



=a - 



Hence 



/ = x cos ^ + y sin 0, or # = #/ cos y-[ sin ^, 
/ = x sin + y cos 0, ?/ = #/ sin + y/ cos 0, 



values which give x 2 + y 2 = a;/ 2 + y^, which equation expresses that the motion is in 
fact a rolling one. 

Imagine the two curves, and the initial relative position given ; say the two 
points A, A l (fig. 2) were originally in contact, then the arcs AI, A^I are equal, and, 
calling each of these s, and X, Y, X l} Fj the coordinates of / in regard to the two 




sets of axes respectively, we have X, Y, X 1} Y l given functions of s, such that 
X 2 + Y 2 = 1, X 1 2 +Y 1 * = 1, the accents now denoting differentiation in regard to s. 
We have, from the figure, 



Y Y 

= tan" 1 -^ tan" 1 -^, ; 

.A JL i 



C. XI. 



14 



106 



ON THE KINEMATICS OF A PLANE. 



[734 



or, what is the same thing, 

tan = ( FX - F/A ) + (* */ + FT/), 
say 

sin 0, cos = FX - F X X ZX + F IV ; 
and then, as before, 

x = a + #1 cos - 2/1 sin 0, 

2/ = /9 + #1 sin + 2/1 cos ; 
or, what is the same thing, 

x - X = cos 6 (x l Xj) - sin (y x - Fj), 
2/ - F = sin ( Xl - X,} + cos 6 (y l - F t ), 

where X, Y, X l} F 1} and therefore also 6, denote given functions of s. The formulae 
will be of a like form if X, Y, X 1} Y 1 are given functions of a parameter t. 

A well known but very interesting case is when two points of the moving plane 
describe right lines on the fixed plane. This may be discussed geometrically as 
follows: Suppose that we have the points A, G (fig. 3) describing the lines OA , 
OC , which meet in 0; through A, G, describe a circle, centre O n and with centre 

Fig. 3. 




and radius =200i, describe a circle touching the first circle in a point /; and suppose 
that A , CQ denote points on the second circle. Then it is at once seen that, considering 
the first or small circle as belonging to the moving plane, and the second or large 
circle as belonging to the fixed plane, the motion is in fact the rolling motion of 
the small upon the large circle ; and, moreover, that each point of the small circle 
describes a right line, which is a diameter of the large circle. In fact, the angle 
IOiG at the centre is the double of the angle IOC at the circumference; that is, 



734] ON THE KINEMATICS OF A PLANE. 107 

it is the double of the angle /0(7 ; and therefore (the radius of the small circle 
being half that of the large circle) the arcs 1C, IC are equal, so that the rolling 
motion will carry the point C along the radius OC , and will, in like manner, carry 
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 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 O l 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 , OA, 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 x lt y,, the axes 1 x 1 being in the direction 0,B 
and 0^ at right angles to it; calling the length AB=2c, we have 1 A = O l B = c, 

and the angle ABO may be called (but this angle was previously taken with a 

contrary sign). We have then for the point P, having in regard to O,A\ and O.y, the 
coordinates (x 1} y^, 

x = a + XT_ cos i/! sin 01 
y = /3-x 1 s m0-y l cos 0} 

where the sign of y, has been changed, and a = ccos0, /3 = csin0: the equations thus 
become 

x = (c + Xj) cos y 1 sin 0, 

y = (c #1) sin y 1 cos 0, 

where observe that c + x ly c-a\ are the distances M.A, M,B respectively. And we 
have, conversely, 

x l = x cos y sin c cos 20, 

y l = x sin y cos + c sin 20. 
If, in particular, y l = 0, then 

. a?, y = (c + O cos 0, (c-#,)sm0; 
or we have 

of y n - 



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 + #0, 
(c-x l \ each taken positively; if a? 1 * + yi 8 = c s , viz. if P be on the circle having AB 
for its diameter, then y? = (c + #1) (c %i), and we have 

y + x = -(c-a; 1 )(smd -- ^ cos 6 ) -=- y x ( sin - - OOitfj, =- (c-x,) + y lt 
\ c Xi j \ y\ i 

viz. as mentioned above, the curve on the fixed plane is a right line. 
In the general case, we have 

x(c a?0 + 2/2/i = (c 2 - #i 2 - 2/i 2 ) cos 0, 

x yi + y (c + a?i) = (c 2 - #i 2 - /i 2 ) sin 0, 
and thence 

{x (c - x,) + yytf + [xy, + y(c + x,)} 2 = (c 2 - x? - 2A 2 ) 2 ; 

or, what is the same thing, 

oc- {(c - xtf + y:-} + bxycy l + f {(c + serf + y?} = (c- - x? - yff. 

Considering (x 1 , 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 (x lt 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 l write 
x l d ; the equation thus becomes 



x 2 {(x, - 2c) 2 + 2/r} + toyixy + y- (x, 2 + y?) = (x? + y? - 2c^) 2 ; 
or, what is the same thing, 

Oi 2 + yi - 2c#0 2 - (x 2 + y 2 ) (x, 2 + 2/j 2 ) + 4c# (xx l - yy,} - 4c 2 ^- = ; 
and if we suppose herein x = 0, it becomes 

(#i 2 + 2/i 2 - 2c^) 2 - y- (x? + y/) = ; 
or, writing #1 = 7*1 cos 1} y 1 = r 1 sin6 1 , where a = angle QAB, this is 

(r, - 2c cos 00 2 - 2/ 2 = 0, 
or say it is 

TI = 2c cos 0! - y, 

which is the polar equation of the curve described on the moveable plane by the 
point 8, whose coordinates in respect to Ox and Oy are (0, y). 

There is no loss of generality in assuming x = 0. In fact, starting with any point 
8 whatever of the fixed plane, if we draw OS meeting the small circle in A, and 



734] ON THE KINEMATICS OF A PLANE. 109 

through 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 , 05 ; 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 , OB thereof (at right angles to 
each other) pass always through the points A and jB respectively, and the curve is 
that described by the point S on the line OA ; the point describes the circle on 
the diameter AB (the small circle), equation r z = 2c cos 6 l ; and OQ having a given 
constant value =y, we have for the curve described by the point 8 the foregoing 
equation r t = 2c cos 6 l y; or writing y = /, that is, taking S on the other side of 
at a distance OS=f, the equation is r x = 2c cos &i+f , viz. this is a nodal Cartesian 
or Lima^on, the origin being an acnode or a crunode according as /&gt; or &lt;2c; and 
if f=2c, then we have the cuspidal curve or cardioid r x = 2c (1 + cos #1), =4ccos 2 1 . 
The general conclusion is that the centre of the large circle describes on the 
moving plane a small circle (centre Oi), and that every other point of the fixed plane 
describes on the moving plane a Limac,on having for its node a point of the small 
circle, and being, in fact, the curve obtained by measuring off along the radius vector 
of the small circle from its extremity a constant distance. 

Considering in connexion with the point, coordinates (a? 1} y^, (x, y), a second 
point, coordinates (X lt F,), (X, Y), in regard to the two sets of axes respectively, 
we have 



x=(c + ocj cos 6 y l sin 6, X = (c + XJ cos Y l sin 6, 
y = (c XT) sin B y 1 cos 6, Y=(c Xj) sin 6 F : cos 6 ; 

from the first two equations we have 

cos 6 : sin 9 : 1 = a? (c a?j) + yy-i : xy 1 + y(c + x 1 ) : c 2 x? yf ; 

and substituting these values in the second set, we find 

X : Y : 1 

= x {c 2 + c (X, - x,) - X& -Y 1 y 1 }+y{ c (y l - F,) + y,X, - ^ Y,} 
:x{ C (y 1 -Y 1 )- y,X, + x, Y,} 



or the points (x, y), (X, Y), considered as each of them moving on the fixed plane, 
are homographically related to each other. 

To find the curve enveloped on the fixed plane by a given curve of the moving 
plane, we have only in the equation f(x l} 2/0 = of the curve in the moving plane 
to substitute for x l , y t their values in terms of x, y, 6, and then considering 6 as 
a variable parameter, to find the envelope of the curve represented by this equation. 
And, similarly, we find the curve enveloped on the moving plane by a given curve 
of the fixed plane. 




110 ON THE KINEMATICS OF A PLANE. [734 

Thus, in the particular case of motion above considered, writing, as before, 

x = (c + Xj) cos - y 1 sin 6, 

y = (c- n\) sin 6 7/1 cos 6 ; 

or conversely 

x l = x cos 6 y sin c cos 26, 

y l = x sin 6 y cos 6 + c sin 20 ; 
the envelope on the moving plane of the line 

Ax + By + C = 
of the fixed plane is given as the envelope of the line 

[A (c + #0 - Byj\ cos0+{-A+B(c- xj} sin 8 + C= ; 
viz. this is 

{A (c + x,) - By,Y + {Ay, -B(c- atf? - C* = ; 
that is, 

(A 2 + B?) (x* + y* + c 2 ) + 2 (A* - #0 G x l - ^ABcy, = 0, 
a circle. 

But the envelope on the fixed plane of the line 

A^ + Byi + C-Q 

of the moving plane is given as the envelope of the line 

C + (Aas + By) cos 6-(Ay + Bx) sin d - AC cos W + BO sin 20 = 0, 

which can be obtained by equating to zero the discriminant of a qtiartic function, 
and is apparently a sextic curve. 



735] 111 



735. 

NOTE ON THE THEORY OF APSIDAL SURFACES. 



[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 

pp. 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 

(a, y, z\ (x&gt;, y , z \ (X, Y, Z\ (X , Y , Z } 
for the coordinates of corresponding points on the four surfaces respectively. 

The condition of reciprocity gives xx + yy + zz 1 = 0, and (the equations being 
U = 0, V = 0) x , y , z proportional to d x U, d y U, d z U, and x, y, z proportional to 
d x U , dy&gt; U , d? U ; or, what is the same thing, we must have 

x dx 4- 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 4- yy + zz 1 = 0. 



112 NOTE ON THE THEORY OF APSIDAL SURFACES. [735 

The other two surfaces will therefore be reciprocal if only we have the like 
relations between the coordinates (X, Y, Z) and (X 1 , Y , Z }; that is, if 

XX + YY + ZZ -1 = 0, 
X dX+ Y dY+Z dZ=0, 
XdX + YdY + ZdZ = 0. 

To find the apsidal surface, we consider an arbitrary section as cos a + y cos /3 + z cos 7 = 
of the surface U = 0, and seek to determine the apsidal radii thereof, that is, the 
maximum or minimum values of R- = x 2 + y 2 + z- when x, y, z vary subject to these 
two conditions. Writing x , y , z to denote functions proportional to d x ll, d y U, d z ll, 
we thus have the set of equations 

x + \x + fj&gt; cos a = 0, 
y + \y + fi cos /3 = 0, 
z +\z + p cos 7 = 0, 

where X, p are indeterminate coefficients ; taking then X, Y, Z as the coordinates of 
the extremity of the line drawn at right angles to the plane, we have R- = X- -f Y- + Z-, 

X Y Z 

and cos a, cos /3, cos 7 = -p- , -p , -p ; substituting these values in the equation 

x cos a + y cos $ + z cos 7 = 0, 

we have Xx + Yy+Zz = 0, and substituting in the other equations, and instead of 
\, fj, introducing the new indeterminate coefficients p, a-, we obtain 

X , Y, Z = px + &lt;rx , py + a-y , pz + crz . 
Hence these last equations, together with 

R 2 = X 2 + Y 2 + Z 2 = a? + y 2 + z\ 
and 

Xx+ Yy + Zz=l, 

contain the solution of the problem. If for convenience we introduce R 2 to denote 
aP + y ^ + z *, and imagine the absolute values of x , y , z determined so that xx + yy + zz = 1, 
then substituting for X, Y, Z their values in the equations X- + Y 2 + Z 2 = R 2 and 
Xx + Yy + Zz= 1, we find 

R* = p*R- 
and thence 

^- 

or, finally assuming 

p ~ 
we have 

X, 7, Z = x- J2V, y - R 2 y , z - R-z, 
each divided by 

* - 1), 



735] NOTE ON THE THEORY OF APSIDAL SURFACES. 113 

where I recall that x, y , z are proportional to d x U, d y U, d z U, and are such that 
xx + yy + zz = 1 : they in fact denote 

d x U, d y U, d z ll, each divided by xd x U + yd y U + zd z U ; 

and that R- and R- denote x 2 + y* + z 2 and &lt;v a + y a + z 2 respectively. The coordinates 
X, Y, Z of the point of the apsidal surface are thus determined as functions of x, y, z. 

For the apsidal of the reciprocal surface, we have in like manner 

X 1 , Y, Z = x -R *x, y -R *y, z-R *z, 
each divided by 

- */(R 2 R 2 - 1), 

and then the two sets of values give, not only 



as is obvious, but also 

X dX + Y dY+ Z dZ = 0, and XdX + YdY + ZdZ = 0. 

In fact, writing for a moment p, p instead of R-, R 2 , and *J(R-R 2 1) = \/(pp I), = CD, 
then 

X dX +Y dY+ Z dZ 



ft) ft) 

_ x xp (dx pdx x dp (x x p) dco] 

~ f "T* OCCt 

CO ( CO CO ) 

= { x dx + y dy + z dz 

ft) 2 

p (x dx + y dy + z dz) 

(x z + y 2 +z 2 )dp 

p (xdx + ydy + zdz ) 
+ pp (xdx -f ydy + zdz ) 

+ p (xx + yy 4- zz ) dp} 

dco ( . . . 

xx +yy + zz 

&) 

- p ( 2 + y 2 + z - ) 

+ pp (xx + yy + zz )}, 
or, since the terms in { } are 

- p . | dp - p dp -p.^dp + 0+ p dp, = - % (pdp + p dp), 
and 

this is 

= { ^(pdp +p dp)+ codco}, =0, 

in virtue of ar=pp -l. And similarly the other equation XdX + YdY + ZdZ = 
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 x z = n 2 , where 
7i 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 + h) 2 = n 2 , that is, 



= rc 2 , h = --, and a + h = 
that is, the successive values are 

_ a- + n- _ a? + n 2 

1 o &gt; ^2 ri &gt; i 

2a 2ttj 

and the question is, under what conditions do we thus approximate to one determinate 
root (selected out of the two roots at pleasure), say n, of the given equation. 

The nearness of two values is measured by the modulus of their difference ; 
thus a nearer to n, than a l is to n, means mod. (a n) &lt; mod. (a x n), and so in 
other cases; in the course of the approximation a, a lf a, ... to n, any step, for 
instance a to a 1} is regular if a-i is nearer to n than a is, but otherwise it is 
irregular; the approximation is regular if all the steps are regular, and if (after one 
or more irregular steps) all the subsequent steps are regular, then the approximation 
becomes regular at the step which is the first of the unbroken series of regular 
steps. 

We do by an approximation, which is ultimately regular, obtain the value n, if 
only the assumed value a is nearer to n than it is to n ; or, say, if the condition 
mod. (a n) &lt; mod. (a 4 n) is satisfied, and the approximation is regular from the beginning 



736] 



APPLICATION OF THE NEWTON-FOURIER METHOD. 



115 



if mod. (a n) &lt; f mod. n, viz. this condition is a sufficient one*; the first step a to c^ 
will moreover be regular under a less stringent condition imposed upon a; and it would 
seem that, without the condition mod. (a n) &lt; mod. n being satisfied, the subsequent 
steps will in some cases be also regular; that is, that the last-mentioned condition 
is not a necessary condition in order to the approximation being regular from the 
beginning; it is, however, the necessary and sufficient condition, to be satisfied by the 
modulus of a n, in order that the approximation may be regular from the beginning. 
All this will clearly appear from the geometry. 




We take N, N (fig. 1) to represent the values n, n; and similarly A, A 1} &c. 
to represent the quantities a, a lt ... ; we have then 



AN = mod. (a n), 



= mod. (^ n) ..., 



so that the approximation is measured by the approach of the points A, A-^ to N. 
The line NN joining the points N, N passes through, and is bisected at, the origin 
; drawing then QQ through at right angles to NN the condition 

mod. (a n) &lt; mod. (a + n) 

means that the point A, which represents the imaginary quantity a, lies on the 
-ZV r -side of QQ , and it will be assumed throughout that this is so. Take now on the 
line ON, OM = ON, and on N M as diameter, describe a circle, which may be called 
the "circle of unfitness"; regarding as an area the segment hereof which lies on 
the JV-side of QQ , say this is the "segment of unfitness." It will be shown that 
if according as A is situate inside, on the boundary of, or outside the segment of 

* In the Smith s Prize Examination, Jan. 28, 1879, I gave the theorem under the following form : "If a, n 
are imaginary quantities, the latter of them given, and the former assumed at pleasure, subject only to the con 

dition mod.(a-?j) &lt;f mod. n; then if i =a -^. ^^t^" &C&gt;? sh W that the terms a Ul a -"" wil1 conver g e 
to the limit ?i." 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, A^N will be greater than, equal to, or less than AN. It may be added 
that, if A be within or upon the boundary of the segment of unfitness, then A^ 
will be outside it, but this by no means hinders that the next point A.,, or some 
later point, shall be within the segment of unfitness; arid, further, that when A is 
outside the segment of unfitness, then the next point A a , or some later point, may 
very well be within the segment of unfitness; the conclusion is, that A being inside 
the segment of unfitness, A^ is less than AN, but that it does not thence follow 
that A,N is less than A,N, A,N than A*N,...; the approximation although regular 
at the first step, may then, or afterwards, for a step or steps, cease to be regular. 

If, however, AN be less than $ON, that is, if the condition mod. (a - n) &lt; f mod. n 
be satisfied, then the point A lies within the circle centre N and radius NM, and 
is consequently outside the segment of unfitness ; A^N being less than AN, the point 
A l is a fortiori outside the segment of unfitness, and the like for all the subsequent 
points A z , A 3 ,..., that is, in this case, the approximation is regular throughout. The 
circle, centre N, and radius NM, = f mod. n, may be called the "safe circle"; and 
the conclusion is that, if the point A or any subsequent point be within the safe 
circle, then every subsequent point will be within the safe circle, and the approximation 
will be regular. 

The successive points A, A lt A, ... (or, as it will be convenient to call them, 
A lt A.-,, ...) may be obtained each from the preceding one by a simple geometrical 
construction. 

I recall that any circle through the two (imaginary) antipoints of N, N is a 
circle having its centre on the indefinite line NN ; it is such that the ratio of the 
distances of a point thereof from the points N, N respectively has a certain constant 
value, viz. for the circles with which we are here alone concerned, those which lie 
on the N-side of QQ , the centres lie beyond the point N (further away, that is, from 
0), and the values of the ratio, distance from N to distance from N , are less unity. 

Starting then from the given point AI, for which this ratio Afl : A^ r/ has a 
given value, suppose A 1 N = kA 1 N , we describe a first circle (passing of course through 
_4j) for each point of which this ratio has the value k; let the diameter of this 
circle be V^W-^ Fj being the extremity between and N, W^ (not shown in the 
figure), that beyond N ; we then describe a second circle, for which the ratio is 
= & 2 ; let its diameter be F 2 Tf 2 , F 2 being the extremity between and N (or say 
between V l and N), W 2 , that beyond N (or say between N and Tfj); the point 
A n lies on this second circle, and is determined as the single intersection of the line 
FoAj with the second circle. And of course drawing a third circle, for which the 
ratio is =fc 4 , on the diameter V 3 W 3 , then A 3 lies on the third circle, and is the 
intersection with it of the line V 3 A 2 , 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 2 , A 3 , ..., 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 IMAGINARY ROOT OF AN EQUATION. 117 

for which this is the case, the next point A p+1 is at a greater distance, so that 
NA P+1 &gt;NA P ; it is, however, clear that we always arrive at a point A q , such that 
NA q &lt;$ON, and so soon as such a point is arrived at the approximation becomes regular. 

The point A 2 determined from A lt as above, is a point such that the subtended 
angle NAN is = twice the subtended angle NA^ ; or calling the latter angle &lt;/&gt;, 
the former is = 2$. It is, in fact, this property which gives rise to the construction ; 
for let the values of A^, A^ 7 , regarded as imaginary quantities, be called for a 
moment 

P! (cos 0j + i sin 0^, p/ (cos 0/ + i sin 0/) ; 

and, similarly, those of A. 2 N, A. 2 N be called 

p (cos 2 + i sin 2 ), p 2 (cos 2 + i sin 0, ) ; 
then these are the values of j n, a-^ + n, a n, a. 2 + n respectively, or we have 

= A [cos (0j 0/) + i sin (0 X 0/)| = k (cos + i sin &lt;), 

Cvj "T" * P\ 

- = ^ {cos (0, - 2 ) + i sin (0 2 - 2 )} = ^ 2 (cos 2&lt;f&gt; + i sin 
that is, 

tto ?l 



a 2 + n 
which relation between a. 2t a x is in fact the original relation 



and, conversely, a lt a. 2 being thus connected, then the representative A a is obtained 
from the representative point A l by the foregoing geometrical construction. 

I give the analytical proofs; we may without loss of generality take, and it is 
convenient to do so, the axis of x as coinciding with the line ON, and to put also 
ON = 1. We then in place of the original coordinates x, y of any point take the 
new coordinates k, &lt;/&gt; which are such that 

x + iy - 1 

T-^-TT = 
x + iy + 1 



x iy 1 

- ^L __ _ l-pify 

i " ^ ? 
x %y + 1 



equations which may also be written 



- if = e-** [x 2 +(y 
or, what is the same thing, 

x- + f- - 1 - 2y cot &lt;f&gt; = 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 &lt;, 
taken in the second form, shows that &lt;f&gt; is the angle subtended at the point by N, N . 

It is sometimes convenient to write ke^, ke^ ^^p, q respectively; we then have 

. l+p . 1+q 

~ ~~ 



Suppose for a moment that we have (p l9 qj, (p 2) &lt;? 2 ), (p s , q 3 ) as the (p, q) coordinates 
of any three points, the condition that these three points may lie in a line, is given 
in the form, determinant = 0, where each line of the determinant is of the form 



l+p l+q 



1, 



1-p 1-q 
or, what is the same thing, it is 

lpq+p q, l-pq-p + q, 1 + pq j 
pq-l, p-q, p + q-2, 



or, again 

viz. the condition is 



p 3 q 3 -I, p 3 - q 3 , p 3 + q 3 - 

Suppose the (k, &lt;) coordinates of the three points are (I, a), (m, ft), (n, 7) respectively; 
then this equation is 

I 2 1 , I sin a , I cos a 1 =0, 

m 2 !, m sin ft, m cos ft 1 
n 2 1, n sin 7, n cos 7 1 



I 2 1, I sin a, 1 =0, 
m 2 1, m sin ft, 1 
n 2 1, n sin 7, 1 



viz. it is 

I 2 1, I sin a, I cos a 

m 2 1 , m sin ft, m cos ft 
n" 1, n sin 7, n cosy 

or, what is the same thing, it is 

[(I- - 1) mn sin (ft - 7) + (m 2 - 1) nl sin (7 - a) + (n 2 - 1) foi sin (a - ft)] 

+ [(m 2 - n 2 ) I sin a + (n 2 - I 2 ) m sin ft + (I- - m-) n sin 7] = 0. 

If in this equation 7 is put = TT, and ft = 2a, so that sin (a ft) = sin a, the equation 
will contain only terms in sin a, and sin 2a, viz. it will be 



that is, 



[ (m 2 ri*)l + (m 2 1) nl (n- 1) lm] sin a 
+ [- ( I" - 1 ) m n + m ( n- - 1-) ] sin 2a = 0, 

I (m 1) (n + 1) (m n) sin a + m (m + 1) (n I 2 ) sin 2a = 0, 



736] TO AX IMAGINARY ROOT OF AN EQUATION. 119 

or, what is the same thing, 

(m + 1) sin a {I (n + 1) (m -n)+2m(n- l z } 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 2 ; or, writing 
k instead of I, say if I = k, m = n = k* ; that is, writing also &lt;/&gt; in place of a, the 
three points 

(k, &lt;/&gt;), (k\ 20) and (*- TT) 



are in a right line; viz. the point A 1} circle k, subtended angle &lt;f&gt; ; the point A 2 , 
circle k 2 , subtended angle 20 ; and the point V 2 , same circle, subtended angle IT ; 
are in a right line. 

The equation of the circle of unfitness can be obtained more easily in a different 
manner ; but I have thought it worth while to give the investigation by means of 
the foregoing (p, q) coordinates. 

Suppose that p it q l refer to the point A^ : then we have 

(AW = (x, - I) 2 + y, = fa + ty x -!)(%- i yi - 1), = (\^ - l) f ]& - l 

\i PI / \ i jj 

that is, 



Similarly, if p a , q 2 refer to the point A 2 , then 



?1 2&gt; 



since ^ 2 , q. 2 =pi*, q^. The two are equal if 

(l+p 1 )(l 
that is, 

Pi + qi + 1 = 0. 

Writing for a moment x l + iy l = %, x^ iy l = y, we have 



+ 1 -n 

and the equation is 



that, is, 

or substituting for , 77 their values, the equation is 

that is, 



120 APPLICATION OF THE NEWTON-FOURIER METHOD [736 

the equation of a circle on the diameter N M, which is, in fact, the before-mentioned 
circle of unfitness; viz. A-L being on the circumference of this circle, or say on the 
boundary of the segment of unfitness, then A-^N = AN; whence also, according as 
A l is inside or outside the segment, A 1 N&lt;A^N or &gt;AN. 

Suppose A 1 to be on the circle, that is, p 1 +q 1 +l =0; it is easy to show that 
the locus of A is also a circle. We have in fact (p^ + q^ 1 = 0, that is, 

p z + q z + 2^-1=0, 
or say 



viz. this is 
that is, 

or finally 



3 2 2 
Measuring off from in the direction of ON, a distance OS = ^, 2 (always &gt;, 

-L \" Z/V 

since & 2 &lt;1), the circle in question is that on the diameter N S; this is a circle 
touching at N , and containing within it the circle of unfitness ; if k = 1 (that is, for 
A l on the line QQ ) it becomes identical with the circle of unfitness, but except 
in this limiting case it does not meet the circle of unfitness in any point on the 
JV-side of QQ , that is, A^ being on the boundary of the segment of unfitness A,, is 
never on this boundary ; and it thus appears that A 1 being inside the segment, A., is 
always outside the segment. 

It is to be further noticed, that we have 



or 



that is, 

"7 A V \ _, """" -I- ^~" In./ * HT\n&gt; 



where T is the tangential distance of A 1 from the circle of unfitness; there should, 
it appears to me, be some more elegant formula for the ratio A^ + A 2 N 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) &lt; mod. (a + n) and mod. (a ri) &lt; f mod. n, 

present themselves in the real theory. Making the usual construction by means of 

the parabola y = x i , the first condition means that the point A must be taken on 

the ^-side of (fig. 2) ; the second that, in order to the regularity of the approxi- 

Fig. 2. 



N 




mation, A must be taken at a distance from &gt;%ON; in fact, if (as in the figure) 
OA = ON, then AN=NA l} or the point A 1 is at an equal distance with A from 
N; and thence, according as OA is greater or less than ^ON, the point A l is 
nearer or further than A to or from N. 



C. XI. 



16 



122 [737 



737. 

ON A COVARIANT FORMULA. 



[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 

pp. 224226.] 

STARTING from the equation 

/* 



/&gt;&gt; 



which presents itself in the Newton-Fourier problem, it is easy to see that, if a be 
a root of the equation fx = 0, then 

_ ^(x-a)f x-fx 

contains the factor (#-a) 2 , that is, the equation (x - x^f x -fx = 0, considered as an 
equation in x containing the parameter x lt will have a twofold root, if x l is equal 
to any root a of the equation /# = 0; and, consequently, the discriminant in regard 
to x of the function (x-x^f x-fx will contain the factor fx^ But if fx be of the 
order n, then the discriminant is of the order 2?i-2 in # 1; and there is consequently 
a remaining factor &lt;j&gt;x l of the order n 2. 

The like theorem applies to the homogeneous form 



(xy, - x,y} a + ft /(*, y) - (^ - 

which reduces itself to the foregoing on writing a = I, ft = Q, y = y l = 1 ; or, changing 
the notation, say to the form 



737] ON A CO VARIANT FORMULA. 123 

viz. the discriminant hereof in regard to , ij, being a function, homogeneous of the 
order 2n 2 in regard to x, y, to a, ft, and to the coefficients of /(, 77), will 
contain the factor f(x, y), and there will be consequently a remaining factor of the 
order n 2 in (x, y}, 2n 2 in (a, fi) and 2?i 3 in the coefficients of /(, 17). 

The most simple case is when /(%, 77) is the quadric function (a, b, C$1, T?) 2 . 
The form here is 



-(ay-fix)(a, b, c$f, ^) 2 = (a, b, 
where the coefficients are 

a = 2y (a* + bfi) - a (ay - fix), = a fa + (aa + 2bfi) y, 
b = y(ba + cfi)-x (aa + bfi) -b(ay- fix}, 

= aouc + cfty , 
c = - 2# (&a + c/3) c (ay fix), = (26a + cfi) x cay ; 

and we then have 

ac - b 2 = - (26a/3 + cfi 2 ) ax" 

- {2a6a a + (2ac + 46 2 ) afi + 26c/3 2 } y - ( 2 + 26yS) c?/ 2 

- a^ 2 . a 2 - {- 2aca/3} a;y - c/3 2 . cy 2 , 
which is 

= - (aa 2 + 26a/3 + c/3 2 ) (ax- + 2bxy + cy"). 

The discriminant is in this case 

= (a, b, c$a, fi) 2 .(a, b, cQx, y)*. 
In the case of the cubic function (a, b, c, dj[j~, rff, the form is 

(&/ - ai) {3 (aa + bfi, ba + cfi, ca + d&$g, ^} 

-(ay -fa) (a, b, c, d$f, ^ = (a, b, c, 
the values of the coefficients being 

a= a fix +(2aa+3bfi)y, 

b = aax +( ba + 2cfi) y, 

c = - (2ba + cfi) x + dfi y, 

d = - (3ca + 2dfi) x- da y. 

Attending only to the terms in x 2 , we have 

ac - b 2 = - (aa 2 + 26a/3 + cfi-) ax", 
ad - be = - 2 (6a 2 + 2cafi + dfi 2 ) ax 2 , 
bd-c 2 = {(3ac-46 2 )a 2 + (2arf-46c)a/S-c 2 /3 2 }a; 2 . 

162 



124 



ON A COVARIANT FORMULA. 



[737 



And hence, in 

a 2 d 2 + 4ac 3 + 4b 3 d - 3b 2 c 2 - 6abcd, = (ad - be) 2 - 4 (ac - b 2 ) (bd - c 2 ), 
we have the term 

400* . x [a (bof + 2ca/3 + d/3 2 ) 2 + (aa 2 + 26a/3 + c^ 2 ) {(3ac - 46 2 ) a 2 + (2ad - 46c) a/3 - c 2 /? 2 }] ; 

then, forming the analogous term in y 4 , and assuming that the whole divides by 
(a, b, c, d^x, y) s , and also expanding the ayS-functions within the square brackets, we 
find 

Discriminant = 4 (a, b, c, dQx, y) 3 multiplied by 



3a 2 c -3a6 2 
2a*d + 6abc - 8b 3 
6abd+6ac 2 -I2b 2 c 
Qacd - Qbc 2 
ad 2 - c 3 

Writing down the Hessian of (a, b, c, dQa, /3) 3 , 

H = (ac b 2 , ad be, bd c 
and the cubicovariant 

a 2 d - Babe + 2b 3 

abd - lac 2 + b 2 c 
- acd + 2b 2 d - be 2 
. -ad 2 + 3bcd-2c 3 
it is easy to see that the coefficient of x is 

= 3 (a, b, c$a, @) 2 .(H- 
hence also that of y is 



a 2 d - b 3 
Qabd - Qb 2 c 
6acd + Qb-d I2bc 2 
2 +6bcd-8c 3 



/3) 2 , 



= 3(6, c, dJia, /3) 
and the final result is that the discriminant = 4 (a, b, c, d^jx, y) 3 multiplied by 

{3 (a, b, c, d$a, 0) 3 (*, y)fT+ (ay -#&)*}. 
It would be interesting to calculate the result for the quartic (a, b, c, d, e^, rj) 4 . 



March 14, 1879. 



738] 125 



738. 

NOTE ON A HYPERGEOMETRIC SERIES. 



[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 

pp. 268270.] 

IN the memoir on hypergeometric series, Schwarz, "Ueber diejenigen Falle, &c.," 
Crelle, t. LXXV. (1873), pp. 292335, the author shows, as part of his general theory, 
that the equation 



x .\ x dx x.\x 



which belongs to the hypergeometric series F(, -j^, f, a?), is algebraically integrable, 
having in fact the two particular integrals 



V(- a 5 



where a is a prime sixth root of -1, 6 +l=0, or say a 4 -a 2 +l=0 (see p. 326, 
a being for greater simplicity written instead of S 2 , and the form being somewhat 
simplified). 



It is interesting to verify this directly ; writing first y = ^(Y) and then x = X 3 , 
the equation between Y, X is easily found to be 



F .__ F _ 1 

1 dX 2 l-X 3 * dX 2 (dXj + I-X** 

and the theorem in effect is that that equation has the two particular integrals 



P and Q being linear functions of X : in fact, 

P = a - a.*X, 



126 NOTE ON A HYPERGEOMETRIC SERIES. [738 

Starting say from the equation 



or, as it is convenient to write it, 

F = p4 + $, 

where P and Q are assumed to be linear functions of X, we have 



d*Y -&gt; 3 

A = - jp-ip 2 - iQ i Q -\ 

and thence 

dX" 
Y d ? = | (P + Q ) + iP-^P + 



where P , Q are written to denote the derived functions of P, Q respectively. 

Substituting these values, the resulting equation contains on the left-hand side 
a rational part, and a part with the factor P~%Q~*, and it is clear the equation 
can only be true if these two parts are separately = 0. We have thus two equations 
which ought to be verified ; viz. after a slight reduction these are found to be 

1 9Y- Y 

(QP* + PQ*) + j^ (P + Q ) - f -^ 3 (P + Q) = 0, 



P Q - 1 + Qr-P - + PQP Q + PQ (PQ f + P Q) - P 2 ^ 2 = 0, 



and it is very interesting to observe the manner in which these equations are, in 
fact, verified by the foregoing values of P, Q. 

We have 

p + Q = (a-tf)(i+X), p + y = *-(?, 

and hence 

2X(P + Q )- 

or, in the first equation, the second part 



is 



738] NOTE ON A HYPERGEOMETRIC SERIES. 127 

viz. this is 



We have 

QP * + PQ 2 = a 10 (- a 5 + aX) + a 2 (a - a 5 Z), 

= a 3 - a 15 - (a 7 - a 11 ) X, = (a - a 5 ) X ; 
and 



hence 



and the sum of the two parts is = 0. 

Similarly as regards the second equation, the second part 



s 



Here PQ + P Q is a (a - a 5 X) - a 5 (- a 5 + aX), which is = 1 + 2Z ; and PQ being 

= 1+X + X-, the term in { } is 



hence, outside the { } writing for PQ its value = 1 + X + X 2 , the term is 



which is the value of the second part in question ; the first part is 

(PQ + QPJ-PQP Q , =(1 
and the sum of the two terms is thus = 0. 



128 [739 



739. 

NOTE ON THE OCTAHEDEON FUNCTION. 

[From the Quarterly Journal of Pure and Applied Mathematics, vol. XVI. (1879), 

pp. 280, 281.] 

A SEXTIC function 

U=(a, b, c, d, e,f, g$x, y)\ 
such that its fourth derivative 

(U, If?, = (ae-4,bd + 3c*)x* 
+ 2 (af - 3be + 2cd) a?y 
+ (ag - 9ce + 8d 2 ) afy 8 
+ 2 (bg - 3cf + 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 4 ) of the octahedron function. 

But the equations may be satisfied in a different manner; viz. the first and last 
equations give 

7 _3c 2 ,_3e 2 
S~ 



739] NOTE ON THE OCTAHEDRON FUNCTION. 129 

and, substituting these in the remaining equations, they become 

^ (- 9ce + 8d 2 ) = 0, - 9ce + 8d- = 0, ^ (- 9ce + 8d 2 ) = 0, 

all satisfied if only 9ce + 8d* = 0. Assuming b =/= 2, the values are 

b, c, d, e,f=2, 2^(2), 3, 2V(2), 2, 
and the form is 



xy 



This is, in fact, a linear transformation of the foregoing form XY(X* F 4 ) ; for 
writing 



we have 



and therefore 



or finally 



and the two forms are thus identical. 



c. XL 17 



130 



740. 

ON CERTAIN ALGEBRAICAL IDENTITIES. 

[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 

pp. 281, 282.] 



IF P , P lt P 2 are points on a circle, say the circle 2 + ?/ 2 = l, then it is possible 
to find functions of (P , PO and of (P lt P 2 ) respectively, which are really independent 
of Pj, and consequently functions of only P and P 2 : 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 , Pj) and of (P 1; P 2 ) being 
#03/1 #i2/&lt; an d #i# 2 + y\y^ %\y* %y\&gt; we have 



and another like equation. This depends obviously on the circumstance that the 
coordinates of a point of the circle are expressible by means of the functions sin, 
cos, x = cos u, y = sin u ; and the identity written down is obtained by expressing the 
cosine of u z u , = (w 2 u^) + (i^ w ) m terms of the cosines and sines of u a i^ 
and U^ UQ. 

Evidently the like property holds good for a curve, such that the coordinates of 
any point of it can be expressed by means of " additive " functions of a parameter 
u ; where, by an additive function f(u), is meant a function such that f(u + v) is 
an algebraical function of f(u), f(v) ; the sine and cosine are each of them an additive 
function, because 

sin (u + v) = sin u V(l 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), 
&lt;j&gt;(u),..., where f(u + v), &lt;f&gt;(u + v\... are each of them an algebraical (rational) function 
of f(u}, (u), . . . , f(v), &lt;f&gt;(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 , P 1} P 2 , points on the curve, and (#, y , 2 ), (x lt y lt z^), (# 2 , y 2 , z a ), the coordinates of 
these points respectively, we have in the same way, from u 2 U Q = (u 2 v^) + (MI M O ), 
three equations, of which the first is 



1 - teicfxf (1 - texfxff (1 - l&xfxff - A 2 (d^yo^o - xy\Zif (o^y^i - 
The form of the right-hand side is 

A + Bx^z-i 



C + Dx^y^z-i 

where A, B, C, D are each of them rational as regards a^ 2 ; and it is easy to see 
that the equation can only subsist under the condition that we have separately 

^0^2 _ -4 _ -5 

~ 



implying of course the identity AD- 5(7=0. The values of B and D are found 
without difficulty; we, in fact, have 

B = 2k 2 (x/ - x 2 ) (xfyjsQj&i + 
D = 2A 8 



so that, comparing the left-hand side with B + D, we have the identity 

a *yoV ~ ^o 2 2/ 2 2 ^ 2 = (^ 2 2 - tfo 2 ) (1 - ^a? a V), 
which is right. The comparison with A + C would be somewhat more difficult to effect. 



172 



132 [741 



741. 



ON A THEOREM OF ABEL S RELATING TO A QUINTIC 

EQUATION. 



[From the Proceedings of the Cambridge Philosophical Society, vol. III. (1880), 

pp. 155159.] 

THE theorem in question is given, (Euvres Completes, [Christiania, 1881], t. II., 
p. 266, as an extract from a letter to Crelle dated 14th March, 1826, as follows: 

" Si une equation du cinquieme degre dont les coefficients sont des nombres 
rationnels est resoluble algebriquement, on peut donner aux racines la forme suivante : 



x = c + caaa 
ou 

a = m + n V(l + e-) + V|&gt; (1 + e 3 + \/(l + e-))], 

a^ = m- n V(l + e-) + &gt;J[h (1 + e 2 - V(l + e 2 ))], 

a 2 = m + n V(l + e-} - \/[h (1 + e 2 + V(l + e 2 ))], 

a 3 = TO - n V(l + e 2 ) - VO (1+ e- - V(l + e 2 ))], 
A = K + K a + K"a, + K "aa,, A 1 = K+ JK a, + K"a 3 
A 2 = K + K a, + K"a + K "aa, , A 3 = K + K a, 

Les quantitds c, ^, e, m, n, K, K , K", K " sont des nombres rationnels. Mais de 
cette maniere 1 equation of + ax + b = n est pas resoluble tant que a et b sont des 
quantites quelconques. J ai trouve de pareils theoremes pour les equations du 7 fcme , 



It is easy to see that a; is the root of a quintic equation, the coefficients of 
which are rational and integral functions of a, a 1} a 2 , a s : these coefficients are not 
symmetrical functions of a, a 1} a. 2y a 3 , 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 a 1} a^ into a 2 , 2 into a :i , 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 l , a. 2 , 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 aa^a^, a^^a, 
Oya. A aa l} a-faa-fi*. And conversely, the quartic equation being of this form, x will be 
the root of a quintic equation, the coefficients whereof are rational and integral 
functions of c, h, e, m, n, K, K , K", K ". 

To investigate the form of a quartic equation having the property just referred 
to, let it be proposed to find 7, 7 functions of e, h, such that 7 2 + 7 /2 is a rational 
function of e, h, but that y 2 y 2 , 77 are rational multiples of the same quadric radical 
\/#. Assume that we have 

72 _ 7 2 = ^p V0, 77 = g V0 ; 
then 



(7* +y? 

that 7 2 + 7 2 may be rational, we must have p 2 + q 2 = \"0, or say p 2 + q 3 = h 2 6 ; hence, 

2 &lt;f 
6 = j- + ^ 2 must be a sum of two squares, or, assuming one of these equal to unity 

and the other of them equal to e 2 , say 6 = 1 + e", we satisfy the required equation by 
taking p h, q = he : viz. we thus have 



77 = he Vl + e 2 , 7 2 + y - = 2h (1 +e 2 ); 

and thence also 

r = h (1 + e 2 + VlTe 2 ), 7 /a = h (1 + e* - Vl + e 2 ), 

the roots of these expressions, or values of 7, 7 , being such that 

77 = he Vl + e 2 . 
Taking now a rational, =m suppose, and /3 a rational multiple of 

VI + e 2 , = A Vl + e\ 
suppose ; it is easy to see that the quartic equation which has for its roots 

a, !, 2 , a 3 = a + /3 + 7, a - 



has the property in question, viz. that every rational function of the roots unchangeable 
by the cyclical change a into a 1} a x into a 2 , a., into a 3 , a 3 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 

&lt;(tt, a lf a 2 , a 3 ) + ^(a 1} a. 2 , a ;{ , a)+&lt;(a 2 , a 3 , a, a 1 ) + ^)(a 3 , a, a 1} a 2 ): 



134 ON A THEOREM OF ABEL S RELATING TO A QUINTIC EQUATION. [741 

and if &lt;/&gt;(, c^, a, a 3 ) contains a term a. m fi n ffy v, then the other three functions will 
contain respectively the terms 



viz. the sum of the four terms is 
n [{1 + (- 



This obviously vanishes unless p and q are both even, or both odd ; and the 
cases to be considered are 1, n even, p and q even ; 2, n odd, p and q even ; 
3", n even, p and q odd; 4, n odd, JD and q odd. Writing, for greater distinctness, 
2n or 2w + 1 for n, according as n is even or odd, and similarly for p and q, the 
term is, in the four cases respectively, 



= 2a B /S 2n+1 



The second, third, and fourth expressions contain the factors 

(7 2 -7 /a )&gt; ry (7 a -7 /2 ) ^77^ 

respectively; and the first expression as it stands, and the other three divested of 
these factors respectively are rational functions of a, /S 2 , 7 2 , 7 *, that is, they are 
rational functions of m, n, e, h. But the omitted factors fi (j 2 - &lt;y 2 ), 77 (7- 7 /2 ), 
fiyy , =2wA(l + e 2 ), 2h-e(l+e z ), nhe(l + e 2 ) are rational functions of n, Jt, e; hence 
each of the original four expressions is a rational function of m, n, h, e; and the 
entire function 

&lt;(, a 1} a 2 , a 3 ) + ^&gt;(a 1 , a,, a 3 , a) + (f&gt;(a 2 , a 3 , a, a 1 ) + ^(a 3 , a, a lt ) 
is a rational function of m, n, h, e. 

Replacing a, 0, y, y by their values, the roots of the quartic equation are 
m + n V(l + e 2 ) + J[h (1 + e 2 + V(l + e 2 ))], 
m - n V(l + e 2 ) + J\h (1 + e 2 - V(l + e 2 ))], 
m + n V(l + e 2 ) - v O (1 + e 2 + V(l + e 2 ))], 
m - w V(l + e 2 ) - V[A (1+ e 2 - V(l + e 2 ))]. 



And I stop to remark that taking m, n, e, h = - t +|, 2, -1 respectively, the 
roots are 



741] ON A THEOREM OF ABEL S RELATING TO A QUINTIC EQUATION. 135 

viz. these are the imaginary fifth roots of unity, or roots r, r 2 , r 1 , r 3 of the quartic 
equation # 4 + # 3 + # 2 +#+l = 0; which equation, as is well known, has the group 



Reverting to Abel s expression for x, and writing this for a moment in the 
form 



the quintic equation in a; is 

= (x - c) 5 

+ (x cf . 5 (pr + qs) 
+ (# c) 2 . 5 (p-s + q*p + r-q + s 2 r) 

+ (x c) . 5 (p 3 q + q*r + i^s + s s p) + 5 (p-r- + q-s -) opqrs 
+ (x c) . (p 5 +q 5 + r 5 + s 5 ) 

+ 5 (p s rs + q 3 sp + r^pq + s s qr) 
5 (p 2 q 2 r + q 2 r*s + r-s-p + s~p 2 q). 



If we substitute herein for p, q, r, s their values, then, altering the order of the 
terms, the final result is found to be 



+ (x c) 3 . 5 (A A + -4 1^.3) 

+ (x cf . 5 (A- A !,.., + AfAzdyft, + A.fAzaa-L + A^Aa 

+ (x c) . 5 ( A^A-fi^a-^ -f A^AaM^a + A^A^a 2 ^ + A 



+ (x - c) . - (^ 

+ 5 (A A^Aja&s + A?AiA&& + A^A^Aaa^ + A/AA^aJ (aa 1 a 2 a 3 ) a 
- 5 (A^-A^AM^ + Ai-A 2 A 3 ciM 3 + AfA?Aa& + AfAfA&a^ (aa^a^ 2 ; 
viz. considering herein A, A l&gt; A.,, A s as standing for their values 

K -f K a + K"a, + K "aa, &c. 

respectively, each coefficient is a function of a, a lt a, 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 ". 



136 



[742 



742. 



ON THE TRANSFORMATION OF COORDINATES. 



[From the Proceedings of the Cambridge Philosophical Society, vol. in. (1880), 

pp. 178184.] 

THE formulae for the transformation between two sets of oblique coordinates assume 
a very elegant form when presented in the notation of matrices. I call to mind that a 
matrix denotes a system of quantities arranged in a square form 

( , ft , 7 )- 

, & , y 

a", (3", y" 

see my "Memoir on the Theory of Matrices," Phil. Trans, t. CXLVIII. (1858), pp. 17 
37, [152]; moreover (a, ft, yQx, y, z) denotes ax + @y + yz, and so 

( a , , 7 ]fo 2/&gt; z ) 

* , , 7 

a", 0", y" 

denotes 

(ax + /3y + yz, OL X + $y + y z, a"x + ft y + y"z}, 

and again 

( a , (3 , y J&gt;, y, *$, 1?, S) denotes (a#+/3y + 7 2) 

of Q ry 

*", & , y 



Consequently 

( a &gt; 

", 



7 
7 
7 



y, z~$%, 17, (^ = ( a , a , a 



7, 



742] 



ON THE TRANSFORMATION OF COORDINATES. 



137 



In the case of a symmetrical matrix 

( a, h, g ), 
h, b, f 



the equal expressions 

(a, h, g $a?, y, 
h, b, f 

ff&gt; f&gt; c 
.are also written 

(a, b, c, f, g, 
In particular, if 

then 

( a, h, g $#, y, zf is written (a, b, c, f, g, h\x, y, zf. 

h, b, f 



Two matrices are compounded together according to the law 



&gt; i?, )&gt; =( a &gt; ;i # $f 17. fe y, z \ 
h, b, f 
g, f, c 

f, ff V, )&gt; or (. *?, 

(f, *7, ) = (*, y, *), 



(a, a , a"), (/3, , /S"), (7, 7&gt; 7") 



a , 6 , c 
a , b , c 
a", b", c" 


5 


a , /3 , 7 

, , 7 
a", yS", 7" 


) = (a , 6 , c ) 
(a , b , c ) 
(a", b", c") 


)&gt; ! 

&gt; ;&gt; 
&gt; )&gt; 



viz. in the compound matrix, the top-line is 

(a, b, c$a, a , a"), (a, 6, c$/3, /3 , "), (a, 6, c$ 7 , 7, 7"), 

and the other two lines are the like functions with (a , b , c ), and (a", b", c"), re 
spectively, in the place of (a, 6, c). 

The inverse matrix is the matrix the terms of which are the minors of the 
determinant formed out of the original matrix, each minor being divided by this 
determinant, viz. 



* , P, 7 

", ", 7" 
where V is the determinant 



7 a" 7"a , 7"a 7", 70 7 a 



C. XI. 



18 



138 ON THE TRANSFORMATION OF COORDINATES. 

Coming now to the question of transformation, write 



[742 



x y 



x y z Xi 2/x z l 



X 


1 v fj, I a 


a a" = x 




y 


v 1 X j /3 


ft ft" y 


a w 


z 


/i X 1 | 7 


y 7 " z 




*l 


a /3 7 i 1 


if // *&gt; 
"i /*l *^1 




2/i 


a 7 | "i 


1 Xj y x 


v n, 


ti 


a" " 7" ! Mi 


Xj 1 ^ 





viz. the axes of x, y, z are inclined to each other at angles the cosines whereof are 
X, p, v: those of x l , y l , z l are inclined to each other at angles the cosines whereof 
are Xj , ^ , v l : and the cosines of the inclinations of the two sets of axes to each 
other are a, ft, 7 ; a , ft , 7 ; a", ft", 7" : as is more clearly indicated in the diagram, 
the top-line showing that cosine-inclinations of x to 



are 



, y, z, i, 2/i&gt; z i&gt; 

1, v, p, a, a, a", 



respectively, and the like for the other lines of the diagram. The letters H, Oj, F, 
W are used to denote matrices, viz. as appearing by the diagram, these are 



( 1, V, /* ), ( 1 , W1 , 

i;, 1, X i/j, 1, 

U., X, 1 | /*!, Aa, 



), ( a , ft , 7 ), ( a, a , a"), 



A P, P 
i -jt 

y, y , 7 



respectively. 

The coordinates (x, y, z) and (^ x , y ly z^) form each set a broken line extending 
from the origin to the point ; hence projecting on the axes of x, y, z and on those 
of #i&gt; 2/i 2 i respectively, we have two sets, each of three equations, which may be 
written 

y, z} = (W^ L x l , y,, z,}, 



where of course each set implies the other set. 
We have 



(x , y , z) = 



y lt z,), = 
y, z), = 



, y lf z,}, 
; , y , *), 



the first giving in two forms (, ?/, ^) as linear functions of (X, y ly z^, and the 
second giving in two forms (x 1} y l , z^ as linear functions of (x, y, z}; comparing 
the two forms for each set, we have 

n- 1 w = F- 1 H! , w~ i n = nr 1 F, 



742] ON THE TRANSFORMATION OF COORDINATES. 

or, what is the same thing, 



139 



where in each equation the two sides are matrices which must be equal term by 
term to each other; but, the matrices being symmetrical, the equation thus gives (not 
nine but only) six equations. Writing 



and 



we have 



(a, b, c, f, g, h) = (l-A, 2 , l-/u 2 , 1 - v-, fjuv-\, V\-/JL, \fj,-v), 



h, b, f 
g. f &gt; c 
The first equation, written in the form 

V( a, h, g )W = 
h, b, f 



denotes the six equations 

(a, b, c, f, g, h) (a , , 7 ) 2 

( , P, yy 

(" fi", 7") 2 

(a , , 7 )(", r, 



(a, 



and 
then 



And, similarly, writing 

1 

hi, bj, fj 

and the second equation, written in the form 



=K , 

=K , 



gi, 



182 



140 ON THE TRANSFORMATION OF COORDINATES. [742 

denotes the six equations 

(a,, D!, c x , f 1} gj, h x ][a, a , a") 2 =K^ , 

08, ff, PJ = #! , 

(y&gt; y&gt; 7") 2 = Ki 

03, /3 , r$7&gt; y, 7") 



The two sets each of six equations are, in fact, equivalent to a single set of six 
equations, and serve to express the relations between the nine cosines 

(a, 0, 7, , P, 7 , "&lt; ". 7"), 

and the cosines (X, //., y) and (X^ ^, ^). Observe that the nine cosines are not 
(as in the rectangular transformation) the coefficients of transformation between the 
two sets of coordinates. 

From the original linear relations between the coordinates, multiplying the 
equations of the first set by x, y, z and adding, and again multiplying the equations 
of the second set by (x, y 1} ^) and adding, we have 



(fl $&gt; , y , z)* 

(rVft^, y lt z 1 ) 2 = (VQx, y 



But 

(W^, ?/!, ^$a;, y , z} 
and 



denote one and the same function ; hence 

(n$&gt;, y, &lt; gr) 2 = (n i $a? J , ?/!, ^) a , 
that is, 

(1, 1, 1, X, fjL, v^x, y, 4 2 =(1, 1, 1, X 1? A*!, ^$i, y^ ^) 2 , 

or the linear relations between (x, y, z) and (a?!, y 1} ^) are such as to transform 
one of these quadric functions into the other: the two quadrics, in fact, denote the 
squared distance from the origin expressed in terms of the coordinates (x, y, z) and 
Oi, 2/i, *i) respectively. 

Since the nine cosines are connected by six equations, there should exist values 
containing three arbitrary constants, and satisfying these equations identically : but, 
by what just precedes, it appears that the problem of determining these values is, in 
fact, that of finding the linear transformation between two given quadric functions: 
the problem of the linear transformation of a quadric function into itself has an 
elegant solution; but it would seem that this is not the case for the transformation 
between two different functions. 



742] ON THE TRANSFORMATION OF COORDINATES. 141 

The foregoing equation 

# = (a, b, c, f, g, h$ a , 0, 7 ) 2 , 

is a relation between X, /j,, v, the cosines of the sides of a spherical triangle, and 
(a, 0, 7) the cosines of the distances of a point P from the three vertices: it can 
be at once verified by means of the relation A + B + C = 2?r, and thence 

1 cos- A cos 2 B cos 2 C + 2 cos A cos B cos (7=0, 

which connects the angles A, B, C which the sides subtend at P. Writing a, b, c 
for X, fi, v, and /, g, h for a, /3, 7, the relation is 



+ 2 (be a) gh -f 2 (ca b) hf+ 2 (ab c)fg, 
viz. this is 

1 - a- - b 2 - c 2 -/ 2 - g 2 - h? + 2abc + 2agh + 2bhf+ 2cfg 

- a 2 /- - &y - c*h? + 2bcgh + 2cahf + 2abfg = ; 

where (a, b, c, f, g, h) are the cosines of the sides of a spherical quadrangle ; 
(a, b, c), (a, h, g), (h, b, f), (g, f, c) belong respectively to sides forming a triangle, and 
the remaining sides (/, g, h), (b, c, f), (c, a, g), (a, b, A) are sides meeting in a vertex. 

The equation 

J ft r i/ 1 = (a, b, c, f, g, h][a, /3, 7) (a , /3 , 7 ) 

is a relation between X, /j,, v, the cosines of the sides of a spherical triangle ; a, /&, 7, the 
cosines of the distances of a point P from the three vertices ; of, /3 , 7 , the cosines of 
the distances of a point Q from the three vertices ; and v l , the cosine of the distance 

PQ. 

Drawing a figure, it is at once seen that 

Vl = oa + VlT-^o 2 Vl - a 2 cos (0 - ), 
where 

cos = . , 

and therefore 

sin = 



- _ 
V 1 - a 2 V 1 - v* 

also 



- _ _ - 

Vl - a 2 Vl - v * 
and therefore 

Vv~ 

sm ff - . ____ . ___ , 
Vl - a 2 Vl - iX 



i/, = aa! + j J {(/3 - avljtft - ccV) + Vv V j. 



A = (a, b, c, f, g, hja, /8, 7?. K" = (ft, ...Jo , , 7 ) - , 
give 



142 ON THE TRANSFORMATION OF COORDINATES. [742 

the values of V, V being 

V = l _ a 3 - /8 s - i/ a + 2a/9i/, 

V = 1 - a " - ff* - v * 
the resulting value of j/, is therefore 

The equations 

A = (a, b, c, f, 

(ga + f/3 + 07 ) 8 = 

(ga + f/3 + e7 ) 2 = 7vV : 
and we therefore have 

(ga + f/8 + c 7 $ga + f/9 -f cy ) = ^ V V V ; 
recollecting that 1 - i/ 2 = c, the formula thus is 

"i = + c |(/3 - oW ~ a v ) + g (g + W + cy$g + f/? 4- cy )| , 
or say, 

Kv t = K*a + c {A" (ft - av-%/3 - ct v) + (ga + f/3Jga + f/3 )} + g (ay + a y) + f (fty + /3 y) + cyy. 
The sum of the first and second terms is readily found to be 



and the equation thus becomes 

Kv, = (a, b, c, f, g, h$a, ft, 7 $a , /3 , 7 ), 
as it should do. 



743] 143 



743. 

ON THE NEWTON-FOURIER IMAGINARY PROBLEM. 



[From the Proceedings of the Cambridge Philosophical Society, vol. in. (1880), 

pp. 231, 232.] 

THE Newtonian process of approximation to the root of a numerical equation 
y(w) = 0, consists in deriving from an assumed approximate root a new value 

/"() 
= f,--j-A which should be a closer approximation to the root sought for: taking 

the coefficients of f(u} to be real, and also the root sought for, and the assumed 
value , to be each of them real, Fourier investigated the conditions under which 
is in fact a closer approximation. But the question may be looked at in a more 
general manner: may be any real or imaginary value, and we have to inquire in 
what cases the series of derived values 



J: =*_ .-* 

/ () 

converge to a root, real or imaginary, of the equation f(u) = 0. Representing as usual 
the imaginary value , =x + iy, by means of the point whose coordinates are x, y, 
and in like manner ,, = #i + ty 1 , &c., then we have a problem relating to an infinite 
plane; the roots of the equation are represented by points A, B, (7,...; the value 
| is represented by an arbitrary point P; and from this by a determinate geometrical 
construction we obtain the point P x , and thence in like manner the points P 2 , P 3 , ... 
which represent the values ,, ,, :! ,... respectively. And the problem is to divide 
the plane into regions, such that, starting with a point PI anywhere in one region, 
we arrive ultimately at the root A ; anywhere in another region we arrive ultimately 
at the root jB; and so on for the several roots of the equation. The division into 
regions is made without difficulty in the case of a quadric equation ; but in the next 
succeeding case, that of a cubic equation, it is anything but obvious what the division 
is : and the author had not succeeded in finding it. 



144 [744 



744. 

TABLE OF A^O 71 - n (m) UP TO ra = n = 20. 



[From the Transactions of the Cambridge Philosophical Society, vol. xin. Part I. (1881), 

pp. i_4. Read October 27, 1879.] 

THE differences of the powers of zero, A m O n , 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 10 is 
given in Herschel s Examples (Camb. 1820), and is reproduced in the treatise on 
Finite Differences (1843) in the Encyclopedia Metropolitana. But, as is known, the 
successive differences A0 n , A 2 W , A 3 n , ... are divisible by 1, 1.2, 1.2.3,... and 
generally A^O" is divisible by 1.2.3...W, = II (ra) ; these quotients are much smaller 
numbers, and it is therefore desirable to tabulate them rather than the undivided 
differences A0 n : moreover, it is easier to calculate them. A table of the quotients 
A w O n -r- II (m), up to m = n = 12 is in fact given by Grunert, Crelle, t. xxv. (1843), 
p. 279, but without any explanation in the heading of the meaning of the tabulated 
numbers C n k , = A w O fc -f- II (n), and without using for their determination the convenient 
formula C n k+l = nC n k -f C n _f given by Bjorling in a paper, Crelle, t. xxvni. (1844), 
p. 284. The formula in question, say 



U(m) 



= m 



is given in the second edition (by Moulton) of Boole s Calculus of Finite Differences, 
(London, 1872), p. 28, under the form 

A m O n = m (^m-iQn-l + AmQn-l). 

It occurred to me that it would be desirable to extend the table of the quotients 
A m O n -f- II (m), up to m = n = 20. The calculation is effected very readily by means 



744] 



TABLE OF A m O"-7-n(m) UP TO m = w = 2 



145 



of the foregoing theorem, which is used in the following form ; viz. any column of 
the table for instance the fifth, being 

A, then the following column is A, 

B, ... 25 + 4, 

C, ... 3C+B, 

D, ... 4D + C, 

E, ... oE + D, 

+ E- 

and then we obtain a good verification by taking the sum of the terms in the new 
column, and comparing it with the value as calculated from the formula, 

Sum = 2A + SB + 4(7 + 5D + QE. 

Observe that, in the two calculations, we take successive multiples such as 4.D and 
5D of each term of the preceding column, and that the verification is thus a safe 
guard against any error of multiplication or addition. 

TABLE, No. 1, OF A W 0" + II (m\ 



&lt;1 



a 


O 1 


O 2 


O 3 


O 4 


O 5 


O 6 


O 7 


O 8 


O 9 


O 10 


O 11 


O 12 


O 13 


O 14 


1 


1 


1 


1 


1 


1 


1 


1 


1 


1 


1 


1 


1 


1 


1 


o 




1 


3 


7 


15 


31 


63 


127 


255 


511 


1 023 


2047 


4095 


8 191 


3 






1 


6 


25 


90 


301 


966 


3025 


9330 


28 501 


86526 


261 625 


788 970 


4 








1 


10 


65 


350 


1 701 


7 770 


34 105 


145 750 


611 501 


2 532 530 


10 391 745 


5 










1 


15 


140 


1 050 


6 951 


42 525 


246 730 


1 379 400 


7 508 501 


40 075 035 


6 












1 


21 


266 


2 646 


22 827 


179 487 


1 323 652 


9321 312 


63 436 373 


7 














1 


28 


462 


5 880 


63 987 


627 396 


5 715 424 


49 329 280 


8 
















1 


36 


750 


11 880 


159 027 


1 899 612 


20 912 320 


9 


















1 


45 


1 155 


22 275 


359 502 


5 135 130 


10 




















1 


55 


1 705 


39 325 


752 752 


11 






















1 


66 


2 431 


66 066 


12 
























1 


78 


3 367 


13 


























1 


91 


14 




























1 


15 






























16 






























17 






























18 






























19 






























20 































C. XI. 



19 



146 



TABLE OF A m O M -r II (m) UP TO l = U = 20. 



[744 



&lt;I 

13 

a 

i i 


O 15 


Ql* O 17 


O 18 O 19 


O 20 


1 


1 


1 


1 


1 


1 




2 


16 383 


32 767 


65 535 


131 071 


262 143 


52428 


3 


2 375 101 


7 141 686 


21 457 825 


64 439 010 


193 448 101 


580 606 44 


4 


42 355 950 


171 798 901 


694 337 290 


2 798 806 985 11 259 666 950 


45 232 115 90 


5 


210 766 920 


1 096 190 550 


5 652 751 651 


28 958 095 545 


147 589 284 710 


749 206 090 50 


6 


420 693 273 


2 734 926 558 


17 505 749 898 


110 687 251 039 


693 081 601 779 


4 306 078 895 38 


7 


408 741 333 


3 281 882 604 


25 708 104 786 


197 462 483 400 


1 492 924 634 839 


11 143 554 045 65 


8 


216 627 840 


2 141 764 053 


20 415 995 028 


189 036 065 010 


1 709 751 003 480 


15 170 932 662 67 


9 


67 128 490 


820 784 250 


9 528 822 303 


106 175 395 755 


1 144 614 626 805 


12 Oil 282 644 72 


10 


12 662 650 


193 754 990 


2 758 334 150 


37 112 163 803 


477 297 033 785 


5 917 584 964 65 


11 


1 479 478 


28 936 908 


512 060 978 


8 391 004 908 


129 413 217 791 


1 900 842 429 48 


12 


106 470 


2 757 118 


62 022 324 


1 256 328 866 


23 466 951 300 


411 016 633 39 


13 


4 550 


165 620 


4 910 178 


125 854 638 


2 892 439 160 


61 068 660 38 


14 


105 


6020 


249 900 


8 408 778 


243 577 530 


6 302 524 58 


15 


1 


120 


7 820 


367 200 


13 916 778 


452 329 20 


16 




1 


136 


9 996 


527 136 


22 350 95 


17 






1 


153 


12 597 


741 28 


18 








1 


171 


15 67 


19 










1 


19 


20 















Writing down the 

1 2 3 

(0) (2) (4) 



sloping lines as columns thus: 

45 6 

(6) (8) (10) 



7 
(12) 



8 etc. 
(14) etc. 



1 
















1 


1 














1 


3 


1 












1 


6 


7 


1 










1 


10 


25 


15 


1 








1 


15 


65 


90 


31 


1 






1 


21 


140 


350 


301 


63 


1 




1 


28 


266 


1 050 


1 701 


966 


127 




1 


36 


462 


2 646 


6 951 


7 770 


3 025 




1 


45 


750 


5 880 


22 827 


42525 


34 105 




1 


55 


1 155 


11 880 


63 987 


179 487 


246 730 




1 


66 


1 705 


22 275 


159 027 


627 396 


1 323 652 




1 


78 


2431 


39 325 


359 502 


1 899 612 


5 715 424 




1 


91 


3367 


66 066 


752 752 


5 135 130 


20 912 320 




1 


105 


4550 


106 470 


1 479 478 


12 662 650 


67 128 490 




1 


120 


6 020 


165 620 


2 757 118 


28 936 908 


193 754 990 




1 


136 


7 820 


249 900 


4 910 178 


62 022 324 


512 060 978 




1 


153 


9 996 


367 200 


8 408 778 


125 854 638 


1 256 328 866 




1 


171 


12 597 


527 136 


13 916 778 


243 577 530 


2 892 439 160 




1 


190 


15 675 


741 285 


22 350 954 


452 329 200 


6 302 524 580 





20 19 



17 



16 



15 



14 



13 etc. 



744] 



TABLE OF A w O n -5- II (w) UP TO m = 7l = 



147 



it appears by inspection that, in the second column the second differences, are constant, 
in the third column the fourth differences, in the fourth column the sixth differences, 
and so on, are constant; and we thence deduce the law of the numbers in the 
successive columns : viz. this can be done up to column 7, in which we have 14 
numbers in order to find the 12th differences : but in column 8 we have only 13 
numbers, and therefore cannot find the 14th differences. The differences are given in 

the following 

TABLE, No. 2 (explanation infra). 



&lt; 

"O 

3 


1 


2 


3 


4 


5 


6 


7 





1 


1 


1 


1 


1 


1 


1 


1 




9 


6 


14 


30 


62 


126 


2 




1 


12 


61 


240 


841 


2 772 


3 






10 


124 


890 


5 060 


25410 


4 






3 


131 


1 830 


16 990 


127 953 


5 








70 


2 226 


35 216 


401 436 


6 








15 


1 600 


47 062 


836 976 


7 










630 


40796 


1 196 532 


8 










105 


21 225 


1 182 195 


9 












10 930 


795 718 


10 












945 


349 020 


11 














90090 


12 














10395 



We have, by means of this Table, the general expressions of A r O r , 
up to A 1 " 6 ^, viz. the formulae are 

A1K -5- n (r) = 1, 

/ r _ o 

A-&gt;0&gt;-- n (r-l)-l+l 



+1 



A r- 2Q r + n ( r - 2) = 1 + 6 ( T 3 Y + 12 



10 



r-3\ 3 



table ; and where for shortness 



is written to denote the binomial coefficient 



-3y 

2 y "V 3 ) "\ 4 
&c., &c., 

where the numerical coefficients are the numbers in the successive columns of the 

/ /* i 

k 
rjfTj. . For instance, r = 10, we have 

A 8 10 -r- n (8) = 1+ 6 . 7 + 12 . 21 + 10 . 35 + 3 . 35, = 750, 

agreeing with the principal Table. It will be observed that, in the successive columns 
of the Table, the last terms are 1, 1, 1.3, 1.3.5, 1.3.5.7, 1.3.5.7.9, and 
1.3.5.7.9.11. This is itself a good verification: I further verified the last column 
by calculating from it the value of A 14 20 -r- II (14), = 6 302 524 580 as above. The 
Table shows that we have A r ~ z O r -f- II (r - m) given as an algebraical rational and 
integral function of r, of the degree 2m. But the terms from the top of a column, 
A0 =l, A 2 - 1 . 2 = 2 r ~ 1 - 1, &c., are not algebraical functions of r. 
22 October, 1879. 

192 



148 [745 



745. 



ON THE SCHWAEZIAN DERIVATIVE, AND THE POLYHEDRAL 

FUNCTIONS. 



[From the Transactions of the Cambridge Philosophical Society, vol. xin. Part I. (1881), 

pp. 568. Read March 8, 1880.] 



dii 
the second order, ~r^ 2 +p -f^ + qy = Q, is determined by a differential equation of the 



THE quotient s of any two solutions of a linear partial differential equation of 
second 

third order 

tfo /d*s\ n - 

da? / da? \ l ( dp 

&- f l* ;r -K F+2 e 

dx \dx/ 

where the function on the left-hand is what I call the Schwarzian Derivative; or 
say this derivative is 



where the accents denote differentiations in regard to the second variable x of the 
symbol. 

Writing in general (a, b, c ,\\X, Y, ZJ- to denote a quadric function 
(a, b, c, ( a -b-c), i(- a +b-c), \ (- a - b + c)$X, F, Z}\ 



then, if the equation of the second order be that of the hypergeometric series, 
generalised by a homographic transformation upon the variable x, the resulting differ 
ential equation of the third order is of the form 



{s, x] =(a, b, c . .)( 
v 



111 



x a x b a cj 



745] ON THE SCHWARZIAN DERIVATIVE. 149 

and, presenting themselves in connexion with the algebraically integrable cases of this 
equation, we have rational and integral functions of s, derived from the polygon, the 
double pyramid, and the five regular solids. They are called Polyhedral Functions. 

The Schwarzian Derivative occurs implicitly in Jacobi s differential equation of the 
third order for the modulus in the transformation of an elliptic function (Fund. Nova, 
1829, p. 79, [Ges. Werke, t. I., p. 133]) and in Rummer s fundamental equation for the 
transformation of a hypergeometric series (Kummer, 1836 : see list of Memoirs) : but it 
was first explicitly considered and brought into notice in the two Memoirs of Schwarz*, 
1869 and 1873. The latter of these, relating to the algebraic integration of the 
differential equation for the hypergeometric series, is the fundamental Memoir upon the 
subject, but the theory is in some material points completed in the Memoirs by Klein 
and Brioschi. 

The following list of Memoirs, relating as well to the Polyhedral Functions as to 
the Schwarzian Derivative, is arranged nearly in chronological order. 

a /? 
Kummer, Ueber die hypergeometrische Reihe 1 + +... Crelle, t. xv. (1836), 

pp. 3983 and 127172. 
Schwarz, Ueber einige Abbildungsaufgaben. Crelle-Borchardt, t. LXX. (1869), pp. 105 120. 

Ueber diejenigen Falle in welchen die Gaussische hypergeometrische Reihe 

eine algebraische Function ihres vierten Elementes darstellt. Do. t. LXXV. (1873), 
pp. 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 Amvendung der Invariantentheorie. Crelle-Borchardt, 
t. LXXXI. (1875), pp. 97142. 

Klein, Ueber binare Formeii mit linearen Transformationen in sich selbst. Math. Ann. 
t. ix. (1875), pp. 183209. 

Brioschi, Extrait d une lettre a M. Klein. Math, Ann, t. XL (1877), pp. Ill 114. 

Klein, Ueber lineare Differentialgleichungen. Math. Ann. t. XL (1877), pp. 115 118. 

Brioschi, La theorie des formes dans 1 integration des Equations differentielles lineaires 
du second ordre. Math. Ann. t. XL (1877), pp. 401411. 

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, Ges. Werke, t. n., p. 351, remarks that the Derivative occurs implicitly in a memoir by 
Lagrange, " Sur la construction des cartes ggographiques," (1779), (Euvres, t. iv., p. 651.] 



150 ON THE SCHWARZIAN DERIVATIVE [745 

Klein, Ueber lineare Differentialgleichungen. Math. Ann. t. xn. (1877), pp. 167179. 

Weitere Untersuchungen liber das Icosaeder. Math. Ann. t. XII. (1877), 

pp. 503560. 

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. Math. 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 



\s, x\ = (a, b, c .*.) I , f , - ) i 

\x-a x-b x c) 



and he shows (by very refined reasoning founded on the theory of conformable figures, 
which will be in part reproduced) that this equation is, in fact, algebraically integrable 
for 16 different sets of values of the coefficients a, b, c. It may I think be taken 
to be part of his theory, although not very clearly brought out by him, that these 
integrals are some of them of the form, x = rational function of s ; others of the form, 
rational function of x = rational function of s ; the rational functions of s being in fact 
the same in the last as in the first set of solutions: they are quotients of Polyhedral 
functions. 

But as regards the second set of cases, the solution of these, introducing for con 
venience a new variable z in place of s, may be made to depend upon the solution 
in the form, x = rational function of z, of an equation of a somewhat similar form, but 
involving two quadric functions of x and z respectively, viz. the equation 



and we have the theorem that the solution of this equation depends upon the determ 
ination of P, Q, R rational and integral functions of z, containing each of them 
multiple factors, which are such that P + Q + R = 0. Using accents to denote differ 
entiation in regard to z, this implies P + Q + R = 0, and consequently 

QR - Q R = RP - R P = PQ - P Q. 

Further, they are such that the equal functions QR - Q R, RF - RP, PQ - P Q contain 
only factors which are factors of P, Q or R. 

In fact, writing /, g, h = b c, c a, a b, the required relation between x, z is 
then expressed in the symmetrical form f(x a) : g (x b) : h(x c) = P : Q : R. 



745] AND THE POLYHEDRAL FUNCTIONS. 151 

The last-mentioned differential equation is considered by Klein and Brioschi: the 
solutions in 13 cases, or such of them as had not been given by Schwarz, were 
obtained by Brioschi : and those of the remaining 3 cases, subject to a correction in 
one of them, were afterwards obtained by Klein. 

The first part of the present Memoir relates, say to the foregoing equation 

f ) / K \f l l l V 

[g, x\ = (a, b, c . .) , , , , 

\ae - a x b x cj 

although the other form in {x, 2} may equally well be regarded as the fundamental 
form. 

We consider in the theory: 

A. The Derivative {s, x}, meaning as above explained. 

B. Quadric functions of any three or more inverts -. . 

SO *"** 6 

C. Rational and integral functions P, Q, R having a sum = 0, and which are 
such that QR -Q R, =RP -R P, =PQ -P Q, contains only the factors of P, Q, R. 

D. The differential equation of the third order. 

E. The Schwarzian theory in regard to conformable figures and the corresponding 
values of the imaginary variables s and x. 

F. Connexion with the differential equation for the hypergeometric series. 
The second part of the Memoir relates to the Polyhedral Functions. 

The paragraphs of the whole Memoir are numbered consecutively. 

PART I. 

The Derivative {s, x}. Art. Nos. 1 to 7. 

., T , s" d (, ds\ , , , dp i , 

1. It p = = ^r { loff -=- , then Is, # = -^ i-. 

s dx \ 6 dx) dx 2/ ^ 

2. The derivative {s, x} may be transformed in regard to either or both of the 
variables. 

Suppose, first, that s is a function of the new variable S, (hence also S is a 
function of x) : using subscript numbers to denote differentiations in regard to S, and 
the accents as before for differentiations in regard to x, we have 

s = S s 1} 
whence, differentiating the logarithms, 



15: 



. &gt;- 



-I :: 



i If m 1 

- 

-7 



.- -.- 



*fcr *. 



:- 



raioe 



-.-.-. 



: -,:-_ 



m fiKt, mdadec Ifce 
= 0= &gt;, x r we hare t 



._- -:- 



-V = V T 



- - 

V .^ * 2 

" 



5. Wiisg= 

-,. : v 

-:-.:- 



~ -r T - " ^ _ --. ~. . 







-if : --.- 
- & 






4 *f- 



745] AND THE POLYHEDRAL FUNCTIONS. 153 

Hence in the first formula {S, x} = {s, x}, that is, 

(as + b } , . 



viz. we may, in the derivative {s, x], write for s any homographic function (as + b) -r- (cs + d) 
of s. 



6. Again, if X ^ , then from the second formula 



that is, 



and here, changing s into (as + b) -=- (c-s + d), we have finally 

(as + b ax + 0} _ (yx + S) 4 
[cs + d 7 tf 



which is the formula for the homographic transformation of the two variables s, x. 

7. Let s be a given function of x, the equation {8, x} = {s, x} is a differential 
equation of the third order in S, and by what precedes, its general integral is S= aS+ . 

CS -p Ct 

The direct process is as follows : we have a first integral - = ^7 -^ 7 ; a second 

S s cs + d 

integral log S = log s - 2 log (cs + d) + const., that is, &--** and thence a final 

(cs + d)- 

A 

integral 8 = B ~j , which is equivalent to the foregoing value of S. 

The Quadric Function of three or more Inverts. Art. Nos. 8 to 15. 

8. We consider a quadric function of any number of inverts - all 

x OL x - /3 

of them different: it is assumed that the constant term is =0, and also that the 
sum of the coefficients of the linear terms is =0. We have therefore square terms 

TaT^ttS P ro( l uct terms - - , and linear terms , where the sum of the 

\" w / X a . X p x Ct 

h 



coefficients A is =0. Any product term - a is expressible in the form of a 

x ct . x p 

_--^ --___ _ 



difference ^ _--^ --___ _ of two linear terms, and (the coefficients of these 



C. XI. 



154 ON THE SCHWARZIAN DERIVATIVE [745 

being equal), after it is thus expressed, the sum of the coefficients of the linear terms 
is still = 0. The function is thus always expressible in the form 



where the sum A + B + ... is = : this may be called the reduced form. 

9. Observe that any particular invert - may disappear altogether from the 

00 " OL 

reduced form : this will be the case if a = 0, that is, if the original form contains no 

term in , and if also A = 0. An invert thus disappearing from the reduced 

(x - a) 2 

form is said to be non-essential: and the inverts which do not disappear are said to 
be essential. The original form contains in appearance the non-essential inverts, but 
it is really a quadric function of the essential inverts only. 

10. Imagine the original function expressed as a rational fraction, the denominator 
being the product (x -a) 2 (x- /S) 2 (^~7) 2 of tne squared factors corresponding to all 
the inverts (non-essential as well as essential): the numerator will be in general of a 
degree less by 2 than that of the denominator, but the coefficients of any one or 
more of the higher powers of x may vanish, and the numerator will then be of a 

lower degree. But this numerator will for any non-essential invert - - contain the 

factor (x 7) 2 , or, dividing the numerator and denominator each by this factor, the 
difference of the degrees of the numerator and denominator will remain unaltered; 
that is, the difference will have the same value whether we do or do not attend to 
the non-essential inverts; or say it will have the same value for the original form and 
for the reduced form. 

A 7? C 1 

11. It is to be remarked that the linear terms 1 -5 + - -+..., where 

x a. xp x 7 

A+B + C + ... =0, can be (and that in a variety of ways) expressed as a sum of 

differences n &gt; tnat is. as a sum f product-terms -5 . Hence the 

x a. x p x a.x p 

quadric function can be (and that in a variety of ways) expressed as a homogeneous 

/ Y 1 1 \ 2 

function (a, ... , - 5 &gt;); w ^ must have in the form all the essential inverts, 

\ S^X Of X P / 

and we need have these only. Supposing that this is so, and that the number of 
the essential inverts is =w, then the number of constants is = ^n(n + 1), whereas the 
number of constants in the reduced form is only = 2n 1 : hence the coefficients are 
not determinate; or, what is the same thing, we may have different quadric functions 
having each of them the same reduced function; these quadric functions, as having 
the same reduced function, can only differ by multiples of the evanescent expressions 

5 H - - H 5 , &c. 

xp.x y x ry.x a. xOL.x p 



745] AND THE POLYHEDRAL FUNCTIONS. 155 

In particular, if the number of essential inverts is = 3, then the quadric function is 
of the form 



/ Y 1 1 1 \ 2 

(a. b, c. f. er, hO , ~ , I , 

V * X*- x-ft x-j 





which contains one superfluous constant, and equivalent functions differ only by a 
multiple of 

_j3-7 _,_ y-ct + a- ft 



x ft.x y xy.x a x a. as ft 

12. A quadric function such that the degree of the numerator is less by 4 than 
that of the denominator is said to be "curtate." 

The conditions, in order that the function 

( v. f uX 1 1 1 \ 2 

a, b, c. f, Q, h - , -- = , 1 

V X*- x-ft x-yj 

may be curtate, are easily found to be 

a + b -f c + 2f + 2g + 2h = 0, 



and by reason of the superfluous constant we are at liberty to assume a third condition : 
the three conditions may be taken to be a + h + g, h + b + f, g + f+ c each =0; and 
this being so the values of f, g, h are =( a -b-c), (-a + b-c), H~a-b + c) 
respectively. Hence the form is 



1 1 1 \ 2 



a, b, c, (a - b - c), (- a + b - c), (- a - b + c) 

A#- #-/* # -ry 

which, as already mentioned, we denote by 



, 

a x 



We have thus the theorem that a curtate function of any number of inverts, but with 
only the three essential inverts 

1 1 ^_ 

x a x ft x y 

is always expressible in the foregoing form 

1 



,,..-_, , __ 

AX - a x - ft x - y 

13. It may be remarked that the function (a, b, c .-.$X, Y, Zf is a function of 
the differences of the variables Z, Y, Z; and similarly, in the case of four variables 
a function (a, b, c, d, f, g, h, 1, m, nX, Y, Z, W)\ for which 

a+h+g + 1, h + b + f+m, g + f+ c + n, 1+m + n + d, 

202 



156 OX THE SCHWARZIAN DERIVATIVE [745 

are each =0, is a function of the differences of the variables X, Y, Z, W: and so in 
general. Any such function is said to be " diaphoric " : and it is easy to see that, 
taking for the variables any inverts whatever, a diaphoric function is always curtate. 



14. The function 

a b c 



a b c 2 

+ + ~ + 



~ 

a? - a 8 p x-y 

where the coefficients a, b, c, . . . satisfy the relation a + b + c+... = 2, is diaphoric, 

and therefore curtate. In fact, forming the sum, coeff. . - ^ + 1 coeff. - - -5 + . . . , 

(x a) 2 x a . x p 

this is a a 2 ab ac ..., = a(2 + a + b + c +...), which is =0; and similarly 
the other conditions are satisfied. 

15. The function 

/ , Y a Oi b &i c GI Y 

a, b, c /.A- -+-+..., ~z + - :L a+-~&gt; - + -+..., 

\ Ax --! fi x & x yx y! J 

regarded as a function of the inverts 

11 1 



where 

a + tti + . ..=6 + 6j +... = 0-1-0!+..., = & suppose, 

is diaphoric, and therefore curtate. In fact, the condition in regard to - - is 

CC ^~ CL 

a (a 2 + aa x + aa 2 + . . .) + 1(- a + b - c) (a6 + a6 x + . . .) + |(- a - b + c) (ac + acj + . . .) = ; 

that is, 

ak {a + i (- a + b - c) + (- a - b + c)} = 0, 

which is satisfied. And similarly the other conditions are satisfied. 



The functions P, Q, R. Art. Nos. 16 to 20. 

16. We consider P, Q, R, rational and integral functions of z, such that P + Q + R = : 
hence, using the accent to denote differentiation in regard to z, we have also P + Q + R = ; 
and therefore QR - Q R = RP f - RP = PQ - P Q, = @ suppose : and we require to find 
P, Q, R such that the function @ contains only the factors of P, Q, R. 

17. It is to be observed that, effecting upon a solution P, Q, R any linear sub 
stitution (az + /3) H- (yz + 8), and omitting the common denominator, we have a solution ; 
but this is regarded as identical with the original solution. The three functions, if 




745] AND THE POLYHEDRAL FUNCTIONS. 157 

not originally of the same order, can thus be made to be of the same order ; or by 
taking account of the root z= oo , we may in the original case regard them as being 
of the same order, and it is convenient so to regard them : say they are taken to 
be of the same order 8. And there is clearly no loss of generality in taking the 
three functions to be prime to each other ; for any common factor of two of them 
would divide the third, and might therefore be struck out. 

18. We may therefore write 

-iy?, Q = GTl(z-m)&lt;i, R 



where (z l) p is taken to denote the distinct simple or multiple factors of P, and 
the like as regards Q and R ; the factors z I, z m, z n are thus all of them different. 
And we have 8 = ^p, = ~2q, = 2r. 

19. It is at once seen that is of the degree 28 2, and moreover that it 
contains the factors II (z iy~ l , II (z m) 9 " 1 , II (z n) r ~^ ; hence it contains the factor 

n (z - ly- 1 (z - m)?- 1 (z - nf-\ 

Suppose the number of distinct indices p is = &lt;r 1 , that of distinct indices q is &lt;r 2 , and 
that of distinct indices r is &lt;r 3 ; then the degree of the factor is = 3S o-j &lt;r. 2 a- 3 ; 
and if this be = 28 2, then can have no other variable factor : viz. if the numbers 
o-j, &lt;7 2 , 0-3 of the distinct indices p, q, r respectively are such that o-j + o- 2 + cr 3 = 8 + 2, 
a relation which is henceforth taken to be satisfied, then we have 

= Kli (z - I)?- 1 (z - m)?- 1 (z - n) r ~\ 

As already in effect remarked, the conclusion extends to the case where P, Q, R are 
not of the same degree ; the equation P + Q + R = here implies that two functions, 
say P, Q, are of the same degree, and the third function R of an inferior degree ; 

(z \ 
1 -- j of 

the degree t proper for raising its degree up to that of P or Q. 

20. Solutions are given in the following PQR-Tsible : in which, where required, 
the proper factor (1 -- J has been added; the first column headed Ref. No. (Reference 

Number) will be explained further on. The Annex to the same Table will also be 
explained. 



158 



ON THE SCHWARZIAN DERIVATIVE 



^ + 



&gt; 





-r + 






OJ + 






+ 1 










+ 

a 



745] 



AND THE POLYHEDRAL FUNCTIONS. 



159 





1 i 1 II 

| S H * 
11 1 l|l| 

o o -S o -^ o M 
CH ft H ft 


^0 

C n ^ ^ n *^ * 

o O O 
C .^C H -"* 

i ^ 1 3 
,3 S * ^ 

I jlll 

H O ft 


o 
55 




^VN^^J^VX-^S^^^^VN 




- a 




a 




8 S^ 88 ^^**^ 






^ ^ *^ ** ^ M 
t^ ** N isi 




*"* rH CO rH. O3 


*& ^5 r ^ (jq |iQ ^**rH C&lt;11O CM CMiOrHGM ^* |ut) W 25 SI 




S co loo co loo co ;oo co 100 
I 


^,^.Mii.ii.i 




U U - 

1 | * IO5 -^ IOS * OS 
1-1 Ji- 


^ 25 ^ sis ^ 53 ^ 53 ^ ^ 


k 

O 

._ 
O 

a 
.0 

la 
"3 


O O O O O 

II II II II II 

rH O3 CO ^^ W5 

o o w o o 


1-10 OS 

^ u" 5 n" 5 t 

rHOcnWirHO (NU5 ml 1 * 
(MC CD (MC&gt;) O rHIN O lc&lt;) 
II II II II II II II 
&lt;M CO &lt;N IO rH IO CO 
O O U O O O O 

yj 00 ^ 00 rH |O ^jOOWUSC^WrHOfMlOCd Oi lOrH IO 

u n u u H i u u i \F ^^ 

(MW(N&lt;NrHrH(MrHrHIMlM 
OOOOC5OOOOOO 


3 
o 


O O 

II II II II II 

.a ,a ,a ^3 ja 


O 
II II 

O OCOIOOO O OCOIOOO 

II II II II II II II II II II II 

&lt;N&lt;M&lt;NC&lt;rHC&lt;)(N(M(Mi-ie&lt;I 




o o o o o 
II II II II II 


O 

II 

rH ICO ^^ ^^ W O rH &lt;O ^5 
II II II II II 11 II II 
r-irHrHi-l CO O3 COCO 

c8 c3 c3 c3 ci c3 c3 c3 
**^ iCft ^^ JA ^* |Oi ^j (o ^ **^ 35 ^^ M^ IO5 ^^ ^* IOi ^?J* ICTi 

u II H II u u u II il u II 




c3 03 * rt S 


cacSc3cSo3c3e3e3c3c8c3 


* 

"o 

JO 

3. 


O ^ ICO ^ IO ^ ^-, ,_) |(-g 

r ... 

1 co loo co 100 co oo co 100 

rH 

ft--- 
| | * ICS -* ICS -# IO5 


eo loo eo 100 eo loo co loo co i E E : E r r 

^^rnls^i^SS^io 5 " " - = - " 


6 
&lt;*-. 


rH =0 ^ U5 


n^pa^^^aS^g 



t? o 



- ^ 

* -x 

o \&lt;y\ a 

nlco g 

T3 -3 

a 

dS %_i 



XI 



160 ON THE SCHWARZIAN DERIVATIVE [745 

The Differential Equations [x, z] and {s, x}. Art. Nos. 21 to 45. 

21. In reference to what follows, it is convenient to put P=XP , P ^X^, 
where P is written for IT (z l) p ~ l , the G.C.M. of P and P ; and X is consequently 
= F multiplied by the product II (z 1) of the several factors taken each with the index 
unity; and so for Q and R: viz. we write 

P, Q, R = XP , YQ , ZR , 
P , Q , R = X 1 P , FjQo, Z^RQ, 

and the foregoing value of then is 



We come now to the investigation of the leading theorem. Take a, b, c arbitrary, 
f, g, h = b c, c a, a b ; P, Q, R functions of z as above ; and write 

f(x-a) : g(x-b) : h(x-c) = P : Q : R, 

equations, which are consistent with each other and determine x as a rational function 
of z. Using, as before, the accent to denote differentiation in regard to z, and taking 
the coefficients (a, b, c) arbitrary, it is required to find the value of 

{x, z}+x 2 (a, b, c .-.I) - , - j, - ) . 
V Ax ax bx cJ 

22. Calculation of the first term {x, z}. 

t p \ / p \ (P } 

We have x = & function \ a n + fi) + (y^p + &}, and thence {x, z} = \^, z\ , = {, z} 

\ JK J \ i ] \1\ J 

for a moment; then 

RP -R P P Q R P Q, 



T}O y r/*) T&gt; f) y F7-i D * 

M 2 Z 2 M - Z 2 M 

Substituting the values 

we have 



" _p-l q-\ r + l 

--, z* r T *-i ~ ft~ , 

z I z m z n 



and thence 



-2 



z I z m z n 

or say 

p 1 PI I q 1 &lt;?i 1 r+l n 4- 1 

(.z-J) 2 "^-^) 2 "" "" (2- m) 2 ~ ( - m^ 2 ~ " + ( - n) 2 + (2 - Wj) 2 

\ ^ ^ -s ^ z m z nii z n z n^ " ) 



745] AND THE POLYHEDRAL FUNCTIONS. 161 

where it is to be observed that 

(p - 1) + 2 (q - 1) - S (r + 1), = 8 - o-j + S - &lt;7 2 - (8 + o- 3 ) = 8 - o-j - o- 2 - o- :5 = - 2 ; 
consequently the function is diaphoric, and therefore curtate. 

It is to be remarked that the function, although presenting itself in a form 
unsymmetric in regard to the factors of P and Q, and of R, is really symmetric 
as regards the three sets of factors ; this is obvious a priori, and it will be presently 
verified. 

23. For the calculation of the second term 

/ Y 1 1 1 Y 

x - ( a, b, c . . , ^ , , 

\ AX ax ox cj 

we have 

/(a? -a), ff(sn-b), h(a;-c)=flP, SIQ, flR, 

where H is a determinate function of z\ hence 

x x 1 x r _p ir &lt;? rr R n 

& "* r\ i f\ n &gt; r&gt; r 



a,_ x-b x-c~ P IT Q I 

J hen substituting these values, by reason that the function is diaphoric, the terms 
in ^r disappear, and we have 



&gt;( v, X l l l V 

c-(a, b, c ()-_ - _i , -^-) 



F 
= a, b, 



which is 

r 



-a b 
a., u, 



z m z n 



We have S/) = 2g = 2? , = 8 : and hence by what precedes, this function, considered as 

a function of the inverts ; , &c., is diaphoric, and therefore curtate. 

z I 

24. We have therefore 



/ Y 1 1 1 \" 

\x, z\ + x - a, b, c /. , f , - = 

V AX -a x-b x- c] 



-2 



^ I z m z n 



, b, c .-.()2- 7) 

* V &gt;v / 



z I z m z n 

where the whole function on the right-hand side is curtate. 

c. XL 21 



162 ON THE SCHWARZIAN DERIVATIVE [745 

25. We have to bring the function on the right-hand side into the reduced form 

a A 

(z a) 2 z a 

for the purpose of getting rid of the non-essential inverts (if any). 

We write 

^ p-1 __ -1^1-1 



viz. z I here denotes any particular factor, and zl-^ represents any other factor of 
the same set; and so in other like cases. 

26. The whole coefficient of . is 



- I) 2 + a.p s , = i 

an expression which, regarded as a function of a and p, is represented by (ap): the 
parentheses are used only to avoid ambiguity, and are omitted when p is a number, 
thus al = a, a2 = f + 4a, and so in other cases. 

27. The whole term in , comes from 

z l 



_ 
z I \ z l-i z m z n 



-- - - - - 

zl[ z l z m zn 

viz. each term such as - -, - r is to be replaced by ,- [ , , 1 , givi 

z-l.z-l, J l-li\z-l z-lj 



vng 



rise to the term , T -- , , or contributing the term 7 .- to the coefficient of 
l ^ z l l l^ 

The whole coefficient thus is 



m n 



&gt; -- - - ,--. 

I m I n 

28. Suppose first that z l is a multiple factor of P, viz. a factor with an index p 

ry n 

greater than 1: then, for z = I, we have Q+R = 0, Q + R = 0, and thence ^ = -7^, 

y Vt 

Q T 

that is, 2 , = S , . We have therefore 
I m l n 



p (- a - b + c) 2 -3- .+p (_ a + b - c) 2 



r 



i 

I n 



I in I nj 



745] AND THE POLYHEDRAL FUNCTIONS. 163 

*-&lt; Q 1* 

moreover, in the top line, the terms 2, r ~- - and S 7 destroy each other. The 

I m l n 

whole coefficient of ,, when z I is a multiple factor of P, thus is 

z ~~ I 

2 l -2 - 1 

I m l n 



n 



a form which is now symmetrical in regard to the inverts , and , -- . 

I - m l-n 

29. The value just obtained must be equal to 

(1 -? + &*) 12 * - - 1 + 2 if - 1 - 2 -U ; 

V i W 471 l lj 

viz. comparing the two forms and reducing, they will be identical if only 

(1 -p + 2ap) J2 t+f _ 2 * (1 +^&gt;g-^ - 2 tlL^TP]. = 0, 
( t x t m t n } 

and it can be shown that the function inside the { } is in fact = 0. 

O 1* 

30. We have, as before, S 7 - = 2 / --- 5 or writing each of these quantities 

I m l n 

the equation to be verified is 



I h 1 l m ^ ln 

We have 

that is, 



The first derived function of the numerator is X^ (z l)+ X 1 pX / , which for 
z=l is X 1 pX , which is =0; and, for the denominator, it is X (z l) + X, which 
is also =0. Passing to the second derived functions, we find 

Pi ^XJ 



z- 1, 2X X 

From the equation 



= - - 

X z I z I 

212 



164 ON THE SCHWARZIAN DERIVATIVE [745 

we find in like manner 



" I - I, X 
and we thence obtain (z being always = I) 



** i "&&amp;gt; 

z LI X 
so that the equation to be verified becomes 



m l n 



31. But from the equation @, = PQ - P Q, = KP Q R , we find XY 1 -X 1 Y = 
and then, differentiating, XY-! + X Y l X^Y X 1 Y = KR : writing in these equations 
z = I, they become 



X Y.-X. Y-X.Y^KR,, 

so that, dividing the second by the first, 

_Z F 1 X T _R^ 
X, Y "*" X, + F ~ M 

Y 
or, recollecting that X 1 =pX f and = we nave 



,-Bn 

that is, 



= (w + !)&lt;!&gt; p"Z^ 2 , , 

I m L n 

the required relation. 

32. The result is that, zl being a multiple factor of P, the coefficient of the 

1 

term , is 
zl 



33. In the case where z l is a simple factor of P we have p=I, and the 
coefficient is 



22 -S.-- -2--(b-c)(2. - -2, 
t tj fc m l n) \ l m l 



745] AND THE POLYHEDRAL FUNCTIONS. 165 

34. Of course the formulae for the coefficients of , -- /v , and - , give at once, 

\Z li) Z ~ (j 

by a mere change of letters, those for the coefficients of rr, - , and 

( Z ~~ tivj" & iff/ 

__ , --- ; and the function in question, 
(z -rif z n 

{a?, jrl+a^fa, b, c . .Q -- - - i, --- ) , 
V /js ax o x cJ 

is now obtained in the required form 



Jb?) M ^L _* -_ 

(z-l) 2 " (z-mY" (z-rif" z-l" z-in" z-n" 

where (ap) denotes (1 - p -) + ap 2 , and the like for (bg) and (c? ); and where, z-l 
being a multiple factor of P, the coefficient A contains the factor (ap); and similarly 
for B and C. 

35. Suppose that the coefficients a, b, c are no one of them =0; we have 
al, =a, which does not vanish; that is, z I being a simple factor of P, the 

expression contains -- 7V&gt; , or the invert - , is essential : and similarly, z m being 

\Z ~~ i)~ Z ~~ I 

a simple factor of 0, or z n a simple factor of R, the inverts and 

z m z n 

are essential. But for z I a multiple factor of P, the coefficient (ap) of the term 

(ZL) 

may vanish, viz. this will be the case if a = | ( 1 -- J ; and, when this is so, the 
coefficient A of the corresponding term ; also vanishes; that is, , is a non- 

% ^~ v & v 

essential invert. And similarly for any multiple factor z-m of Q or z n of R, the 

invert ---- or - may be non-essential. 

z m zn 

36. If P, Q, R contain each of them only multiple factors of the same index, 
say of the indices p, q, r for the three functions respectively, viz. if the functions 
are F(U(z-l))*, G (U (z - m})* , H(U(z-n)) r } the result contains only the six terms 

written down : and then, if a, b, c are = (l - - ] , | f 1 - -J , if 1 " ^) respectively 
the result is = : viz. we then have 

\x, z\ + x- ( a, b, c . . I) , r i - 1 = 0, 

V X ax b as c) 

or we in fact have, for the values in question of a, b, c, a solution 

f(x-a) : g(x-b} : h(x-c) = P : Q : R 
of this differential equation of the third order. 



166 ON THE SCHWARZIAN DERIVATIVE [745 

37. The reasoning applies directly to lines 2, 3, 4, 5 of the PQR-Tab\e : and 
with a slight variation to line 1 ; viz. here the factors of R (= 1 + 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-Table gi ye eacn f them a result = 0, for the values of (a, b, c) 
appearing by the table itself, and shown explicitly in the corresponding line of the 
Annex. 

Thus line 3 shows that the function x, determined by 

f(x-a) :g(x-b) : h(x-c) = (z* + 2 V- Sz- + I) 3 : - 12 V-3 (z 5 - z)- : - (z 4 - 2 \/-Sz- + 1), 
satisfies 

J ) , 



,9 8 9"X#-a x-b x-c; 
and so for any other of the five lines. 

38. The indices of the factors of P, Q, R may be such that, for proper values 
of the coefficients a, b, c, there are in all only three essential inverts, say 

7^7^ z _ belonging to the three functions P, Q, R respectively, or it may be 

two, or three, of them to the same function. When this is so, the function of these 
inverts is, by what precedes, a curtate function, and it is consequently a function 

1 1 1 



where a lf b l5 c x are the values of the three which do not vanish in the series of 
expressions (ap), (b#), (cr). 



The remaining lines (III, V, VII, VIII) and IX to XV of the P&R-Table give 
such values of P, Q, R, the values of (a, b, c); and the calculation of the values of 
(& l} b 1} Cj) is shown by the corresponding lines of the Annex. And we have thus 
values of x determined by the equations 

f(x-a) : g(x-b) : h(x-c} = P : Q : R, 
and giving 

1 * 



* a&gt; b, , ,, 

V Ax -a x-b x-c -z-a: z- z - c 

39. For instance, from line IX we have 

f(x-a) : g(x-b) : h (x - c) = (z - 4) 3 : - (z - \}(z + 8? : 27 z- ( 1 - ) , 

\ oo / 

4 3 12 
the values of (a, b, c) are -, -, ^ ; and since P, Q, R contain factors with the 

exponents 3; 1, 2 ; and 1, 2 respectively, the coefficients which present themselves 
on the right-hand side are 

a3; bl, b2; cl, c2, 



745] AND THE POLYHEDRAL FUNCTIONS. 167 

which are 

3 12 21 

= 0; g, 0; 25, ^ respectively. 

3 12 21 11 

Hence writing a 2 , b 1; c 1 = -^, ^, , the corresponding inverts are -- , ~ 

-: and the result is 
z 



f l "&gt;/* ? i? X - 1 1 _J_V_/ 3 1? 21 X ! J !Y 

^ ^ \9 8 ZS^JU-a 1 x-V -c/ U %&gt; 25 SO^JU^T 0-x ^J 1 

40. It is hardly necessary to remark that an expression 



a-i z 
in fact denotes 

ai P! - 



(z - a,) (z - b^ 

The particular form of the z inverts is immaterial ; we could by a general linear 
transformation upon the z make them to be -- , j- , with the (a lt 6 1; d) 

& ~~~ 6^1 Z ^"~ C/j Z ~^ Cj 

arbitrary; or we can give to the a 1} b ly Cj any particular values we please: there 
would be a propriety in making the inverts to be in every case (as in the foregoing 

example) - , - , - - ; but the numerical work would be troublesome, and it is 

Z Z 00 Z 1 

not worth while to effect it. 

41. The conclusion is that lines (III, V, VII, VIII) and IX to XV of the 
PQR-Table, give, for determinate values of (a, b, c) and (a 1( b ls Cj), solutions 

f(x-a) : g(x-l)} : h(x-c) = P : Q : R 
of the equation 

/ i. X ! ! ] V / v X ! l l V 

{x, *}+aP[&, b, c . .O -, -T, - = a : , bj, c x .-. -- , 7-, 

- 



where a, b, c, a 1; 6 1} d are or can be made arbitrary, but without any real gain of 
generality herein. This is the Differential Equation \x, z}. 

42. Recurring to the results from the Arabic lines of the PQTZ-Table, but for 
convenience writing s instead of z, we have 

f(x-o} : g(x-b) : h(x-c} = P : Q : R, 
where P, Q, R are now functions of s, a solution of 

( , (faV f i, X l l l V A 

, s } + (-J--) a, b, c . . , r, ---- =0. 

VO*/ V A* a x o x cj 

But we have 



168 ON THE SCHWARZIAN DERIVATIVE [745 

and the foregoing is therefore a solution of 

r i. x l l l v 

[8, x\ = a, b, c . . H --- , - T , I , 
V AX a # 6 x- c) 

a differential equation of the third order. This is the Differential Equation [s, x}. 

43. From the Roman lines, if we assume 

f(x-a) : g(x-b) : 7i(a?-c) = $ : d : ?R, 

where S 4&gt;, d, 9t are functions of z, not the same functions that P, Q, R are of s, 
since they belong to a different line of the Table : we have, as before, 

(dx\"i , Y 1 1 1 V / , Y 1 1 IV 

\x, z + j- a, b, c . . , , , - = a^ b,, c, /. ft - - , - r , - 1 . 
\dzj \ Axax b x c) \ AZ a* z b^z cj 

44. We may combine any such result with a properly selected result of the 
preceding system, the two results being such that (a, b, c) have the same values in 
each of them. (See as to this the foot-note referring to the Annex to the PQR- 
Table.) The last equation then becomes 

dx / , 

C 



or since 



this is 



1 1 IV- 

a^ rrv *- J 

/dr\~ 

f -v / \AI**J \ i &gt; / * 

fa Zl + (dz) ** X &gt; = fa * * 

, , / , Y 1 1 IV 

[s, z\ = (a l , bj, Ci /.(J - , T-, I , 
1 V A.z-0^ z h z-cj 



the corresponding relation between s, z being of course obtained by the elimination 
of x from the two sets of equations 

f(a; - a ) : g (x - b) : h(x-c} = P : Q : R, and f(x - a) : g (x - b) : h (x - c) = $ : Cl : SR; 

viz. the required relation is 

F : Q : - $ : : 91, 

where P, Q, R are functions of s ; *$, d, 3J functions of z ; and, in virtue of 



the relations are equivalent to a single equation between z and s. And writing 
finally x in place of , that is, now considering ty, d, $ as functions of a, we have 

$ : d : R=P : Q : 12 
as a solution of 



/ Y 1 1 \ 2 

fi/v A- 1 - \ 

[, J = [a,, b n c x /.I) , 7 , - 1 , 

V X* o x b^ x cj 



a differential equation of the third order of the foregoing form, {s, x} = given function 
of x, but with different values of the coefficients, (a^ bj, Cj) instead of (a, b, c). 



745] 



AND THE POLYHEDRAL FUNCTIONS. 



169 



45. It thus appears that there are in all 16 sets of values of (a, b, c), for 
which the equation is solved, viz. the 16 sets of values are shown in the right- 
hand column of the Annex. For greater clearness I exhibit the integral equations 
as follows: 





Functions of x. 


Functions of s. 




1 


f t rp ft\ . ft 1 ry* A\ A / ~ ft\ 

J \t*j Uj J . y \*AJ Uj . 11 \JLt } 


= P : Q : R (1) 


Polygon 


I 


5) 


(2) 


Double Pyramid 


II 


,, 


(3) 


Tetrahedron 


III 


Ix : -(x+iy : (x-l) 2 


(3) 





IV 


/ / \ / j_\ Z. / \ 

j (x Ojj g (x o ) \ fi (x c) 


(4) 


Cube and Octahedron 


V 


(x-\r -(x + iy : 4* 


" ( / 





VI 


f (x a) g (x b) : h (x c) 


(5) 


Dodecahedron and Icosahedi-on 


VII 


A. y* (nf* _]_ 1 )* \ ** 1 i^ 


(5) 





VIII 


(x-1) 2 -(x+iy- : x 


(5) 


,, 


IX 


P Q : R (IX) 


(5) 


,, 


X 


(X) 


(5) 





XI 


(XI) 


(5) 





XII 


(XII) 


(5) 


5? 


XIII 


(XIII) 


(5) 


II 


XIV 


(XIV) 


(5) 





XV 


(XV) 


(5) 


J) 



The values of the P, Q, R as functions of x, or of s, are taken out of the 
PQR-Table: only in the lines III, V, VII, VIII, where P, Q, R are given as 

= 4,2, -O + l) 2 , O-l) 2 , 

and where, as regards V and VIII, there is a transposition of P and R, I have 
inserted the actual values of the ay-functions. (See as to this the foot-note referring 
to the Annex.) 



The Schwarzian Theory. Art. Nos. 46 to 62. 
46. Considering the foregoing equation 



= a 



, l .-- , __ __ 

as a particular case of the equation {s, x} = Rational function of x, =R (x) suppose, 
then we have in 1, I, II, IV, VI solutions of the form x = Rational function of s. 
G. xi. 22 



170 ON THE SCHWARZIAN DERIVATIVE [745 

Consider, in general, a solution of this form, x = F (s) a rational function of s : then 
s is an irrational function of x, and if s i} s 2 are any two of its values, {s l) x} = R(x), 

{s 2) x}=R(x); that is, {s 2 , x} = {s lt as], and therefore (ante, No. 7) s. 2 = -, . And 

CSi ~r Ct 

then x = F(s 2 )=F( - -), = F(s l ): viz. F(s) is a rational function of s, transform - 
\csi T a/ 

able into itself by the transformation s into 7 : and it is moreover clear that 

cs + a 

between any two roots s whatever of the equation x = F (s) there exists a homographic 
relation of the form in question. Further, it is clear that these homographic trans 
formations form a group; and consequently that F (s) is a rational function of s, 
transformable into itself by the several homographic transformations of a group of 
such transformations : viz. taking x to be a rational function of s, it is only in the 
case x = F(s), a function of the form in question, that {s, x] can be equal to a 
rational function of x. 

47. We may, in any equation between x and s, consider these as imaginary 
variables p + qi and u + vi respectively ; considering then (p, q) and (u, v) as rect 
angular coordinates of points in different planes, we have a first plane the locus of 
the points x, and a second plane the locus of the points s: there is between the 
two planes a correspondence which is in fact the correspondence of conformable 
figures : to the infinitesimal element dx drawn from a point x of the first figure 
corresponds an infinitesimal element ds drawn from the corresponding point s of the 
second figure, these elements being in general connected by an equation of the form 
ds = (a + bi) dx, where a and 6 are functions of x or s ; and this signifies that, to obtain 
the pencil of infinitesimal elements or radii ds proceeding in different directions from 
the point s, we alter in a determinate ratio the absolute lengths of the infinitesimal 
elements or radii proceeding from the corresponding point x, and rotate the pencil 
through a determinate angle : this ratio and angle of rotation, or say, the Auxesis 
and the Streblosis, being of course variable from point to point. Or, what comes to the 
same thing, if dx and d-^x be consecutive elements of the path of the point x, and 
ds, d^s the corresponding consecutive elements of the path of the point s, then the 
ratio of the lengths of the elements dx, d^x is equal to that of the lengths of the 
elements ds, d^s; and the mutual inclination of the first pair of elements is equal 
to that of the second pair of elements. In particular, if at any point the path of x 
is a curved line without abrupt change of direction, then at the corresponding point 
the path of s is a curved line without abrupt change of direction. In what precedes, 
we have the relation at ordinary points; but there may be critical corresponding 
points (x, s), the relation at a critical point between the corresponding elements dx, 
ds being of the form ds = (a + bi) (dx)*, (X a positive integer or fraction) : here the 
angle between two elements ds is = X times that between the two elements dx ; or, 
if the path of the point x through the critical point is without abrupt change of 
direction, say if the angle between the two consecutive elements is the flat angle TT, 
then the angle between the two consecutive elements ds is = A/TT : viz. there may be 
in the path of the point s an abrupt change of direction. 



745] 



AND THE POLYHEDRAL FUNCTIONS. 



171 



48. I consider the foregoing equation {s, x} = R (x), where R (x) is a rational 

function, and is now taken to be a real function of x : we may assume s = ip 6 e 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 



6" 



&" f& \* 



"& 



and thence 



. 

pV p 

Putting this = R (x), and assuming that as is real, we have 



{p, 



The last equation gives p"& = 0, that is, ff = 0, which gives s = 0, and may be 
disregarded ; or else p" = 0, therefore p , a real constant, = 7 suppose, and {p, x] = : 
hence for the solution of the equation {s, x} = R (x), we have s = iy0 e i9 , a real 
quantity determined by {0, x} +0 2 = R(x): and then, integrating the equation for s , 
we have s = a + fii + ye, a, 0, y real constants. 

49. The conclusion is that, if {s, x} = R (x), a real function of x, and if x be 
real, that is, if the point x move along a right line (say the #-line), then s - a + @i + &lt;ye ie 
(6, and the constants a, /3, 7, being real), that is, the point s moves in a circle, 
coordinates of the centre a, /3, and radius =7. 




50. Suppose a, b, c are any real values of x representing points a, b, c on the 
; and A, B, C any given imaginary values of s representing points A, B, C 

222 



172 ON THE SCHWARZIAN DERIVATIVE [745 

in the s-plane : since {s, x}=R (x) is a differential equation of the third order, the 
integral contains three arbitrary constants, and we may imagine these so determined 
that to the values x=a, b, c shall correspond the values s = A, B, C respectively. 

If there is not on the #-line any critical point, as the point x moves continu 
ously along this line the point s will move continuously along a circle, which (in 
asmuch as a, b, c and A, B, C are corresponding points) must be the circle through 
the three points A, B, C*. 

51. If however the points a, b, c are critical points, such that the element ds 
at the corresponding points A, B, C are equal to multiples of (cfo) x , (doty* (dx) v re 
spectively, then to the flat angles TT at a, b, c correspond in the path of s the 
angles XTT, /XTT, VTT at the points A t B, C respectively: and, assuming that a, b, c 
are the only critical points on the #-line, the path of s is made up of the three 
circular arcs CA, AB, BG meeting at angles XTT, /JLTT, vir respectively. The arcs are 
completely determined by these conditions ; for supposing the arc BC to make with 
the chord BG, at the points B and C, the angles /, /, and similarly the arcs CA 
and AB to make with the corresponding chords the angles g, g and h, h, then the 
conditions give XTT, fiir, V7T = Z.A + g + h, ^B + h +/, ^ C +f+g, where the angles 
referred to are those of the rectilinear triangle ABC: we have thus the values of 
/, g, h; and the arc BC is the arc on the chord BC meeting it at angles f, f: 
and the like as regards the arcs CA and AB respectively. 

52. The foregoing equation 

r i f v X ! ! * V 

{s, x] = a, b, c /. fl - -- , -- T , - , 

V AX a x b x cj 

where a, b, c have the values \ (1 - X 2 ), \ (1 - /A 2 ), \ (1 - v 2 ), and X, p, v are real and 
positive, has x = a, b, c for critical points of the kind in question. In fact, writing 
x a = h, the equation is of the form 



- X 2 ) a 
, -j + -j? + a 1 + aJi+ ..., 

which is satisfied by 

d , ds 1 + X , , 7 , , 

a^a -r-^ +M+MF+ - J 

we thence obtain an integral of the form 

s = M~ x (1 + kji + k 2 h* +...), =k&lt;}&gt; for shortness. 
This is a particular integral, but we have from it the general integral 



* Since there is no critical point on the x-line there can be no abrupt change of direction in the path 
of s, that is, the path of s cannot consist of circular arcs meeting at an angle: but it is in the text 
further assumed that the path of s cannot consist of different arcs of circle, the one continuing the other 
without any abrupt change of direction. 



745] AND THE POLYHEDRAL FUNCTIONS. 173 

If A be the value of s corresponding to h = 0, then /3 = BA , and we find 



l4 __ - A+ ^, 

"l S 7J.M ^^/-t] -* 1 T 5.7 " ~ "7 T 5 



__ 

i 7 A. " S 7J. ^/-t -* T 5.7 " ~ "7 

7 + ofc&lt;p V bk&lt;pj \ OK&lt;p/ OK 9 

viz. reducing - to its principal term A A , and then writing ds, dx for s A, and h(=x a} 

respectively, we have ds = K (dx) K , or x = a is a critical point with the exponent A, ; 
and similarly x = b and x = c are critical points with the exponents yu, and v respectively. 

53. Hence in the equation 

/ Y 1 1 1 \ 2 

fi/i A- 1 - * * i 

[8, X \= a, D, C . .(I -- , -- r, -I, 

V Ax ax bx c] 

as the point x, passing successively through a, b, c, describes the #-line, the point s, 
passing successively through A, B, C, describes the sides AB, BC, CA of the curvilinear 
triangle ABC. To points x indefinitely near the -line correspond points s indefinitely 
near the boundary AB, BC, CA of the triangle, viz. to points x indefinitely near to 
and on one side, suppose the upper side, of the #-line, correspond the points s 
indefinitely near to and within the boundary of the triangle : and in like manner to 
whole series of the points x on the same upper side of the #-line, correspond the 
whole series of points s inside the triangle. 

54. We have attended so far only to one of the points s which correspond to 
a given point x, but considering the set of points s which correspond to the same 
point x, we have in the s-plane entire circles forming by their intersections curvilinear 
triangles ABC, ABC , &c. ; we have thus two systems, say ABC, &c., and ABC , &c., 
of triangles, such that to a point x on the upper side of the #-line correspond 
points s, one of them within each of the triangles ABC, &c., and to a point x on 
the lower side of the #-line correspond points s, one of them within each of the 
triangles ABC , &c. ; and so consequently that, to the two half-planes on opposite sides 
of the #-line, correspond the two sets of triangles ABC, &c., and ABC , &c., respectively. 

55. In order that the relation s and x may be an algebraical one, it is necessary 
that the two sets of triangles should completely cover, once or a finite number of 
times, the whole of the s-plane : and this implies that the angles XTT, JJLTT, VTT have 
certain determinate values; and, in fact, that dividing the surface of a sphere into 
triangles, each with these angles, the curvilinear triangles ABC, ABC , &c., are the 
stereographic projections of these triangles. It was by such considerations as these 
that Schwarz, in the Memoir of 1873, p. 323, obtained the series of values I to XV 
of X, /*, v, giving for a, b, c, = (1-V), ^ (I - ^), % (1 - v -), the series of values 
mentioned in the Annex of the PQR-T&lale : and thus showed a priori that the equation 



{*, }=(, b, c .*.Q , =-, - } 

\ \x ax bx c) 



is algebraically integrable for these values of a, b, c; and only for these values, or 
for values reducible to them. 



174 ON THE SCHWARZIAN DERIVATIVE [745 

56. As an instance, take the double pyramid form: the integral equation is 
f(x-a) : g(x-l) : h(x-c) = 4,s n : -(s l -l) 2 : (s n + l) z , 



or say 



(c - a) (x - b) _ Q n -l) 2 . 
(a-b)(x-c)~ (s w +l) 2 



s n _ ]\2 

or if, for greater simplicity, we assume a, b, c = 1, 0, oo , this is #= n or say 

1 -f V ic 

_( s w 1) = V#(s n + l), that is, 5 n = - f -, a solution of the differential equation 

1 + V x 



I, .i - f _ in - 
l, -(^, 2( 

In particular, if n = 3, we have x = 

3 4 3 



or s 3 = 



, a solution of 



1 \ 2 



a; _ 



57. We have here the spherical surface divided by the equator and three meridians 
into twelve triangles, each with the angles \ir, \TT, ^TT : and then, projecting from the 
South pole on the plane of the equator, we have the annexed figure of the s-plane, 



JB\ 




B, 



divided into 12 curvilinear triangles, each with these same angles 90, 90, 60 ; the 
plane is divided by the shading into two systems, each of 6 triangles. The figure 
of the #-plane is by the #-line divided into two half-planes, one shaded, the other 
unshaded ; and we have on the line the point c at oo , a at the origin, and b at 
the distance unity. 



745] AND THE POLYHEDRAL FUNCTIONS. 175 

58. Take x real ; then, if x is positive and less than 1, 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 + i sin 6 ; whence s is of the same form, or the locus of the 

1 + ki 

~^ Y y&gt; 

point s is a circle radius unity. Writing s 3 = -._ , and supposing that the point x 

1 + V# 

moves along the #-line from b through a to c at oo , and then from c at + oo to 6, 
the point s describes the sides BA, AC, CB of the shaded triangle marked K. 

59. Suppose that the point x is at k, in the shaded half-plane at an indefinitely 
small distance from a ; say we have x = %K 2 i, (K small), then taking for *Jx the value 

1 K. ( 1 i) 
*(l t), we have s 3 = = -^ /, =1 2(1 i) nearly, and hence a value of s is 

JL "T" fC ^ -L "~~ %J 

= 1 f/c + f/tt, which belongs to a point K near A, and within the shaded triangle: 
we have thus, in respect of this value of s, the shaded half of the #-plane corre 
sponding to this shaded triangle. To the same value x = 2fc 2 i, correspond in all six 
values of s, giving six points K each lying near a point A within one of the shaded 
triangles; and hence the shaded half-plane corresponds to the six shaded triangles, and 
the unshaded half-plane corresponds to the six unshaded triangles. 

60. Suppose the equation is 

i i / u X * 1 1 \ 2 

{s, as = a, b, c /.Q -, = , -) , 

V \x ax bx c) 

that is, 

_ - (b c) (c - g.) (a b) / a b c \ 

x a.xb.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 x is the circle passing through the points a, b, c, then the path of s is a circle 
passing through the corresponding three points. 

61. We may find a, /3, 7, , lf 0,, such that a, b, c are =a + @i + f y& i , a + fti + ye 9 ^, 
a + *+7e &lt; (this is, in fact, finding a and ft the coordinates of the centre, and 7 the 
radius of the circle through the three points a, b, c) : we then have x a. + @i + 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 ei - er 8 *), = ^ &lt; e + &lt;&gt;&gt; {e &lt;-&gt; - e~* {e ~ e ^} ; the second factor is 
isin(0 ), =iP suppose, or the equation is x a = iPye* (0+e }i , say 

x a = iPy expi (0 + ). 

Similarly x - I . = iQy expi 1 (6 + 6,), and as- c = iRy expi (6 + 2 ); where P, Q, R denote 
sin |(0 - # ), sin (6 - 0J, sin (0 - 2 ) respectively. In like manner, we have b - c, c-a, 
a - b, = iFry expi | (0! + 2 ), iGV expi (0 3 + ), ^ 7 expi (0 + 0,), where JP, G, 5" denote 

-^) respectively. 



176 ON THE SCHWARZIAN DERIVATIVE [745 

We have 

b c . c a .a b FGH \ /A t A . om 
- -.-..- o = -PQR expl * (e " + * + * - 3 "&gt; 



with the like values for - r and - , - . Hence the right-hand side of 

c a.x b a b.scc 

the equation is 

FGH /a b 



62. Considering now the left-hand side of the equation, we have 



substituting for x its value = a + fii + &lt;ye ei , this becomes 



that is, 

1 

7 2 

Assume s = L + Mi + Ne &i , L, M, and N constants ; then using the accent to denote 
differentiation in regard to 0, we find without difficulty {s, 6} = [, #}+|- 2 , and the 
value of {s, x} becomes 



= - - 2 ({0, 6} + $&* - i) expi (- 20). 



Hence, substituting the values of the two sides of the equation, the imaginary 
factor expi ( 20) divides out, and the equation becomes 



AI irvs i _ a b c 

~ 



an equation, in which everything is real and which thus determines S as a real 
function of d : and we have therefore the theorem in question. 



Connexion with the differential equation for the hypergeometric series. Art. Nos. 63 to 68. 

63. Take p, q given functions of x, and y a function of x determined by the 
equation 



745] AND THE POLYHEDRAL FUNCTIONS. 177 

again P, Q given functions of z, and v a function of z determined by the equation 



d?v 
j-. &gt; 
dz i 

and assume 



y = wv. 



Substituting this value of y in the first equation, we obtain for v an equation 
of the second order (the coefficients of which contain w), and we may make this 
identical with the second equation ; viz. comparing the coefficients of the two equations, 
we thus have two equations each containing w ; and by eliminating w we obtain a 
differential equation of the third order between z and x. This is, in fact, the basis 
of Kummer s theory for the transformation of a hypergeometric series : the equation 
between z, x will be found presently in a different manner. 

64. But if with Schwarz, instead of making the equation obtained for v as above 
identical with the given equation for v, we merely assume that the two equations are 
consistent, then there is nothing to determine the value of z, which may be regarded 
as an arbitrary function of x] y and v are then functions of x, and w denotes the 
quotient y-r-v of these two functions, and as such satisfies an equation the form of 
which will depend on the assumed relation between z and x. In particular, if P and 
Q denote the same functions of z that p and q are of as ; and if we assume z = x, 
P, Q will become =p, q respectively : the given equation in v will be 

d 2 v dv 



and w will thus denote the quotient of any two solutions of the equation 

d-y 



viz. writing X = p- + 2 -^- 4&lt;q, then, by what precedes, the equation for w will be 
doc 

{w, x}=- 



65. Returning now to Kummer s problem, and considering y, v as solutions of 
the two differential equations respectively, w is a function independent of the particular 
solutions denoted by these letters : we have y = ivv, and taking any other two solutions 

77 ?J 

we have y 1 = wv 1 , so that = -; calling each of these equal quantities s, we have s 

y\ v i 

denoting the quotient of two solutions of the equation in y, and also the quotient 

dp 
dx 



of two solutions of the equation in v ; whence, writing as before X =p 2 + 2 ~- 4&gt;q, 



dP 

and similarly Z = P 2 + 2 -^ -- 4Q, we have 



and since in general 

^ ^ = () ^ z ^ + & * * 

C. XL 23 



178 ON THE SCHWARZIAN DERIVATIVE [745 

we obtain 



-I 



as the required equation for the determination of z as a function of x. The process 
does not give the value of w, but this can be found without difficulty, viz. 



-. 
ax 

If z, x are regarded each of them as a function of the new independent variable 
9, then the equation is 



66. Jacobi s differential equation of the third order for the transformed modulus X, 
Fund. Nova, p. 78, [Ges. Werke, t. i, p. 132], is 



where the accents denote differentiations in regard to an independent variable 6 : viz. 
dividing by 2fc 2 X 2 , this becomes 



which is thus a particular case of Kummer s equation, k, X corresponding to x, z 
respectively, and the values of X, Z being 



67. In the case of the hypergeometric series, the two differential equations of the 

second order are 

d?y 7-(a + ff + 1) x dy __ aff.y _ 



Hence 



__ 
da? x .Ia; dx x.l x 

drv y- (of +^ + 1)2 dv _ _affv_ = Q 
dz- z . 1 z dz 2.1 z 

^ ))-( + /3 + l)A = 7 7-g-^-l = _- ayg 

. 1 X ~ X I X * X.I X 

;ind hence 






1.9 _ , . 

^ * q ~ 



x.l-x 

viz. writing 

X 2 = (1 - 7 ) 2 , a = i (1 - X 2 ), 



745] AND THE POLYHEDRAL FUNCTIONS. 

and putting in the formula x\, = (!#), we have 



-a + b c 



.x x oo x I/ 

with a like formula for ( P 2 + 2 -, 4Q ] . We then have 

V az J 

y = wv, 

~dz 
and the differential equation of the third order for the determination of z is 

/ , Yl 1 1 \- fdz\ 9 / , Yl 1 1 V 

{z, } + (%, b 1} c x .*.(}-, -, -^r-J j- - (a, b, c .-.-, -, - -T =. 

V )(z z en z 17 Vc?^/ V X x oo a; I/ 

where a u b ls Cj are the same functions of a, ft , y which a, b, c are of a, ft, y r 
This is, in effect, Rummer s equation for the transformation of the hypergeometric series. 

68. And in like manner the Schwarzian equation for the determination of s, the 
quotient of two solutions, is 



r i / u X 1 

[s, x = a, b, c ..()-, 

V X 



-, = . 
oo a; IJ 



PART II. THE POLYHEDRAL FUNCTIONS. 

Origin and Properties. Art. Nos. 69 to 80. 

69. The functions in lines 1, ...,5 of the PQJ?-Table are connected with the 
geometrical forms : 

1. Polygon or 

^2. Double Pyramid *, 

3. Tetrahedron, 

4. Octahedron and Cube, 

5. Dodecahedron and Icosahedron, 

(these figures being regarded as situate on a. spherical surface), and with the stereo- 
graphic projections of these figures. 

Prof. Klein regards 1 as belonging to the polygon and 2 to the double pyramid : it seems to me 
that the fundamental figure, to which 1 and 2 each of them belong, is the polygon. 

232 



180 ON THE SCHWAEZIAN DERIVATIVE [745 

Consider a spherical surface and upon it any number of points: take at pleasure 
any point as South Pole, this determines the plane of the equator ; and the stereo- 
graphic projection of any point is the intersection with the plane of the equator of 
the line joining the point with the South Pole. 

To fix the ideas take the radius of the sphere as unity: let the axes of x and y 
be drawn in the plane of the equator in longitudes and 90 respectively, and the 
axis of z upwards through the North Pole : the position of a point on the sphere 
is determined by means of its N.P.D. 6 and longitude /: moreover we take X, Y, Z 
for the coordinates of the point on the surface, and x, y for those of its projection ; 
and we then have 

X, Y, Z=sinO cos/, sin 6 sin/, cos 6 ; 

X 

tf = 

and conversely, 

X, Y, Z=2x, 2y, \-a?-y\ - (1 +x ~+ y-}. 

We represent the point (X, Y, Z) on the spherical surface by means of the 
magnitude x + iy, = tan (cos/+ i sin/), or say by the linear factor, s (x + iy): and 
similarly any system of points on the surface by means of the system of magnitudes 
x + iy, or say by the function IT [s (x + iy)}, denoting in this manner the product of 
the linear factors which correspond to the different points respectively. 

70. It will presently appear that, if (considering a different stenographic pro 
jection, that is, a different position of the South Pole) we take x, y as the coordinates 
of the new projection of the point, then x + iy is a homographic function 

a (x + iy) + b -i- [c (x + iy) + d} 

of x + iy: and consequently that the functions of s, which belong to different pro 
jections, are linear transformations one of the other: but at present we consider a 
single projection. 

It may be proper to remark that the figures in question are spherical figures 
having summits which are points on the spherical surface, edges (or sides) which 
are arcs of great circle joining two summits, and faces which are portions of the 
spherical surface : the mid-points of the sides, and the centres of the faces are of 
course points on the spherical surface. 

71. (1), (2). Considering a regular polygon formed by n summits on the equator, 
the longitude of one of them being 0, then the stereographic projections correspond 
with the points themselves, and the values of x + iy are 

27T . . 2-7T (n-l)2w . . 

1, cos M sin , .... cos h i sin 

n n n 

The corresponding function of s is s n 1. 



AND THE POLYHEDRAL FUNCTIONS. 



181 



The values of x + iy for the mid-points of the sides are 

TT . TT 3?r . . STT ( 2n-l}-rr . (2/1-1) TT 

cos h i sin - , cos --- h i sm . .... cos - - - -- (- 1 sm 2 - - . 
n n n n n n 

The corresponding function of s is 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 5=00. We have thus 



as the function corresponding to the double pyramid. 

72. (3). Considering for a moment the tetrahedron as a figure with rectilinear 
edges, this is so placed that two opposite edges are horizontal, and that the vertical 
planes passing through the centre and these two edges respectively are inclined at 
angles 45 to the meridian: viz. the upper edge has the longitudes 135, 315, 
and the lower edge the longitudes 45, 225. We thus explain the position of the 
spherical figure. 

Corresponding to the summits we have the function s* 2i V3 s 2 + 1. 

In fact, the equation s* 2i V3 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 


x + iy 


1 


I I 


l+i 


V3 
135 - 

225 - 

21 5 -u 


V3 V3 


V3-1 


V3 + 1 
-1 -t 


V3-1 
l + i 



Corresponding to the centres of the faces, or summits of the opposite tetrahedron, 
we have the function s*+ 2tV3s 2 + l. 

Corresponding to the mid-points of the sides, we have the function 



viz. the points in question are the North Pole s = 0, the South Pole s = oo , and 
the four points s=l, s=i on the equator at longitudes 0, 90, 180, 270 
respectively. 



182 ON THE SCHWARZIAN DERIVATIVE [745 

78. (4). The octahedron is placed with two of its summits as poles, and the 

other four summits in the equator at longitudes 0, 90, 180, 270 respectively: 

the values of s are, as in the last case, 0, oo , +1, i, and the function is 



The function for the centres of the faces, or summits of the cube, is s 8 + 14s 4 + 1. 
The function for the mid-points of the sides of the octahedron or of the cube is 

s 12 - 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-- 



The function for the centres of the faces of the Icosahedron, or summits of the 
Dodecahedron, is s 20 - 228s 15 + 494s 10 + 228s 3 - 1. 

The function for the mid-points of the sides of the Icosahedron or the Dodecahedron 
is 

s 30 - 522s 25 + 10005s 20 + Os 15 - 10005s 10 + 522s 5 + 1. 

I give for the present these results without demonstration. 

tJT 

75. Writing - for s so as to obtain homogeneous functions (*]#, y) n , it will be 

(7 

recollected that the x, y of these functions have nothing to do with the x, y of 
the foregoing values x + iy the forms which have thus presented themselves may be 
denoted as follows : 

(3): /3 = (1, -2V3, 1* 2 , tf-r, 
A3 = (l, +2tV3, l&r 2 , y-}-, 
3 = xy O 4 - y 4 ), 



(4): f* 

/i4 = (l, 14, l^ar 4 , ?/ 
4 = (1, -33, -33, 

(5): /5 = ay(l, 11, - 

k5 = (I, - 228, + 494, + 228, - 1$V, y 5 ) 4 , 

*5 = (1, -522, 10005, 0, -10005, 522, 1$V, y 5 ) 6 , 

where observe that /4 is the same function as 3. In each set of functions /, h, t, 
we have h and t covariants of f, viz. disregarding numerical factors, 

h is the Hessian, or derivative (/, /) 2 , and t is the derivative (f, h). 



745] AND THE POLYHEDRAL FUNCTIONS. 

76. Since /4 is the same function as 3, we have of course /4, A4 and t4&gt; 
themselves covariants of f 3 : but it is convenient to separate the two systems. 

77. It is to be observed that f 3 is a quartic function having its quadrinvariant 
(/) = ; but independently of this, that is, qua quartic function, it has only the 
covariants A3 and 3 (the Hessian and the cubicovariant respectively), viz. every other 
covariant is a rational and integral function of f3, A3 and 3. In particular, A4 and 
t are rational and integral functions of /3, A3 and 3 ; but inasmuch as f 3 and 
A3 are not covariants of /4, this is not a property of A4 and 4 considered as 
covariants of f4&gt;, and the relation in question need not be attended to. 

78. It has just been stated that /3 qua quartic function has (in the sense 
explained) only the covariants A3 and 3 : f4&gt; qua special sextic function and fo qua 
special dodecadic function have the like property, viz. /4 has only the covariants A4 
and 4 ; f5 only the covariants A5 and to. Hence fS, f4&gt;, fo are " Prime-forms " in 
the sense defined in the paper by Fuchs, of 1875, viz. a Prime-form has no covariant 
of a lower order than itself, and also no covariant of a higher order which is a power 
of a form of a lower order. 

79. The same functions have also the property that they are functions trans 
formable into themselves by means of a group of linear transformations, and in this 
point of view they were considered in the nearly contemporaneous paper by Klein, of 
1875; it is in this paper shown that the functions so transformable into themselves must 
be Polyhedral functions as above, the linear transformations in fact corresponding to 
the rotations whereby the spherical polyhedron can be brought into coincidence with 
its own original position. This theory will be presently given. 

80. It is to be observed that, if U, V are functions (*$#, y) n of the same 
order n, then using the accent to denote differentiation in regard to x, UV - U V 
and (U, V) differ only by a numerical factor: and further that, writing as before 

cc 
*=-, and in the expression UV - U V regarding U, V as functions (*}[s, l) n , and 

t/ 

the accent as denoting differentiation in regard to s, we have UV U V and (U, V) 
differing by a numerical factor only. We have in the PQP-Table, lines 3, 4, 5, 
P, Q, R equal to given numerical multiples of W, F, f a , the indices a, /3, 7 being 
such as to make these to be functions of the same degree: hence, neglecting 
numerical multipliers, PQ - P Q is equal to a function (/*?, F), which is = A"" 1 ^- 1 (A, t) : 
and the theorem that PQ - P Q, = QR - Q R, = RP - R P, contains only factors of 
P, Q, R is in fact the theorem that (A, t\ (A, /), and (t, /) are each of them equal 
to a term or product of /, A, t : which is a result included in the theorem that / 
has only the covariants A and t. And by this last theorem we know already how 
from R, assumed to be known, we can derive P and Q : viz. R is a power of /; 
and we thence have A = (/, /) 2 and t = (h, /), equations giving the functions A and t, 
upon which P and Q depend. 



184 ON THE SCHWARZIAN DERIVATIVE [745 



Covariantive Formulce. Art. Nos. 81 to 84. 

81. The various covariantive formulae will be given with their proper numerical 
coefficients. 

Tetrahedron function. /, h, t stand for the before-mentioned values, 
/3, A3, $ (P, Q, R = h 3 , -12iV3. 2 , -f 3 ). 



For /3. 



^_ o 

(a, b, c, d, e) = l, 0, -- , 0, 1. 



(/, A) = 32i V3 . t, (/, /) 4 = 5767= 0, (/ A) 4 = 1152/ = 1152 . 



--, 



&-/- 12* V S^ -O, 

/A = (l, 14, l$a-, 2/ 4 ) 2 
It is convenient to remark that t 2 , f 3 , h 3 being of the same order we have 

t- (f 3 , h 3 ) +f 3 (k 3 , Z 2 ) + h 3 (t\ f 3 } = 0, 
that is, 

t 2 . 3 . 3/ 2 A 2 (/ h) +f 3 . 3 . 2hH (h, t) + h 3 .2. 3tf 2 (t, /) = 0, 

an equation which, substituting for (/, h), (h, t), (t, /) their values, reduces itself to 
the before-mentioned relation h 3 f s I2iJ 3P= ; and we have thus a verification oi 
the values of (/, K), (h, t) and (t, /). The like remark applies to the other two 
cases, which follow. 

82. Hexahedron function. /, h, t stand for the before-mentioned values 
/4, A4, *4 (P, Q, R = h 3 , -t\ - 



For /4. 

(a, 6, c, d, e,f, #)=(0, i 0, 0, 0, -, 0). 

y = 0, I (/ /) = (720) 2 . f, 



(/, = - 12A 2 , | (, O 2 = 2* . 3MP . 

(A, t) = - 1728/ 3 , 

7i 3 - 2 -108/ 4 =0. 



745] AND THE POLYHEDRAL FUNCTIONS. 185 

83. Dodecahedron function. /, h, t stand for the before-mentioned values 

/5, h5, to (P, Q, R=h s , -t\ -1728/ 5 ). 
For /5. 

(a, b, c, d, e,f, g, h, i, j, k, I, m) = (0, &, 0, 0, 0, 0, , 0, 0, 0, 0, -&, 0). 

$ (/ /) = - 121/&gt;, \ (f, fY = 0, \ (/, /) = \ (924)* (720)" . if/*, 

I (/, /) 8 =0, i (/, f) w = 0, J (/, fr = i (924)* (720)&lt; . ff*, 

(/ A) = - 20*, i (A, A) 2 = 173280/ 3 , 

(/*) = - 30#, | (*, *) 2 = 9082800/ 3 /*, 
(A, $) = - 86400/ 5 , 



84. We have 

t = (x w + y w ) (1, 522, -10006, -522, 

Write 

^ = (^ + 2 / 2 ).(l, 2, 6, -2, l$a;, y) 4 , 

then 

= f(l, -10, 45$f,/). 

Or putting 

(^ + y 2 )(i&gt; 2. 6, -2, i$a? f 



V/ Vy ( 10 + 

that is, ^ = jjV/. then 



- 10p 3 + 45^ = -y^j . (Klein.) 



Investigation of the forms fb and ho. Art. Nos. 85 and 86. 

jfy _ ^ 
85. Writing for shortnessf A: = tana = ^ &gt; and ff = cos 36 + * sin 36 then tne 

values of x + iy corresponding to the summits of the Icosahedron are 

0, 

k, kg*, kg*, kg 6 , kg*, 

k-&gt;g, &-y, k-y, k-y, k-y, 

oo; 
and the function /5 is thus 



* The numerical coefficients - \ and ff are Klein s B and A: the latter of them is the ordinary 
quadrinvariant of a dodecadic function; the former is an invariant linear as regards the coefficients of /, 
and existing only for the special form / in question : viz. writing for a moment 



then (/, /) 6 contains the factor X-, and (/ containing the factor X) the form is 

4(/./) 6 
which is linear as regards X. We have also 



say 4 = !fX 2 , U=- \X; or 8iB*=A. Of course in the case of a general dodecadic function /, we have 
(/, /) ti , an irreducible covariant, not breaking up into factors. 

t a is the a, 7 is the 7, and / the a-/3 of the Table, No. 99. 

c. XL 24 



186 ON THE SCHWARZIAN DERIVATIVE [745 

where the product of the last two factors is s w + (/c~ 5 -k 5 ) s 5 -I. We have 

kr* = ^ (80 V5 + 170), = I (5 V5 + 11), 

fr = ^(80^5-176), =|(5V5-11), 
and consequently k~ 5 -k 5 = Il; or the function is 



86. Similarly, writing for shortness* I tan 7, =tan|7 , where 

5 + 2V5 . , 10-2^5 cos 7 3 + V5 

cos 2 7 = - , sm 2 7 = -- -- - ; and therefore -. * = /- ; 
15 15 sm7 4 



, = 

15 15 sin 7 4 

and # = cos 36 + i sin 36 as before, then the values of x + iy for the summits of the 
dodecahedron are 



l~\ l-y, l-y, i-y, i-y. 

The function ho is therefore 

= 5 10 + S 5 (I s - I *) + 1 . s 10 4- S 5 (l s - l ~ 5 ) - 1. 

We have 

7_ 5 75 (1 + cos 7) 5 - (1 - cos 7) 5 2 cos 7 . 
~ * = ^5 ~^ = 5 - (5 + 10 cos 2 7 + cos 4 7) 

sin 8 7 sin 5 7 f 

_ 2 cos 7 384 + 64 V5 128 cos 7 

- sin*7 ~ ~4T = T5 sin^7 (6 + Vo) = 114 + V ; 

viz. this last identity depends on 

ff (3 + V5) (6 + V5) = (114 + 50 v/5) sin 4 7) 
that is, 

160 (3 + V5) (6 + V5) = (114 + 50 V5U120 - 40 Vo), 
or 

2 (3 + V5) (6 + V5) = (57 + 25 V5) (3 - V5), 
or finally 

(7 + 3V5)(6 + V5)= 57 + 25V-5, 
which is right. 

Similarly 

l -*-r* = 114-50^5, 

and observing that the sum and product of 114 + 50^-5, 114 - 50 x/5 are =228 and 
496 respectively, the required function of s is 

( S io -i)2_228 (s 15 -s 5 ) + 496s 10 , 
= s 20 - 228s 15 + 494s 10 + 228s 5 + 1, 
which is the required value of /to. 

* a is the a, 7 is the 7, and 7 the a-/3 of the Table, No. 99. 



745] AND THE POLYHEDRAL FUNCTIONS. 187 

Invariantive property of the Stereographic Projection. Art. Nos. 87 to 93. 

87. The before-mentioned theorem that the functions derived from two different 
stereographic projections of the same point are linear transformations one of the other, 
may be thus stated : 

Considering on the surface of a sphere, two fixed points A and B; and determining 
the position of a point C, first in regard to A by its distance and azimuth / and 




x 



next in regard to B by its distance & and azimuth / , the azimuths from the great 
circle ABx which joins the two points A and B, then we have 

tan \Q (cos / + i sin / ), and tan \& (cos / + i sin / ), 

homographic functions one of the other : calling them s, s , and putting the distance 
AB=c, the relation between them in fact is 

. s tan ic 

o - _ *L 

1 +stan |c 
or, what is the same thing, 

tanc(l + ss) = s s ; 
or, observing that 

ss = tan \Q tan \& (cos (/+/ ) + i sin (/+/ )}, 
we have the two equations 

tan |- c [1 + tan |0 tan %0 cos (/+/ )} = tan \Q cos/ - tan # cos/ , 
tan c { tan \Q tan \& sin (/+/ )} = tan \6 sin/ tan \& sin/ . 
88. If we denote the angles of the spherical triangle by C, A, B, and the 
opposite sides by c (as before), a, b, then 6, = b, a; /, f = A, ir B, whence 

s, s = tari | b (cos A + i sin A), tan |a (cos B i sin B) : 

or we have between the sides a, b, c and angles A, B of a spherical triangle the 

relations 

tan c {1 tan |a tan |6 cos (A B)} = tan ^b cos A + tan \a cos B, 
tan \c { tan \a tan ^b sin (A B}\ = tan ^b sin A tan |a sin B; 

242 



188 



ON THE SCHWARZIAN DERIVATIVE 



[745 



equations which may be verified by means of the ordinary formulae of Spherical 
Trigonometry. 

89. But it is interesting to give the proof with rectangular coordinates. 

Taking (X, Y, Z), (X lt Y 1} ZJ for the coordinates, referred to two different sets 
of axes, of a point on the spherical surface: also x, y, x l , y l for the coordinates of 
the corresponding stereographic projections, we have 

(x l} r lv 4)-(, , 7 

, 13 , y 



X : Y : Z : 1 = 2aj : 2y : 1 - a? - f : 1+ a? + f , 

X, : Y, : Z, : 1 = 2^ : 2y x : l-a;*-y* : l+tf + yf, 
and thence 

a?j : y l : 1 = 2or + 2/3y +7 (1 a? y 2 ) 

: 2a f a; + 2j3 y + 7 (l-x 2 - y 2 ) 

: 1 + x 2 + f- + 2a"x + 2@"y + 7" (1 - a? - f). 

/y a I f** ni 

90. Introducing z, z^ for homogeneity, or writing -, - and , in place of 

Z Z&gt; j Z \ 

x, y and x lt y lf respectively, we have 

x l = 2ax + 2fiy +7 (z- - y? - y 2 ), = ( - 7 , - 7 , 7 , , a , 0$, y, 0)-, 

/ (^-^-.V 2 ), =( ~7&gt; -7, y,ff,*,01i ) 2 , 



- y 



+ 7", 



and thence without difficulty 
* = --77 {(1 + 7") 



7 + 7" 



{(1 + 7") 

K! - 7") 



iy}}, 
ty)J, 



^ _ iy, = _ /2 {(1 - 7 ") ^ - (a" + iff ) (x - iy)} {(1 + 7") * + &lt; a" - iff ) (x + iy)}, 

viz. the form is z l : x l + iy t : x 1 iy 1 = MN : NL : LM (L, M, N linear functions of 
z, x 4- iy, x iy) : showing that the relation between two stereographic projections of 
the same spherical figure is in fact that of a quadric transformation, the fundamental 
points in each figure being an arbitrary point and the two circular points at infinity: 
or, what is the same thing, to any line in the one figure there corresponds a circle 
in the other figure, which is the " circular relation " of Mobius. 

91. The actual values are 

xi + iyi = 1 + j" (1 - y ) z - (a" - iff ) (x + iy) 
z, 7 + 7 (1 + 7") * + (" - ") ( x + *y) 
xi - iyi = !+ 7" (1 - y") z - (a" + iff ) (x - iy) 
Zi 7 - 7 f t (1 + 7") z + (" + iff) (x - iy) 



74 5 J AND THE POLYHEDRAL FUNCTIONS. 189 

np \ *i 7/ 

viz. attending only to the former of these, we have - a homographic function of 

z \ 

- , which is the before-mentioned theorem. 

z 

92. Supposing that the transformation from (X, Y, Z) to (X lt Y l} Z^) is made by 
a rotation, the coordinates of which are \, ji, v: that is, if /, g, h are the inclinations 
of the resultant axis to the axes of x, y, z respectively, and 6 the angle of rotation, 
putting X, /j,, v = t3ui^6cosf, tan|0cos#, tan |# cos h: then the coefficients of trans 
formation are 



, P, 7 
", 0", 7" 



2 (i/X + //,) , 2 (// - X) , 1 - X 2 - p? + v- 



Substituting these values, the formulae become, after an easy reduction, 

i] (x + iy} + (X, + ip) z 



Zi (X - t (a; + iy) + (v-i)z 

\ ij/i _ (v i) (x iy) + (X ip) z 
z^ (\ + i/j,) (x - iy) +(v + i)z 

attending to the former of these, and writing for greater simplicity 

x l + iy-, x + iy 


respectively, we have 

or writing this 



_ (v + i) s + (\ + if 



then A : B : C : D = v i 



v . 



93. I call to mind that the condition, in order that the homographic transformation 
s 1 = (As + B) -f- (Cs + D) may be periodic of the order n, is 

(A + D)- - 4 (AD - BC) cos 2 = 0, 

in being an integer different from zero and prime to n. In particular, when n = 2, it 
is ^1+D = 0: w = 3, it is A*+ AD + D- + BC = : n = 4, it is A 2 + D 2 + 2BG= : and 
n = 5, it is (A + D)* - i (3 V5) (AD - BC) = 0. 

Groups of homographic transformations. Art. Nos. 94 and 95. 

94. The formulae just obtained serve to connect the theory of the rotations of 
a polyhedron with that of the homographic transformations s into (As + B) + (Cs + D) : 
and, corresponding to the rotations which leave the polyhedron unaltered, we have 
groups of homographic transformations. We have thus, corresponding to the cases of 
the tetrahedron, the cube and the octahedron, and the dodecahedron and icosahedron 
respectively, groups of 12, of 24, and of 60 homographic transformations s into 



190 ON THE SCHWARZIAN DERIVATIVE [745 

(As + B) + (Cs + D). The group of 60 and the group of 24 include each of them as 
part of itself the group of 12 : it is further to be remarked that the group of 12 
may be regarded as that of the positive substitutions upon four letters abed, the 
group of 24 as that of all the substitutions upon the four letters, and the group of 
60 as that of the positive substitutions upon five letters abode. 

95. I call to mind that a group of functional symbols 1, a, /3, ... can always 
be expressed in the equivalent form 1, ^aS-" 1 , ^/S^" 1 , ... where ^ is any functional 
symbol whatever : clearly, a, /3, . . . being homographic transformations, then, S- being 
any homographic transformation whatever, the new symbols ^a^-" 1 , S-/3S-" 1 , ... will also 
be homographic transformations ; and thus the group of homographic transformations 
can be expressed in various equivalent forms : these correspond to the different 
positions of the polyhedron in regard to the axes of coordinates: and there are in 
fact three cases which it is proper to consider, viz. attending for the moment to the 
dodecahedron, we may have the axis of z passing through the midpoint of a side, 
through the centre of a face, or through a summit; that is, in the language 
presently explained, the cases are 1, Pole at a point ; 2, Pole at a point A ; 
3, Pole at a point B. 

The regular Polyhedra. Art. Nos. 96 to 103. 

96. We require a theory of the regular Polyhedra considered as systems of points 
on a sphere. I refer to my two papers [375] and [679]. In the latter paper, I 
remark that, considering the five regular figures drawn in proper relation to each 
other on the same spherical surface, the only points which have to be considered are 
12 points A, 20 points B, 30 points @, and 60 points 3&gt;. Describing these by 
reference to the dodecahedron, the points A are the centres of the faces, the points 
B are the summits, the points are the midpoints of the sides, and the points 3&gt; 
are the midpoints of the diagonals of the faces. Or describing them by reference to 
the icosahedron, the points A are the summits, the points B are the centres of the 
faces, the points are the midpoints of the sides: viz. each point is the common 
midpoint of a side of the dodecahedron and a side of the icosahedron, which there 
intersect at right angles: and the points 4&gt; are points lying by threes on the faces 
of the icosahedron, each point &lt;& of the face being given as the intersection of a 
perpendicular AS of the face by a line BB joining the centres of two adjacent 
faces and which intersects A at right angles. 

97. The points 3&gt; are comparatively unimportant, and it is proper in the first 
instance to attend only to the 12 points A, the 20 points B, and the 30 points 0: 
these form 6 pairs of opposite points A, 10 pairs of opposite points B, and 15 pairs 
of opposite points . Considering the diameters through each pair of opposite points 
, we have thus a system of 15 axes, which in fact form 5 sets each of 3 rect 
angular axes: attending to any one of such sets, the diametral plane at right angles 
to one of the three axes contains of course the other two axes : it contains also 
two axes each through a pair of opposite points A, and two axes each through a 
pair of opposite points B. If instead of the plane we consider its intersection with 
the sphere, we have thus on the sphere 15 circles each containing 4 points , 



745] 



AND THE POLYHEDRAL FUNCTIONS. 



191 



4 points A and 4 points B. The fifteen circles intersect by fives in the pairs of 
opposite points A, by threes in the pairs of opposite points B, and by twos in the 
pairs of opposite points ; the mutual inclinations of successive circles at the points 
A, B, being =36, 60 and 90 respectively. The whole number 15.14, =210, of 
the intersections of the circles two and two together is thus made up of the 12 
points A each counting 10 times, the 20 points B each counting 3 times, and the 
30 points each counting once ; 210 = 120 + 60 + 30. 

98. The angular magnitudes which present themselves are all obtained from 
the dodecahedral pentagon, as shown in the annexed figure, in which the angle 
subtended by a side at the centre is = 72, and the angle between two adjacent 
sides is =120. 




We write A = a, B=@, AB = j, B,B^ = x, ^B 1 B 4 B = 0, B 4 =g, Z 

From the triangle AB, the angles of which are 36, 90, 60 and the opposite 
sides ft, 7, a, we find the values of a., ft, 7, and these are such that a + ft + 7 = TT. 

From the triangle B 4 BB^, where the sides B 4 B, BB L , and the included angle are 
2/3, 2/3, 120, we have the opposite side x, and the other two angles each = 0. 

From the triangle B 4 BS, where the sides B 4 B, B, and the included angle are 
2/3, ft, 120, we find the opposite side g, the angle BB 4 , = &lt;, and the angle 
B 4 %B, =45. 

Hence each of the angles B 4 SB, B a B lt being =45, the angle B t @B a is =90: 
in this triangle the hypothenuse B 2 B 4 is =x, and each of the other two sides is 
= g: whence we have cos x = cos 2 g, as is in fact the case, and moreover the values 
give x + 2^ = 180. Also each of the other angles is found to be =60; that is, we 
have Z B 2 BJ& = 60, or the whole angle at B 4 being =120, the sum of the remaining 
angles B 3 B 4 B n _ and BB 4 is =60 C : that is, + = 60. 

From the triangle J^ where the two sides and the included angle are 
/3, ft, 120, we find = 36. 



192 



ON THE SCHWARZIAN DERIVATIVE 



[745 

And from the triangle 5 4 @", where the two sides and the included angle are 
g, g and (120 - 2&lt;=)20, we find 08" = 60. 

99. We thus arrive at the following Table: 



where as above 



+ + 7 = 90, 
x + 2g = 180, 
6+&lt;f&gt; =60. 



cos 







01 o JO 


/5-V5 


/5 H- ^5 


A 


a 




ol 4-o 

OA KK 


V 10 
,/-! 


V 10 

V5+1 


_ofe) 


P 




2V3 


2^/3 


A n 




owo 9 .-y 


/ 10 -2^/5 


/5 + 2V5 


(BB) 
(&) 

D B D 


y 

X 

9 


70* 32 
54 44 

.7 &lt;lfi 


V 15 

2^/2 
3 

5/2 

V3 

x/3 


V 15 
1 
3 
1 

T3 

J* 


JiJtJb 
T)(3i T) 


V 


92 14 


2^2 

-s/3 (V5 - 1) 


1J2 
V5 + 3 




9 

2a 

2/8 

9,/ 


63 26 
41 50 
74 44 


4^/2 
2 

%/ 

2 

3 

2(^/5 + 1) 


4x/2 
1 

N/5 

^5 
3 

4-^/5 




Jy 




3^5 


3^/5 




n R 




/5-2V5 


/10 + 2^/S 




a p 




V 15 


V 15 






18 


V5-1 


/5 + N /5 








4 


V 8 







36 


/5-V5 


^5 + 1 








V 8 


4 



100. We now construct three figures of the points A, B, ; viz. these are 
stereographic projections, each showing the Northern hemisphere projected on the plane 
of the equator by lines drawn to the South Pole: hence, for any pair of opposite 
points not on the equator, only the point in the Northern hemisphere is shown: 
but for a pair of opposite points on the equator the two points are each of them 
shown. In fig. 1 the North Pole is taken to be a point ; in fig. 2 it is a point 
A ; and in fig. 3 it is a point B. The position of any point on the sphere is 
determined by its N.P.D. and its longitude, measured from an arbitrary origin, 
say from the point E of the centre left-handedly : then, in the three figures, the 
positions are as follows. 



194 ON THE SCHWARZIAN DERIVATIVE 

102. Fig. 2. Pole at A. 



e 




N.P.D. s 



Longitudes. 



A 








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 


5 


90 - a + ft = 79 12 


36 108 180 252 324 


5B 


90 + a - /? = 100 48 


72 144 216 288 


bB 


180 - y = 142 38 


72 144 216 288 


5 


o= 31 43 


72 144 216 288 


5 


90 - a = 58 17 


36 108 180 252 324 


10 


90 


(36 108 180 252 324) + 18 


5 


90 + a = 121 43 


72 144 216 288 


5 


180 -a= 144 17 


36 108 180 252 324 



745] 



AND THE POLYHEDRAL FUNCTIONS. 



195 



103. Fig. 3. Pole at B. 




KP.D. s 



Longitudes. 



3A 


y= 37 22 


30 150 270 




3A 


90- a + = 79 12 


90 210 330 




3A 


90 + a - /3 = 100 48 


30 150 270 




3A 


180 - y = 142 38 


90 210 330 




B 





__ 




3B 


2/3- 41 50 


90 210 330 




6B 


x= 70 32 


(30 150 270) + 


= 37 46 


QB 


180- x= 109 28 


(90 210 330) + & 


= 37 46 


3B 


180 -2)8= 138 10 


30 150 270 




B 


180 







3 


/8 = 20 55 


90 210 330 




6 


g= 54 44 


(90 210 330) + &lt;^&gt; 


- 22 14 


3 


90- = 69 5 


30 150 270 




6 


90 


60 120 180 


240 300 


3 


90 + /?= 110 55 


90 210 330 




6 


180 -g = 125 16 


(30 150 270) + &lt;f&gt; 


= 22 14 


3 


180 -0= 159 5 


30 150 270 





252 



196 ON THE SCHWARZIAN DERIVATIVE [745 



The groups of homographic transformations, resumed. Art. Nos. 104 to 117. 

104. The axes of rotation for the dodecahedron and the icosahedron are 15 axes 
each through a pair of opposite points , 6 axes each through a pair of opposite 
points A, and 10 axes each through a pair of opposite points B\ or say 15 -axes, 
10 jB-axes and 6 .4-axes : the corresponding angles of rotation are 180, 72 and 120 ; 
so that (excluding in each case the original position or that of a rotation 0) we have 
in respect of each -axis 1 position, in respect of each A-axis 4 positions, and in 
respect of each 5-axis 2 positions; in all, including the original position, 

1 + 15 + (6 x 4) + (10 x 2), = 60 positions, 
that is, a group of 60 rotations. 

To find, in any one of the three forms, the group of homographic transformations, 
we can in each case obtain from the foregoing tables the values cos/, cosg, cosh of 
the cosine-inclination of an axis of rotation to the axes of coordinates, and thence 
calculate the values of 

X, fj,, v = tan ^S-cos/, tan |S- cos g, tan |S- cos h, 
and thence the values of 

A, B, C, D = v i, 



viz. in the case of a -axis, S- is = 180, (so that here tan S- = oo , or the values of 
A, B, C, D are = v, \+ifji, \-ifju, v, that is, cos h, cosf+icosg, cosficosg, cos A); 
in the case of a JB-axis, the values are ^ = 120, 240, and therefore tan^= + \/3; 
and in the case of an J.-axis, they are S- = 72, 144, 216, 288, and therefore 



V5 - 1 V + 1 

105. The -form was first given in my paper of 1879, but in obtaining it I 
used results given in the paper of 1877. As regards the identification with the 
substitution-symbols, since there is nothing to distinguish inter se the letters a, b, c, d, e, 
any transformation A, B, C, D of the fifth order might have been taken for abode, 
but No. 37 of the group having been taken for this substitution abcde, I do not 
recall in what manner I found that, consistently herewith, the transformation No. 2 
( 1, 0, 0, 1, that is, s into -s) of the second order could be taken for ab.cd. But 
there is no sub-group of an order divisible by 5 ; and hence, these two transformations 
being identified with the two substitutions, the other transformations correspond each 
of them to a determinate substitution. 



745] 



AND THE POLYHEDRAL FUNCTIONS. 



197 



106. Homographic Transformations. The group of 60. Pole at . 

(Ax + B) * (Cx +D) 



2 
3 
4 

5 

6 

7 

8 

9 
10 
11 
12 
18 

14 ! 

15 i 
16 



17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 

37 
38 
39 
40 
41 
42 
43 
44 
45 



-1 


2 
2 
2 
2 
2 
2 
2 
2 



-i 

-1 

1 

i 

1 
-1 



3 + v/S) 



2 
2 
2 
2 
2 
2 
2 
2 
-2 



afc . cd 
ac . bd 
ad . be 
be . de 
ae . be 
ad . ce 

ad . be 
ae . cd 
ab . de 
be . cd 

ab . ce 
ac . be 
bd .ce 

ae . bd 

ac . de 

abc 
acb 
adc 
acd 
adb 
abd 
bed 
bdc 
aec 
ace. 
bed 
bde 
bee 
bee 
aed 
ode 
cde 
ced 
aeb 
abe 

abcde 
acebd 
adbec 
aedcb 
adceb 
acbde 
aedbc 
abecd 
acbed 



198 



ON THE SCHWARZIAN DERIVATIVE 



[745 



46 
47 
48 
49 
50 
51 
52 

n 

54 
M 
M 

57 
68 

59 
00 



l-x/5) 



l-x/5) 



5) 



3-^/5 + ^ l-x/5) 



-1 
2 
2 
2 
2 
2 
2 
2 
2 



abdce 
aecdb 
adebc 
aecbd 
acdeb 
abedc 
adbce 
aebdc 
abced 
adecb 
acdbe 
abdec 
adcbe 
aebcd 
acedb 



107. Taking out of the foregoing group of 60 a group of 12 contained in it, 
viz. that corresponding to the positive substitutions of the four letters abed, it is 
easy to see, that there is a transformation (i, 0, 0, 1), that is, s into is, which can 
be taken for the substitution adbc, and also to complete thence the group of 24. And 
we have thus the following Table. 



Groups of 12 and 24. Pole at @. 



(Ax 



1 


1 








1 


2 


1 








1 


3 





1 


1 





4 





1 


-1 





5 


-i 


i 


1 


1 


6 


- 1 


i 


1 


i 


7 




-i 


1 


i 


8 


i 


-i 


1 


-1 


9 


i 


i 


1 


-1 


10 


1 


i 


1 


i 


11 


-1 


-i 


1 


-i 


12 


i 


-i 


1 


1 


13 


i 








1 


14 


i 








1 


15 





i 


1 





16 





i 


-1 





17 


1 


-1 


1 


1 


18 


/ 


_ i 


1 


i 


19 


? 


1 


1 


i 


20 


1 


1 


1 


1 


21 


_ j 


-1 


1 


-1 


22 


i 


-1 


1 


-i 


23 


-i 


1 


1 


-i 


24 


-1 


1 


1 


1 



1 

ab . cd 

ac . bd 

ad . be 



adbc 
acbd 
cd 
ab 

acdb 

bd 

abed 

be 

abdc 

ac 

a deb 

ad 



745] AND THE POLYHEDRAL FUNCTIONS. 199 

108. The group of 60 was obtained in the J.-forrn by Gordan in his paper. 
The passage from the -form to the J.-form is made as follows: let X, Y, Z be 
the coordinates of a point when the axes are as in the -form, X l} Y 1} Z, the 
coordinates of the same point when the axes are as in the A-form : we may write 

X, Y, Z^bX.-aZ, : Y, : a 
where 



/ 

V 



5-V5 5 + V5 



then, if the equations of an axis of rotation referred to the first set of coordinates 
are X : Y : Z = L : M : N, those of the same axis referred to the second set of 
coordinates are 

bX. + aZ, : Y, : -aX^bZ^L : M : N; 
or taking these to be 

X, : 7, : Z, = L t : M, : N,, 
we may write 

L lt M lt N^bL + siN, M, -aZ 



these values are such that 

I, 2 + Mf + NI- = L 2 + M 2 + N 2 , 

and hence, \, /A, v and \ 1} fj, l , ^ being the rotations, we may write 

L, M, N=$\, v, *v, LI, M lt ^=^\ 15 Vi, X; 

where S- has the same value in each set of equations. From the equations 
A : B : C : D = v i : \+i/j, : \ i/j, : v i, 



we have 

B + C : B-C : D-A : D + A =\ : ip : v : - i 

= L : iM : N : - fb 
and similarly 

B. + C, : B.-C, : D,- A, : D. + A^L, : iM, : N, : - 

Hence we may write 

#+&lt;?!= \&gt;(B + C) + a,(D-A), 
B 1 -C l = B- C, 



or say, 

A , = 



which are the values for a transformation (A,, B 1} C lt A) in the ^-form: of course, 
as only the ratios are material, the values may be multiplied by any common factor. 



200 



ON THE SCHWARZIAN DERIVATIVE 



[745 



109. The results are exhibited in terms of e, an imaginary fifth root of unity : 
taking e = cos 72 + i sin 72, we have 



5 + V5 



where the upper signs belong to e, e and the lower to e 4 , e 3 . It may be remarked 

that 

1_ /5 + V5 1_ /5 - V5 b V5 + 1 a V5 - 1 
a~V~ 2 b~V~ 2 a~ 2 b~ 2 

For instance, we have in the -group (A, B, G, D) = ( 1, 0, 0, 1); ab.cd: and thence 
in the A -group A it B 1} C 1} D 1 = (-2b, 2a, 2a, 2b); ab.cd: or say this is 

-1, |, , l), =(-1, e + e 4 , e + e 4 , 1); 

which in the Table is given as ( e 3 , e 2 + e 4 , e 2 + e 4 , e 3 ) ; ab . cd. 

By effecting the passage to the A -group in this manner, we of course obtain the 
proper substitution corresponding to each transformation : but I found it easier starting 
from two transformations and the corresponding substitutions, to obtain thence by 
successive compositions the entire group. 

110. Homographic Transformations. The group of 60. Pole at A. 
0No. (As +B) ~(Cs +D) 



1 




1 


1 






1 


1 


2 




4 





-1 


1 





ad . be 


3 




13 





-e 4 


1 





ac . be 


4 




9 





-e 3 


1 





ae . cd 


5 




10 





-e 2 


1 





ab . de 


6 




14 





-e 


1 





bd . ce 


7 




6 


e + e 2 


e 4 


1 


-(e+e 3 ) 


ae . be 


8 




5 


e+e 3 


1 


e 4 


-(e+e 3 ) 


be . de 


9 




16 


e + e 3 


e 


e 3 


-(e + e 3 ) 


ac . de 


10 




3 


e + e 3 


e 2 


e 2 


-(e + e 3 ) 


ac . bd 


11 




15 


e + e :s 


r 5 


e 


-(e + e 3 ) 


ae . bd 


12 




12 


-1 


e + e 3 


e 2 + e 4 


1 


ab . ce 


13 




11 


e 


e 3 +l 


e 2 + e 4 


e 


be . cd 


14 




7 


- 2 


1+e 2 


e 2 + e 4 


e 2 


ad . ce 


15 




2 


-e 3 


e 2 +e 4 


e 2 + e 4 


e3 


ab . cd 


16 




8 


-e 4 


e 4 +e 


e^ + e 4 


e 4 


ad . be 



745] 



AND THE POLYHEDEAL FUNCTIONS. 



201 



17 




21 


6+l 


e 


1 


-(e + e 3 ) 


adb 


18 




35 


ea+1 


e 2 


e 4 


-(e + e 3 ) 


aeb 


19 




30 


e 3 + l 


e 3 


e 3 


-(e + e 3 ) 


bee 


20 




34 


e^+l 


e 4 


e 2 


-(e + e 3 ) 


ced 


21 




19 


+ ! 


1 


e 


-(e + e 3 ) 


adc 


22 




33 


e + e 4 


2 


1 


-(e + e 3 ) 


cde 


23 




20 


e + e 4 


e 3 


e 4 


-(e + e 3 ) 


acd 


24 




22 


e + e 4 


e 4 


e 3 


-(e + e 3 ) 


abd 


25 




36 


e + e 4 


1 


e 2 


-(e + e 3 ) 


abe 


26 




29 


e + e 4 


e 


e 


-(e + e 3 ) 


bee 


27 




31 


e 


e 2 + e 4 


e 2 + e 4 


1 


aed 


28 




17 


-e 2 


e 4 + e 


e 2 + e 4 


e 


abc 


29 




27 


-e 3 


e+e 3 


e 2 + e 4 


e 2 


bed 


30 




25 


-e 4 


e s +l 


e 2 +e 4 


e 3 


aec 


31 




23 


-1 


1+e 2 


e 2 + e 4 


e 4 


bed 


32 




24 


-e 4 


1 + 2 


e 2 +e 4 


1 


bdc 


33 




32 


-1 


e 2 + e 4 


e 2 + e 4 


e 


ode 


34 




18 


-e 


e 4 + e 


e 2 + e 4 


e 2 


acb 


35 




28 


-e 2 


e+e 3 


e 2 + e 4 


e 3 


bde 


36 




26 


-e 3 


eO+1 


e 2 +e 4 


e 4 


ace 


37 




44 


e 








1 


abecd 


38 




43 


e 2 








1 


aedbc 


39 




42 


e 3 








1 


acbde 


40 




41 


e 4 








1 


adceb 


41 




38 


e 2 + e 4 


1 


1 


-(e + e 3 ) 


acebd 


42 




46 


e 2 + e 4 


e 


e 4 


-(e + e 3 ) 


abdce 


43 




58 


e 2 + e 4 


e 2 


e 3 


-(e+e 3 ) 


adcbe 


44 




55 


e 2 + e 4 


e 3 


e 2 


-(e + e 3 ) 


adecb 


45 




50 


e 2 +e 4 


e 4 


e 


-(e + e 3 ) 


acdeb 


46 




51 


1+e 2 


e 3 


1 


-(e + e 3 ) 


abedc 


47 




39 


1+e 2 


e 4 


e 4 


-(e+e 3 ) 


adbec 


48 




47 


1+e 2 


1 


e 3 


-(e + e 3 ) 


aecdb 


49 




59 


1+e 2 


e 


e 2 


-(e + e 3 ) 


aebcd 


50 




54 


1+e 2 


e 2 


e 


-(e + e 3 ) 


abced 


51 




56 


-e 2 


e 3 + l 


e 2 + e 4 


1 


acdbe 


52 




49 


- e 3 


1 +e 2 


e 2 + e 4 


e 


aecbd 


53 




37 


-e 4 


e 2 + e 4 


e 2 + e 4 


e 2 


abcde 


54 




45 


-1 


e 4 + e 


e 2 +e 4 


e 3 


acbed 


55 




57 


e 


e+e 3 


e 2 + e 4 


e 4 


abdec 


56 




48 


-e 3 


e 4 + e 


e 2 +e* 


1 


adebc 


57 




60 


-e 4 


e+e 3 


e 2 + e 4 


e 


acedb 


58 




53 


_ j 


e 3 +l 


e 2 + e 4 


e 2 


aebdc 


59 




52 


e 


1+e 2 


e 2 + e 4 


e 3 


adbce 


60 




40 


-e 2 


e 2 + e 4 


e 2 + e 4 


e 4 


aedcb 



C. XI. 



26 



202 



ON THE SCHWARZIAN DERIVATIVE 



[745 



111. Selecting the transformations which correspond to the positive substitutions 
abed, and completing the group of 24 we have 

Homographic Transformations. The groups of 12 and 24. Pole at A. 

(As +B) -HC* +D) 



1 




1 








1 


1 


2 







-1 


1 





ad . be 


| 




e + e 3 


e - 


e 2 


-(e + e 3 ) 


ac . bd 


4 




-e 3 


e 2 + e 4 


e 2 + e 4 


e 3 


ab . cd 


5 




_ e 2 


e + e* 


e 2 + e 4 


e 


abc 


fi 




6 


e+e 4 


e 2 +e 4 


e 2 


acb 


7 




e+e 4 


r ! 


e 4 


-(e + e 3 ) 


/ li 


8 




e 3 + l 


1 


e 


-(e + e 3 ) 


adc 


9 




e+e 4 


e 4 


e 3 


-(e + e 3 ) 


aid 


10 




l+l 


e 


1 


-(e + e 3 ) 


adb 


11 




_ i 


1 + e 2 


e 2 + e 4 


e 4 


bed 


12 




-e 4 


1 + e 2 


e 2 + e 4 


1 


bdc 


11 




1 


l + 2e 4 


l + 2e 


-1 


ab 


14 




- e 2 + e 3 


1 + e + Se 4 


-1-Se-e 4 


6 2_ e 3 


cd 


15 




e 2 -e 4 


3 + e + e 3 


-l-Se-e 3 


-e 2 + e 4 


ac 


16 




-1+e 2 


-l-e 2 + 2e 4 


l + e 2 -2e 3 


1-e 2 


bd 


17 




2+e 3 + 2e 4 


- 2 - 2e 2 - e 3 


2e + e 3 + 2e 4 


2e + 2e 2 + e 3 


ad 


18 




2+2e 2 + e 3 


2 + e 3 + 2e 4 


-2e-2e 2 -e 3 


2e + e 3 + 2e 4 


be 


19 




-2 + e+e 3 


-e + e 3 


-e + e 3 


e + e 3 -2e 4 


abed 


20 




1 


-1 


1 


1 


abdc 


21 




1 


1 


-1 


1 


acdb 


22 




l + e + 3e 4 


e 2 -e 3 


e 2 -e 3 


l + 3e + e 


acbd 


23 




l + 2e 4 


-1 


-1 


-l-2e 


adbc 


24 




3 + e + e 3 


- e 2 + e 4 


-e 2 +e 4 


l + 3e + e 3 


adcb 



As an example of the calculation we have (A, B, C, D) = (0, i, 1, 0); ab. Hence 
A,, B lt G lt A= 



a a 

The second and third coefficients are 



1 . / 
*V 



5 + 



which, in virtue of the values of e and e 4 , are = 1 + 2e 4 and 1 + 2e respectively: or 
the result is as above (1, 1 + 2e 4 , 1 + 2e, 1). 



745] AND THE POLYHEDRAL FUNCTIONS. 203 

112. In like manner for the passage from the -form to the .#-form, if X, Y, Z 
be the coordinates of a point on the spherical surface in regard to the -axes, 
X 2 , Y.,, Z those of the same point in regard to the .B-axes, we may write 

X : Y : Z=X, 

where 



2V3 2V3 

Hence X : Y : Z= L : M : N, being the equations of an axis of rotation in the 
first set of coordinates, those of the same axis in the second set of coordinates 
will be 

J\. o : b JL a -|- 8i/ : a -i o i D == x/ . J.VI : JM , 
or calling these 

X, : Y., : Z, = L 2 : M^ : N.,, 
we have 

L,, M 2 , N, = L : })M-aN : aj 



these values are such that 

i a 2 + M, 2 + N./ = L- + M- + N 2 , 
or X, yu,, v, X 2 , fa, v-2 being the rotations, we have 

L, M, N=^\, V, *v\ L 2, M. 2 , N, = ^\, 

where ^- has the same value in the two sets of equations. We have thus 
B +C : B -C : D -A : D + A = L : 2M : N : -?$, 
B 2 + C, : B, - C, : D, - A, : D, + A, = Z 2 : 2Jf 2 : N, : - ?$, 

and hence 

B, + C, = B + C, 

B,-C,= \)(B-C)-ai(D-A), 
D. -A, = - ai (B-C) + b (D- A), 

D 2 + A,= D + A- 
and thence 

At= Sii(B-C)-}) (D-A) 

B,= b B-C-MD-A 



(D - A) + (D + A}. 
113. As an example of the transformation, take 

(A, B, C, D)= 2, -3 + V5 + i(l-V5), -3 + V5 + i(-l+V5), -2 [bc.de]: 



then 

B-C, B+C, D-A, D + A=i(\-*Jo\ -3 + V5, -2, 0; 

26 2 



204 

and thence 



ON THE SCHWARZIAN DERIVATIVE 



1 

2TV3 

1 
1 



23 



v/5) + ( - 



viz. multiplying by 2 \/3, these are 

8" / /^ [ rt / X \ i_ O / O / O i^ / "* \ si ( \ 9 / P\ \ i O / Q / __ 
j v \ ^" O "i ^ ^ O ) j~ *- M O I ~"~ O *i \f OK Z I O ^^ ^ ^ O ) "T* ^ Y ^ V "" 

that is, 

or since 

2 -f V3 = - 2i&&amp;gt; and - 2 + ^ 3 2ia)-, 
dividing by 4 these are 

2, i (3 - &lt;\/5) &lt;y, i (- 3 + V5) &lt;w 2 , - 2, 
as in the table. 

114. Homographic Transformations. The group of 60. Pole at B. 



-8, 



1 




1 








1 


1 


2 







1 


1 





ac . bd 


3 







u 


1 





ae . bd 


4 







u- 


1 





bd . ce 


5 




2 


i( 3- x /5) 


i( -3 + ^5) 


-2 


ab .cd 


6 




2 


i(-S-^5) 


i ( 3 + ^5) 


-2 


ad . be 


7 




2 


*( 3-^/5)0, 


i( -3 + ^/5)0? 


-2 


be .de 


8 




2 


t ( -- 3 - N /5) w 


i( 3+^5)^ 


-2 


be . cd 


9 




2 


i( 3- N /5)w 2 


i( _3 +x /5) w 


-2 


ad . be 


10 




2 


i(-3- N /5)w 2 


i( 3+^/5)0, 


-2 


ab . de 


11 




2 


(-JB-iJS)* 


(-v/S + zVS)^ 


-2 


ab .ce 


12 




2 


- v /3-i x /5 


-V3 + W5 


-2 


ac . be 


13 




2 


(_ x / 3 _, t - x / 5 ) w 2 


(.^3 + ; ^5)0, 


-2 


ae . be 


14 




2 


s/8-V6 


x/3 + ^5 


-2 


ac . de 


15 




2 


( V3-iV5)w 


( x/S + V^Jw 2 


-2 


ad . ce 


16 




2 


( \/3 i ^/5) w 2 


( ^3 + U /5) W 


-2 


ae .cd 



745] 



AND THE POLYHEDRAL FUNCTIONS. 



205 



17 




W 








1 


ace 


18 




w 2 








1 


aec 


19 




x/3-iV5 


2 


-2 


V3 + W5 


bed 


20 




^/3 i &gt;/5 


2 


-2 


-^3 + 1^/5 


bde 


21 




-x/3-i^S 


2w 2 


-S 


-v/3 + iV5 


bdc 


22 




v/S-i^S 


2w 2 


-2 W 


/o i , /r 

V "T * v 


bed 


23 




-V3-iV5 


2w 


- 2w 2 


-V3 + W5 


abd 


24 




^3-iVS 


2 co 


- 2w 2 


V3 + W5 


adb 


25 




2w 2 


-x/3-iV5 


-V3 + W5 


-2w 


abc 


26 




2w 


-V3-W 5 


- x/3 + 1 v 5 


-2w 2 


acb 


27 




2o&gt; 2 


-X/3-W5 


(-v/S + VS)^ 


-2 


abe 


28 




2 


-v/3-W 5 


(-v/S + i^w 2 


-2w 2 


aeb 


29 




2w 


X/3-W5 


\ ** ~T ^ V 


-2w 2 


acd 


30 




2w 2 


x/3-tVS 


X/ *J ~T 1 *V 


-2w 


adc 


31 




2w 2 


X/3-V5 


( v/o~f~z ^/o) w 


-2 


ade 


32 




2 


v/3 - / x/5 


( v/3 + jV 5 )" 2 


-2w 2 


aed 


33 




2 


-v/3-i^S 


(-v/3 + iV5)w 


-2w 


bee 


34 




2w 


/D , /- 

~ V " ~ * V " 


(- N /3 + tV5)w 


-2 


bee 


35 




SM 


V3-V5 


( V3 + W5) 


-2 


cde 


36 




2 


J3-IJ5 


( ^/a+tvsjo, 


-2o, 


ced 


37 




2 


i( 3 - ^5) w 2 


i(-3 + v /5) 


-2W 2 


adceb 


38 




-V3-W5 


+ 2w 2 


-2 


(-V3 + W5)w 2 


acbde 


39 




V3-W5 


2 


-8w 


( v/3 + W 5 ) w 


aedbc 


40 




2 


i( 3-^/5) 


i(-3 + x /5)w 


-iH 


abecd 


41 




2 


i( 3-V5)w 


&lt;&lt;-8+^/6) 


-2w 


aedcb 


42 




""* /v ^ \&gt; 


2w 


-2 


(-V3 + W5)" 


adbec 


43 




v^o 1&gt; w & 


2 


- 2w 2 


( VS + i^w 2 


acebd 


44 




2 


i( 3-V5) 


i (-3+^5) co 2 


-2w 2 


abcde 


45 




2 


* ( 3 - v/5) w 2 


^(-S + Vo)^ 2 


-9w 


adebc 


46 




&lt;V Z (^&lt;^ 


2w 2 


-2w 2 


( V3 + V6)w 


aecdb 


47 




~~ M & ^ /^ " 


2w 


-2w 


(-v/3 + zV5)w 2 


abdce 


48 




2 


i ( 3 - ^5) w 


i(-3 + x /5) w 


2 co 2 


acbed 


49 




2 


i(-B-^ 


t( 3 + V5)o; 


-2w 2 


acdeb 


50 




jy O 1 ^^ 


2w 


-2w 


( &gt;/3 + i ^/o) w 2 


adbce 


51 




v ~ i v 


2w 2 


-2w- 


(_V3 + i\/5)w 


aecbd 


52 




2 


i ( - 3 - J5) ur 


i( 3 + x /5) W - 


-2w 


abedc 


53 




2 


i(- 3^/5) w 


M 3 + v /5) 


-2w 


aebcd 


54 




-x/3-W 5 


2w 


-2 


(-v/S + iVS)" 


abdec 


55 




x/3-^5 


2 


-2or 


( VS + WS)" 2 


acedb 


56 




2 


t ( - 3 - ^5) 


i( 3 + V5)o; 2 


-2w 2 


adcbe 


57 




2 


f(-8-Vfi) 


i( 3 + ^/5)0; 


-2o&gt; 


adecb 


58 




-s/3-i^/S 


2 


-2w 


(-x/S + i^w 


aebdc 


59 




v/S-i^S 


2w 2 


2 


( ^3 + W5)w 2 


acdbe 


60 




2 


i ( - 3 - ^5) w 2 


i( 3 + ^/5) 


-y 


abeed 



206 ON THE SCHWARZIAN DERIVATIVE 

115. We hence derive 

Homographic Transformations. The groups of 12 and 24. Pole at B. 

(An +B) -=-(&lt;?* +D) 



[745 



l 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 



1 
2 


2 

2w 



2 

2 

1 
2 

2 
1 

1 

N/3( 



3-^/5) 



- x /3 - i , 
2w 



/3 - i 



2w 2 
2W 2 



v /3( 



(-3-J5) 



3-v/5) 
3-^/5) 



1 
i( 



-2w 2 



-2 



3 + ^/5) 



i 

iJ5 



1 
2 

2 

2w 



-2W 2 



-2 
-2 

-1 
-2 
- 2 

1 
1 

N/3( 

v/3( 



1 
at 

1C 

nd 
&lt;ibc 

tld) 

ula 
adl 
acd 
adc 
bed 
Me 

ab 
cd 
ac 
bd 

ad 

be 

ale 
ndi 
a be 
neb 
acd 
adl 



116. I give also the group of 12, (abce), slightly modifying the form: viz. I 
write first ^3 + i^/o=2^2k, and therefore V3 - i V5 = 2 V2. ^ : then for a; I write X^r, 

K 

and divide the A and .8 by X: the A and 5 then contain -, and the G and D 

\ 

X & X 

contain p and assuming - = t, we have ^ = - i. For instance, in the transformation 

corresponding to abc, the Ax + B and Cx + D, 

and - 3+ 2 5 as - 2o&gt; 



become first 2&&amp;gt; 2 # - 2 V2&, and - 2 V2 T - 2o&gt;, and then (omitting also the factor 2) 

Z" "X 7 

a&gt; 2 # - V2 - and - V2 -^ a- - to, viz. when - = i, they are w-a; - i *J2 and i . i \/2 - &&amp;gt; ; that 
is, the values of A, B, C, D are &&amp;gt;- , -z V2, zV2, - w. The group is 



745] 



AND THE POLYHEDRAL FUNCTIONS. 
Group of 12. Pole at B. 



207 



1 








1 


1 


M 








1 


ace 


or 








1 


aec 


1 


-iuJ2 


iuj2 


or 


dbc 


1 


-fV2 


tV 2 


0) 


acb 


1 


-t-w^/2 


/v/2 


- w 


abe 


1 


-i*J2 


i w 2 V 2 


- ur 


aeb 


1 


-tV 2 


z V 2 


or 


bee 


1 


-iV2 


JWv/2 


- w 


bee 


1 


- iw V 2 


f* v/2 


-1 


ab . ce 


1 


- tw 2 ^/2 


?wV 2 


-1 


ae . be 


1 


-iv/ 2 


W 2 


-1 


ac . be 



117. From the Table of the Groups of 12 and 24, -form, it appears that the 
group of 12 is 



X x X x x + l x + l x-l 



i (x + i) (x i) 



xi ,v + i # t a; + t 

and if we proceed to form the product of the twelve factors s x, s -- , s + x, &c., 

vC 

we have first the three products 

1 (x-l\- /a?+l\= /a? + A 3 /a?-iV 

s 2 - ^ . s- - - ; s- + - = . s- + -- ; s- - -- :}.{ - . 
a; 2 \ + l/ w I/ \x-iJ \x + i) 

= s 4 + as- + 1 ; s 4 -I- /3s 2 + 1 ; s 4 + ys* + 1 ; 

if for shortness 

/ 1 \ ar + 6^ + 1 _ a? - Gtf + 1 
* A 7- -*. 2 2 2 



The product of the three quartic functions is 

= (s 4 + 1) 3 + (s 4 + 1) 2 s 2 (a + yS + 7) + (s 4 + 1) s 4 (7 + 7 + a/3) + s 6 . a 2 /3y ; 
and we have 



32^ (V + 1) 



- (a^ 2 - 3-S6- 8 - 33s 4 + 1) 

= 









r-* I i / * I ^A^) o/i ~t \.. u\r, w 

/rJ / /y4 _ _ 1 \ J 
*t/ I X^ J- I 

Hence the product is found to be 

= (s 12 - 33s 8 - 33s 1 + 1) - s 2 (s 4 - 1)- . 



4 (x 1 -- 33s 8 - 33s 4 +1) 

^,-J f ,w 1 \a 

w ^ wx if 



_l_ 



208 ON THE SCHWARZIAN DERIVATIVE [745 

which is 

, fs 12 - 33s 8 - 33s 4 + 1 x - SSafi - 33^ + 1 

o I &* _._ I 

1 



We thus verify that the twelve transformations x into x, into -, &c., give each of 

cc 

them a transformation of the function 

x 12 - 33s 8 - 3&c* +1 

^(o,- 4 -!) 2 
into itself. 

The system of 15 circles. Art. Nos. 118 to 127. 

118. It has been already remarked that we can from the coefficients (A, B, C, D) 
of the homographic transformation pass back to the position of the axis of rotation : 
viz. we have 

A : B : C : D = v i : \ + ifj, : \-ifj, : v i, 
and thence 

\:H .v:\= B + G : -i(B-C) : D - A :i(D + A), 
that is, 

\,l*,v = -i(B+C), - (B-G\ -i(D-A)- + 

The equations of the axis thus are 

x iy z 



B+C~B-GD-A 

and the equations of the central plane at right angles to the axis are 



119. In particular, we may find the equations of the 15 planes at right angles 
to the -axes : these are in fact the before-mentioned 15 planes, intersecting the 
sphere in great circles the projections of which are the circles in the three figures 
respectively. Taking the equation of the plane to be Lx + My + Nz = 0, it is at once 
seen that the equation of the projecting cone (vertex at the South pole) is 

N (a? + y* + z&gt; - 1 ) - 2 (z + 1) (Lx + My + Nz) = 0, 
and hence, writing z = 0, we find 

lY(a; 2 + y- - 1) - 2 (Lx 



for the equation of the circle in the plane figure. We have thus the equations of 
a system of 15 circles related to each other in the manner before referred to. 

120. Taking the -form, the equations of the 15 planes are at once found: and 
we thence obtain the equations of the 15 circles: viz. writing for shortness 



745] AND THE POLYHEDRAL FUNCTIONS. 209 

the equations are 

2 = 0, (ab . cd) fl = 0, 

a; = 0, (ac . bd) # = 0, 
y = 0, (ad .be) y = 0, 

(- 1 - Vo) a? + (- 1 + V5) y + 2^ = 0, (a& . ce) O - [(- 1 - A 



- = 0, (ac? . 6e) and similarly for the other circles. 

= 0, (ab . de) 



121. Observe that the arrangement is in sets of 3 planes, or circles, intersecting 
at right angles. One of the circles is the circle fl, =a? + y&gt; 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 

+ (V5 - 1) [y - | (V5 - 1) x\ = 0, 
fl - (V5 - 1) [y - i (V5 + 3) x\ = 0, 
O - (V5 + 3) [T/ + i (V5 - 1) *] = 0, 
li - (V5 - 1) [y - i (Vo - 1) ^] = 0, 



ft + (Vo + 3) [y + \ (Vo - 1) x] = 0, 

ft + (V5 - 1) [y + (V5 -!)] = 0, 
ft - (V5 - 1) [y + \ (V5 + 3) x-\ = 0, 
ft - (V5 + 3) [y - % (Vo - 1) x] = 0, 

O - (V5 - 1) [T/ + \ (V5 -!)] = 0, 
O + (V5 - 1) [y + i (^5 + 3) ^ = 0, 
+ (V5 + 3) [y - \ (V5 - 1) x\ = 0. 

It hence appears that 4 and 4 circles have with H = the common chords y + 1(\/5 - 1) x = 0, 
y- H\/5-l)# = respectively: and that 2 and 2 circles have with H = the common 
chords y + 1 (V 5 + 3) x = 0, y - i ( V5 + 3) x = respectively. 

c. xi. 27 



210 ON THE SCHWARZIAN DERIVATIVE [745 

122. The equations of the 12 circles are, in fact, 

fl (V5 - 1) [y i (V5 - 1) x] = 0, (V5 + 3) [y | (V-5 -!)] = 0, 
+ (V5 - 1) [y + (V5 + 3) a;] = : 
hence the radii are = V5 - 1, 2 and V 5 + 1 respectively. 

The construction of the 12 circles is as follows. Starting with a circle radius 1. 

Lay down the diameters y |(\/5 - 1)^ = (A A in the figure), and through the 
extremities of each describe 2 pairs of circles with the radii J5 - 1, \/5 + 1 respectively. 

Lay down the diameters y (^5 + 3) a? = (55 in the figure), and through the 
extremities of each describe a pair of circles with the radius 2. 

123. For the .4 -form, the equations of the fifteen planes are at once found to be 







y = o, 


ad 


.be 




a; 


+ (e + e 4 ) z = 0, 


ac 


.Id 


(e + 


e 4 )* 


+ * = 0, 


ab 


.cd 


(e 2 - 


e 3 )* 


- i (e 2 + e 3 ) y =0, 


ac 


.be 


-(e 2 + 


e 3 )^ 


+ i (e 2 - e 3 ) y + 2 (e + e 4 ) z = 0, 


ae 


.be 




- + 


i (e 2 + e 4 - e - e :! ) y+ 2z = 0, 


ab 


.ce 


( ~ 


e 4 )* 


- i (e -t- e 4 ) y =0, 


ab 


.de 


-(e + 


eV 


+ i (e - e 4 ) y + 2 (e + e 4 ) 2 = 0, 


ae 


.bd 


+ (e 2 + e 3 + 


2) a; 


i (e- e 3 ) y + 2z = 0, 


ad 


.be 


(e ~ 


e 4 )* 


+ i (e + e 4 ) y =0, 


ae 


.cd 


-(e + 


e 4 )* 


- i (e - e 4 ) y + 2 (e + e 4 ) z = Q, 


ac 


.de 


(e 2 + 3 + 


2)0 


+ i(e*-e&gt;)y+ 2^ = 0, 


ad 


.ce 


(e 2 - 


e 3 ) x 


+ i (e- + e 3 )y = 0, 


bd 


.ce 


-(e 2 + 


e 3 )* 


- i (e- - e 3 ) y.+ 2 (e + e 4 ) z = 0, 


be 


.de 




a; 


i (e 2 + e 4 - e - e 3 ) y + 2z = 0, 


be 


. cd, 



where, as before, the three planes of each set intersect at right angles. 

124. Passing to the circles, the first plane of each set gives a right line, and 
we have thus five of the circles reducing themselves to right lines inclined to the 
axis of os at angles 0, 36, 72, 108 and 144 respectively. 

The remaining 10 circles form 5 pairs, the circles of a pair having different 
radii, but the two radii being the same for each pair, and so that for the several 
pairs the common chords with the circle fl = 0, are the diameters inclined to the 
axis of x at the angles 18, 54, 90, 126 and 162 respectively. Considering the 
two circles for which the inclination is 90, these arise from the planes x + (e + e 4 )2 = 0, 
= respectively. The equations of the circles thus are (e + e 4 )O+ 2# = 0, 



745] AND THE POLYHEDRAL FUNCTIONS. 211 

2 

ft 2 (e 4- e 4 ) x = 0, or recollecting that 2 (e + e 4 ) = \/5 1 and therefore ^ = V5 4- 1, 

the equations are 

# 2 + ?/ 2 - (V5 - 1) # - 1 = 0, a 2 + ?/ 2 + (V-5 + 1) # = ; 



hence for the first circle the ^-coordinate of the centre is (\/5 1) and the radius is 
= | V(10 2 V-5) 5 for the second circle the ^-coordinate of the centre is = -|-(v 5 + l), 
and the radius = V(10 + 2 \/5). We have thus the construction of these two circles, 
and consequently the construction of all the 12 circles. 

125. For the B- form, after some easy reductions and attending to the relation 
co &&amp;gt; 2 = i\/3, the equations of the 15 planes become 



x =0, 


ac . fca 7 


(- 3 + V5) y + 20 = 0, 


aZ&gt;. cd 


(3 + \/5)y + 20 = 0, 


ad. be 


V3# + \/5 2/ + 20 = 0, 


ac . be 


- (1 + \/5) \/3a? + ( 3 - V5) 2/ + 40 = 0, 


ab . ce 


/ -| | / *" \ /O i / Q / K\ A f\ 


ae . be 


a;+ V3 y =0, 


ae . bd 


-V3tf+ 7/ + (3 + V5) ^ = 0, 


ad. be 


V 3*- 2/ + (3-V5) ^ = 0, 


ab . de 


\/3a; + \/5 y + 20 = 0, 


ac . de 


(1 /\/5) V3^ + ( 3 \/5) y + 40 = 0, 


ad. ce 


(1 + V5) \/3# + (3 \/5) y + 40 = 0, 


ae . cd 


- V3 y =0, 


bd . ce 


V3* + 7/ + (3 + Vo) = 0, 


be . de 


_ ^/3^ _ y + (3 _ ^5) z = o, 


be . cd. 



126. Of the 15 circles, 3 are the lines ac y /v/3 = 0, a?=0, # + 2/\/3 = 0, viz. 
these are lines at inclinations 30, 90, 150 to the axis of oc. The equations of the 
remaining 12 circles are 

ft + (3 - V5) y = 0, 

H- (3 + V5)y = 0, 

(3 + V5) H - 2 (y - x V3) = 0, 



(3 + V5) H - 2 (y + x V3) = 0, 

(3 - V5) ft 4- 2 (y + a V3) = 0, 

272 



212 



ON THE SCHWARZIAN DERIVATIVE 



[745 



viz. these are pairs of circles having, for their common chords with O = 0, the diameters 
at inclinations 0, 60, 120 respectively. And, lastly, we have the circles 

2fi - [(- 1 + V 5) V3a? - (3 + V5) y} = 0, 



5 y] = 0, 



127. The first three of these have, for common chords with H = 0, the diameters 
whose equations are 



viz. these equations are y = ( 2 + \/5) # V3&gt; y = 



= (2 + V5)#\/3. If, as in a 



/O /K /O 

foregoing table, = 37 46 , sin 0=^.^, cos0 = ^^, and therefore tan#= ; then the 

2i y 2i A y L y5 

inclinations of these diameters to the axis of x are respectively 60 9, 6 and 
120 -0, or say 30 -(0-30), 30 + (0-30) and 90 -(0-30), where 0-30 = 7 46 , 
i.e. the inclinations are 30 + 7 46 and 90 - 7 46 . And for the other three circles 
the common chords are the diameters at the same inclinations taken negatively. The 
geometrical construction of the fifteen circles for the .B-case in question is thus not 
so simple as in the @- and A -cases. 



The Regular Polyhedra as Solid figures. Art. Nos. 128 to 134. 

128. I annex some results relating to the polyhedra considered as solid figures 
bounded by plane faces ; or say results relating to the regular solids : s is in each 
case taken for the length of the edge of the solid. 



Tetrahedron. 



Cube. 



Octahedron. 



Dodecahedron. Icosahedron. 



Edge 

Bad. of circum. sphere, R 

Bad. of inters, sphere, p 
Bad. of inscribed sphere, r 
Bad. of circle circum. to face, R 

Bad. of circle inscribed to face, r 
Incl. of adjacent faces 
Incl. of edge to adjacent face 



s 


s 


s 


s 


s 


v/3 


S . W 3 


1 


^3(^/5 + 1) 


fo-\- &gt;Jo 


2v/2 


4 


S V 8- 


1 


1 


s . \ 


3 + v/o 


l + N/5 


"2^2 


8 v/2 


4 


4 


1 


,.i 


1 


/25 + 11^/5 


3 + v/o 


S 2 J2 /3 


S v/2v/3 


V 40 


4J3 


1 


jl 


1 


/5 + J5 

s V io" 


1 


1 

8 2^3 


,.i 


1 


75 + 2^/5 
8 V 20 


1 


cos- 1 = 70 28 


90 


cos- 1 -^ = 109 32 






cos" 1 -4 = 54 46 


90 


cos" 1 - 3 = 125 44 

v rf 







But we require further data in the cases of the dodecahedron and the icosahedron 
respectively. 



745] 



AND THE POLYHEDEAL FUNCTIONS. 



213 



129. For the dodecahedron, taking the edge to be =s as before, then in the 
pentagonal face 

diagonal, g is = s . (V5 + 1), 

altitude, k = s . \ \/(o + 2 
segments of do., e = s . 



where 



130. The section through a pair of opposite edges is a hexagon, as shown in 
the figure, viz. this is constructed by taking the four equal distances 0, = p, 
= s . | (3 + V5), meeting at right angles in ; then drawing the double ordinates BB, 
each =s, through 1 and 3 respectively, and joining their extremities with 2 and 
@ 4 : the sides 2 B and 4 B are then each = k, = s . | V(5 + 2 V5) ; and inserting 
upon them the points A, 4&gt; from the figure of the pentagon, we have several 





geometrical relations; viz. the line A A cuts the parallel sides 50 2 , B 4 at right 

angles, and when produced passes through the intersection of B 1 and 5 4 : we have 

OA, OB, 0@ = r, R, p respectively: the four points 3&gt; form a square, the side of 
which is g, = s . \ (V5 + 1). 



214 ON THE SCHWARZIAN DERIVATIVE 

131. We find also 



[745 



AM=s 



10 

5 + 2 V5 



/25_+JlV5 
S \/ 5 

25 + 11V5 



03f_. /ZV5, 



/2 ( 
"V" 



2 (5 + 2 V5) 



It may be remarked that in the figure B 2 , B* are the projections of pentagonal 
faces, at right angles to the plane of the paper, having their centres at the points 
A, A, and the perpendicular distance between them =AA: the points Q, Q (only 
one of them shown in the figure) determine the directions of the 5 + 5 sides which 
abut on these pentagonal faces respectively ; and the 5 + 5 points B which are the 
other extremities of these sides respectively form two pentagons, centres M, M in the 
planes MB and MB respectively : the remaining 10 sides of the dodecahedron are the 
skew decagon obtained by joining in order these 10 points B. We have thus the 
means of making the perspective delineation of the dodecahedron. 

132. The dodecahedron is built up from the cube, by placing on each face a 
figure of two triangular and two quadrangular faces, the orthogonal projection of 
which on the face of the cube is as in the figure : the side of the square is g&gt; 




= s.(V5 + l): the slope-breadths of the triangular faces are e, = s. \ V(10- 2 \/5) 
and those of the quadrangular faces are /, = s . \ V(10 + 2 V^) 5 the lines represented 
by the other lines of the figure are in actual length each = s. We have thus a 



AND THE POLYHEDRAL FUNCTIONS. 



215 



745] 

section which is an isosceles triangle, base = g, other sides each =/; and the square 
of the altitude is thus =/ 2 - i# 2 = is 2 &gt; or tne altitude =^s; viz. the altitude of the 
ridge-line BB, above the face of the cube is =%s, the half-side of the dodecahedron. 





We have in this result the most simple means of forming the perspective delineation 
of the dodecahedron. 

133. For the icosahedron the section through two opposite edges is a hexagon, 
as shown in the figure (p. 216): to construct it, we take the four distances each 
p = s ,i(l + ^o) meeting at right angles; and then the distances A. 2 , A 4 each 
= s; and complete the hexagon. This gives the sides A 1} A 3 each =s.^V3, the 
altitude of the triangular face, side =s; and then, taking (Dj-B one-third of this, 

= s , we have OB at right angles to A lt and OA, OB, = R, r, p respectively. 
2 \/o 



Moreover, joining 
a point M : we find 



., and OA.,, we have these lines cutting at right angles in 



4(T J = * A * M &gt; 

5_-V5 
"10 



134. It may be remarked that A^.,, A^ are the projections of two pentagons 
in planes perpendicular to that of the paper, their centres being M, M: producing 
OM, OM to the points A. 2 , A 4 respectively, we have a pentagonal pyramid, summit 
A.,, standing on the first pentagon, and an opposite pyramid, summit A 4 , standing on 



216 THE SCHWARZIAN DERIVATIVE AND THE POLYHEDRAL FUNCTIONS. [745 

the other pentagon : the 5 + 5 triangular faces of the two pyramids are ten of the 
faces of the icosahedron, and the remaining ten faces are the triangles each having 
for its base a side of the one pentagon, and for its vertex a summit of the other 




pentagon, viz. the sides are the sides of the skew decagon obtained by joining in 
order the angular points of the two pentagons. We have thus a convenient method 
of forming the perspective delineation of the icosahedron. 



746] 217 



746. 

HIGHEK PLANE CURVES. 

[From Salmons Higher Plane Curves, (3rd ed., 1879) ; see the Preface.] 



ONE chapter and a large number of articles, in the second edition of Salmon s Higher Plane Curves, 
are due to Professor Cayley. Full reference to these is given by Dr Salmon in the preface. 



c. xi. 28 



218 [747 



747. 



NOTE ON THE DEGENERATE FORMS OF CURVES. 



[From Salmon s Higher Plane Curves, (3rd ed., 1879), pp. 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 xy = ; and reciprocally a point-pair can be represented in line- 
coordinates by an equation ^ = 0, but we have to consider how the point-pair can be 
represented in point-coordinates : an equation a? = 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 
u conic, or say as an indefinitely flat conic ; we have to consider conies certain of the 
coefficients whereof are infinitesimals, and which, when the infinitesimal coefficients 
actually vanish, reduce themselves to two-fold lines ; and it is, moreover, necessary to 
consider the evanescent coefficients as infinitesimals of different orders. Thus consider 
the conies which pass through two given points, and touch two given lines (four con 
ditions) ; take y = 0, z=0 for the given lines, # = for the line joining the given 
points, and (x = 0, y az Q), (x = 0, y fiz = Q) for the given points ; the equation of 
a conic satisfying the required conditions and containing one arbitrary parameter 6, is 

n -(y- OLZ) (y - #z) = ; 



747] NOTE ON THE DEGENERATE FORMS OF CURVES. 219 

or, what is the same thing, 

[x + By + 6 VO/3) z} - s + ) yz = ; 



and this equation, considering therein as an infinitesimal, say of the first order, 
represents the flat conic or point-pair composed of the two given points. Comparing 
with the general equation 

(a, b, c, f, g, h^x, y, zf = 0, 
we have 

ft = l, b = 6-, c = 6 2 a/3, /=-0 2 ( a +/3), &lt;7 = 0V(a), h=9, 

viz. a being taken to be finite, we have g and h infinitesimals of the first order ; 
b, c, f infinitesimals of the second order ; and the four ratios V(&) V(c) */(f) ff h 
are so determined as to satisfy the prescribed conditions. 

Observe that the flat conic, considered as a conic passing through the two given 
points and touching the two given lines, is represented by a detwminate equation, 
viz. considering the condition imposed upon 6 (6 = infinitesimal) as a determination of 
6, the equation is a completely determinate one ; but considering the flat conic merely 
as a conic passing through the two given points, the equation would contain two 
arbitrary parameters, determinable if the flat conic was subjected to the condition of 
touching two given lines, or to any other two conditions. 

Generally, we may consider the equation of a curve of the order n; such equation 
containing certain infinitesimal coefficients and, when these vanish, reducing itself to 
a composite equation P a 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 = 0, Q = 0, &c. respectively ; (2) the lines through the singular points of these same 
curves respectively ; (3) the lines through the points of intersection P = 0, Q = 0, &c. 
of each two of the component curves; these points, each reckoned a proper number 
of times, are called " fixed summits " ; (4) the lines from the fixed point to certain 
determinate points called " free summits " on the several component curves P = 0, 
Q = 0, &c. respectively. We have thus a degenerate form of the ?i-thic curve, which 
may be regarded as consisting of the component curves, each its proper number of 
times, and of the foregoing points called summits, and is consequently only inadequately 
represented by the ultimate equation P*Q? . . . = ; the number and distribution of the 
summits is not arbitrary, but is regulated by laws arising from the consideration of 
the penultimate curve, and there are of course for any given value of n various forms 
of degenerate curve, according to the different ultimate forms P*Q? . . . = 0, and to the 
number and distribution of the summits on the different component curves. The case 
of a quartic curve having the ultimate form a?y- = has been considered by Cayley, 
Comptes Eendus, 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 x-y 2 = 0, with nine free summits, 
three of them on one of the lines (say the line y = 0), and which are three of the 
intersections of the quartic by this line (the fourth intersection being indefinitely near 
to the point ao = 0, y = 0), six situate at pleasure on the other line x = ; and three 
fixed summits at the intersection of the two lines." Other forms have been con 
sidered by Dr Zeuthen, Comptes Rendus, t. LXXV. pp. 703 and 950 (September and 
October, 1872), and some other forms by Zeuthen ; the whole question of the degenerate 
forms of curves is one well deserving further investigation. 

The question of the number of cubic curves satisfying given elementary conditions 
(depending as it does on the consideration of the degenerate forms of these curves) 
has been solved by Maillard and Zeuthen ; that of the number of quartic curves has 
been solved by Dr Zeuthen. 



748] 



221 



748. 

ON THE BITANGENTS OF A QUARTIC. 

[From Salmons Higher Plane Curves, (3rd ed., 1879), pp. 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, 187G, pp. 456472 ; and see also Weber s Theorie 
der Abel schen Functionen vom Geschlecht 3," Berlin, 1876. Riemann connects the 
several bitangents with the characteristics of the 28 odd functions, thus obtaining for 
them an algorithm which it is worth while to explain, but they will be given also 
with the algorithm employed p. 231 et seq. of the present work*, which is in fact the 
more simple one. The characteristic of a triple ^-function is a symbol of the form 

a/3% 
a /3Y, 

where each of the letters is = or 1 ; there are thus in all 64 such symbols, but they 
are considered as odd or even according as the sum era + ft ft + 77 is odd or even; 
and the numbers of the odd and even characteristics are 28 and 36 respectively; and, 
as already mentioned, the 28 odd characteristics correspond to the 28 bitangents 
respectively. 

We have x, y, z trilinear coordinates, a, @, 7, a , /3 , 7 constants chosen at pleasure, 
and then a", /3", 7" determinate constants, such that the equations 

x+ y+ 2+ % + 77 + =0, 



a ^j 7 
[* That is, Salmon s Higher Plane Curves.] 



222 



ON THE BITANGENTS OF A QUARTIC. 



[748 



are equivalent to three independent equations; this being so, they determine f, 77, , 
each of them as a linear function of (as, y, z) ; and the equations of the bitangents of 
the curve V(#) + VCy?) + *J( Z %) = (see Weber, p. 100) are 



18 


111 
111 


-0, 


28 


001 
Oil 


y = o, 


38 


Oil 

001 


. = 0, 


23 


010 
010 


f-o, . 


13 


100 
110 


77 = 0, 


12 


110 

100 


r-o, 


48 


101 
100 


x + y + z = ; 


14 


010 
Oil 


f+y + * = o, 


58 


100 
101 


OLX + /3y +yz = 0, 


15 


Oil 
010 


+ ySy + y 2 = 0&gt; 


68 


110 
010 


OLX + yS y + y z == 0, 


16 


001 
101 




, + /3 ?/ + 7 z = i 
a 


78 


010 
110 


&lt;W,9-, + /,-&lt; 


17 


101 
001 


4, + /3"j/ + y"z = I 


24 


100 

111 


a; +77 + z = 0, 


34 


110 
101 


* + y + ?=o, 


25 


101 
110 


ra + | + 7 , = o, 


35 


111 

100 


cw? + ySy + - = 0, 
7 



748] 



ON THE BITANGENTS OF A QUARTIC. 



223 



26 


111 

001 


*+! +y,-o, 


36 


101 
Oil 


a! x + ft y + -, =0, 
7 


27 


Oil 
101 


***- 


37 


001 

111 




av 4- fi"ti -1- - 
* i A-* y &gt; it "i 

7 


67 


100 
100 


* y * 


1 - /8y 1 - 7 l-a/8 


57 


110 
Oil 


\ y \ . . o 


1 -/8 y 1 - 7 V + 1 -a /3 ~ 


56 


010 

111 


x y z 


1-^V l-y V 1-a ^"" 


45 


001 
001 


fc t 

1 ( 


a (1 /3 7 ) /8 (1 7 a) 7 (1 a/3) 


46 


Oil 
110 


* ^ 

S ^ b 


a (1 - /3 7 ) yS (1 - 7 a ) y 7 (1 - a /3 ) 


47 


111 

010 


fc v 

H 1 





The whole number of ways in which the equation of the curve can be expressed 
in a form such as V(#) + V(2/^) + V(^) = is 1260; viz. the three pairs of bitangents 
entering into the equation of the curve are of one of the types 



12 . 34, 13 . 24, 14 . 23 IS 
12.34, 13.24, 56.78 Q \ 
13.23, 14.24, 15.25 



No. is 70 

630 

560 

1260. 



It may be remarked that, selecting at pleasure any two pairs out of a system of 
three pairs, the type is always D or \\ , viz. (see p. 233) the four bitangents are such 
that their points of contact are situate on a conic. 



224 [749 



749. 

SOLID GEOMETRY. 



[From Salmon s Treatise on the analytic geometry of three dimensions, (3rd ed., 1874) ; 

see the Preface.] 

A considerable number of articles in the third edition of Salmon s Treatise are due to Professor Cayley. 
Full reference to these is given by Dr Salmon in the preface. 



750] 225 



750. 

ON THE THEOEY OF RECIPROCAL SURFACES. 

[From Salmons Treatise on the analytic geometry of three dimensions, (3rd ed., 1874), 

pp. 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. 

8, number of nodal edges of the cone. 

K, number of its cuspidal edges. 

p, class of nodal torse. 

ff, class of cuspidal torse. 

6, order of nodal curve. 

k, number of its apparent double points. 

f, number of its actual double points. 

t, number of its triple points. 

; , number of its pinch-points. 

q, its class. 

c, order of cuspidal curve. 

/;, number of its apparent double points. 

0, number of its points of an unexplained singularity. 

%, number of its close-points. 
c. XL 29 



226 ON THE THEORY OF RECIPROCAL SURFACES. [750 

G&gt;, number of its off-points. 
r, its class. 

/3, number of intersections of nodal and cuspidal curves, stationary points on 
cuspidal curve. 

7, number of intersections, stationary points on nodal curve. 
i, number of intersections, not stationary points on either curve. 
C, number of cnicnodes of surface. 
B, number of binodes. 

And corresponding reciprocally to these : 
n , class of surface. 

a , class of section by arbitrary plane. 
S , number of double tangents of section. 
K, number of its inflexions. 
p, order of node-couple curve. 
o- , order of spinode curve. 
b , class of node-couple torse. 
k , number of its apparent double planes. 
/ , number of its actual double planes. 
t , number of its triple planes. 
j , number of its pinch-planes. 
q , its order. 

c, class of spinode torse. 
ti, number of its apparent double planes. 
&, number of its planes of a certain unexplained singularity. 
%, number of its close-planes, 
a) , number of its off-planes. 
r , its order. 

/3 , number of common planes of node-couple and spinode torse, stationary planes 
of spinode torse. 

7 , number of common planes, stationary planes of node-couple torse. 
t, number of common planes, not stationary planes of either torse. 
G , number of cnictropes of surface. 
B , number of its bitropes. 

In all, these are 46 quantities. 



750] ON THE THEORY OF RECIPROCAL SURFACES. 227 

601. In part explanation, observe that the definitions of p and cr agree with 
those already given. The nodal torse is the torse enveloped by the tangent planes 
along the nodal curve ; if the nodal curve meets the curve of contact a, then a 
tangent plane of the nodal torse passes through the arbitrary point, that is, p will 
be the number of these planes which pass through the arbitrary point, viz. the class 
of the torse. So also the cuspidal torse is the torse enveloped by the tangent planes 
along the cuspidal curve ; and a- will be the number of these tangent planes which 
pass through the arbitrary point, viz. it will be the class of the torse. Again, as 
regards p and cr : the node-couple torse is the envelope of the bitangent planes of 
the surface, and the node-couple curve is the locus of the points of contact of these 
planes. Similarly, the spinode torse is the envelope of the parabolic planes of the 
surface, and the spinode curve is the locus of the points of contact of these planes, 
viz. it is the curve UH of intersection of the surface and its Hessian ; the two 
curves are the reciprocals of the nodal and the cuspidal torses respectively, and the 
definitions of p, a correspond to those of p and or. 

602. In regard to the nodal curve b, we consider k the number of its apparent 
double points (excluding actual double points) ; / the number of its actual double points 
(each of these is a point of contact of two sheets of the surface, and there is thus at 
the point a single tangent plane, viz. this is a plane f, and we thus have / =/) ; 
t the number of its triple points; and j the number of its pinch-points these last 
are not singular points of the nodal curve per se, but are singular in regard to the 
curve as nodal/ curve of the surface ; viz. a pinch-point is a point at which the two 
tangent planes are coincident. The curve is considered as not having any stationary 
points other than the points 7, which lie also on the cuspidal curve ; and the 
expression for the class consequently is q = b 2 b 2k 2f 87 6t. 

603. In regard to the cuspidal curve c, we consider h the number of its apparent 
double points; and upon the curve, not singular points in regard to the curve per se, 
but only in regard to it as cuspidal curve of the surface, certain points in number 
6, %, w respectively. The curve is considered as not having any actual double or other 
multiple points, and as not having any stationary points except the points /3, which 
lie also on the nodal curve ; and the expression for the class consequently is 

r = c 2 - c - 2h - 3/3. 

604. The points 7 are points where the cuspidal curve with the two sheets (or 
say rather half-sheets) belonging to it are intersected by another sheet of the surface ; 
the curve of intersection with such other sheet, belonging to the nodal curve of the 
surface, has evidently a stationary (cuspidal) point at the point of intersection. 

As to the points /3, to facilitate the conception, imagine the cuspidal curve to be 
a semi-cubical parabola, and the nodal curve a right line (not in the plane of the 
curve) passing through the cusp ; then intersecting the two curves by a series of 
parallel planes, any plane which is, say, above the cusp, meets the parabola in two 
real points and the line in one real point, and the section of the surface is a curve 
with two real cusps and a real node ; as the plane approaches the cusp, these approach 

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 i the nodal curve crosses the cuspidal curve, being on the side 
away from the two half-sheets of the surface acnodal, and on the side of the two 
half-sheets crunodal, viz. the two half-sheets intersect each other along this portion of 
the nodal curve. There is at the point a single tangent plane, which is a plane i! \ and 
we thus have i = i . 

606. As already mentioned, a cnicnode C is a point where, instead of a tangent 
plane, we have a tangent quadri-cone ; at a binode B, the quadri-cone degenerates into 
a pair of planes. A cnictrope C is a plane touching the surface along a conic; in 
the case of a bitrope B , the conic degenerates into a flat conic or pair of points. 

607. In the original formulas for a (n - 2), b(n 2), c (n 2), we have to write 
K B instead of K, and the formulae are further modified by reason of the singularities 
6 and w. So, in the original formulas, for a (n 2) (n 3), b (n 2) (n 3), c (n 2)(n 3), 
we have instead of 8 to write S C 3&&amp;gt;, and to substitute new expressions for 
[ab], [ac], [be] ; viz. these are 

[ab] = ab-2p -j, 

[ac] = ac 3&lt;r ^ &&amp;gt;, 
[be] =bc- 3/3 - 27 - i. 

The whole series of equations thus is 

(1) a = a. 

(2) /=/ 

(3) i =i. 

(4) a = n (n - 1) - 26 - 3c. 

(5) K = 3n (n - 2) - Qb - 8c. 

(6) 8 = \n (n - 2) (?i 2 - 9) - (n 2 - n - 6) (26 + 3c) + 26 (6 - 1) + 6bc + c(c- 1). 

(7) a (n - 2) = K - B + p + 2&lt;r + 3&&amp;gt;. 

(8) 6(n-2)= p + 2t3 + 3y + 3t. 

(9) c (n - 2) = 2&lt;r + 4/3 + 7 + + a). 

(10) a(-2)(ft-3)-2(a-0-3) + 3(a6~&r-x-8)+2(a6-2p -j). 

(11) 6(w-2)(w-3) = 4& + (ab-2p-j ) + 3(6c -3/8 - 2 7 -i). 

(12) c(w-2)(n-3) = 6A + (ac-3o--x 

(13) q = b*-b-2k- 2/- 87 - 6i. 

(14) r = c 2 - c - 2/i - 3/3. 



750] ON THE THEORY OF RECIPROCAL SURFACES. 229 

Also, reciprocal to these, 

(15) a = n (n - 1) - 26 - 3c . 

(16) K = 3 (n - 2) - 66 - Sc. 

(17) a = in (/* - 2) (n 2 - 9) - (n 2 - n - 6) (26 + 3c ) + 26 (6 - 1) + 66V + f c (c - 1). 

(18) a ( - 2) = * - 5 + p + 2o- + 3ft) . 

(19) & (n -2)= p + 2/3 + 3 7 + 3 . 

(20) c (n - 2) = 2o- + 4/3 + 7 + & + &&amp;gt; . 

(21) a (n - 2) ( - 3) = 2 (V -C - 3a&gt; ) + 3 (aV - 3o- - % - 3a&gt; ) + 2 (a 6 - 2p -/). 

(22) 6 ( - 2) (n - 3) = 4# + (a b -Zp -f ) + 3(6V-3/3 -2 7 W). 

(23) c ( - 2) ( - 3) = 6fc + (aV - So- - % - 3a&gt; ) + 2(6 c -3/3 -2 7 W). 

(24) g = 6 2 - 6 - 2A; - 2/ - 3 7 - 6t . 

(25) r = c 2 - c - 2^ - 3/8 , 

together with one other independent relation: in all 26 relations between the 46 
quantities. 

608. The new relation may be presented under several different forms, equivalent 
to each other in virtue of the foregoing 25 relations; these are 

(26) 2(w-l)(7i-2)(?i-3)-120i-3)(6 + c) + 6 ? + 6r+24 + 42/3 + 30 7 -f^ = S, 

in each of which two equations 2 is used to denote the same function of the accented 
letters that the left-hand side is of the unaccented letters. 



(28) /3 + i&lt;9 = 

+ (_ 66n + 184) 6 
+ (- 93?i + 252) c 



+ 27(4/3+ 7 + &lt;9) 

+ + W 

- 240 - 285 - 27j - 38^ - 73&lt;u 
+ 4C" + 105 + 7/ 
Or, reciprocally, 

(29) /3+|&lt;9= 2i / (w -2)(lln 
+ (- 66n + 184) 6 
+ (- 93??, + 252) c 
+ 22 (2/8 + 3 7 
+ 27(4/3 + y 



- 246" - 285 - 27j - 
+ 4(7 + 105 + 7j + 



230 



ON THE THEORY OF RECIPROCAL SURFACES. 



[750 



The equation (26) expresses that the surface and its reciprocal have the same deficiency; 
viz. the expression for the deficiency is 

(30) Deficiency = %(n- 1) (w - 2) (n - 3) - (n - 3) (b + c) + (q + r) + 2t +l/3+f 7 +;_i0, 
= (n - 1) (n - 2) (n - 3) - &c. 

609. The equation (28) (due to Prof. Cayley) is the correct form of an expression 
for /3 , first obtained by him (with some errors in the numerical coefficients) from 
independent considerations. But it is best obtained by means of the equation (26) : 
and (27) is a relation presenting itself in the investigation. In fact, considering as 
standing for its value n (n 1) 26 3c, we have from the first 25 equations 



6 

+ 2 

- 2 

- 4 
-6 

+ 2 

-3 

_ 2 



a 

3w - c - K 

a (n - 4 2) - K + B - p - 2o- - 3&&amp;gt; 

6(w-2)-p-2/3-37-3* 

c (n - 2) - 2&lt;r - 4 - 7 - - w 



= 2, 
= 2, 

= 2, 
=2, 

= 2, 



+ K - a- - 2(7 - 4&gt;B - 2j - 3^ - 3a&gt; = 2, 

=2, 
&&amp;gt; =2; 



multiplying these equations by the numbers set opposite to them respectively, and 
adding, we find 



- 2?i 3 + 12w 2 + 4w + b (Un - 36) + c (12n - 48) 

-Qq-Qr- 46 - 105 - 4,1/3 - 30 7 - 24* - 7j - 8* + 20 - 4a&gt; = 2, 

and adding hereto (26) we have the equation (27); and from this (28), or by a like 
process, (29), is obtained without much difficulty. As to the 8 2-equations or symmetries, 
observe that the first, third, fourth, and fifth are in fact included among the original 
equations (for an expression which vanishes is in fact = 2) ; we have from them 
moreover 3n - c = 3a - K, and thence 3?i - c - K = 3a - K - K, which is = 2, or we have 
thus the second equation; but the sixth, seventh, and eighth equations have yet to 
be obtained. 

610. The equations (15), (16), (17) give 

n = a (a -1)- 28 -3*, 

c = 3a (a - 2) - 68 - SK, 



From (7), (8), (9), we have 

(a- b- c)(n-2) =K-B-Q/3-4ry-3t-0 + 2(o, 

(a - 2b -3c)(n-2)(- 3) = 2(8 -C)- 8k- I8h - Gbc + 180 + I2y + 6i - fa ; 



750] ON THE THEORY OF RECIPROCAL SURFACES. 231 

substituting these values for K and 8, and for a its value = n (n 1) 26 3e, we 
obtain the values of n, c, b ; viz. the value of n is 

n - n (n - I) 2 - n (76 + 12c) + 46 2 + 8b + 9c 2 + 15c 

- 8& - ISA + 18/3 + 12? + 12i - 9 

- 2(7 - 35 - 30. 

Observe that the effect of a cnicnode C is to reduce the class by 2, and that of a 
binode B to reduce it by 3. 

611. We have 

(n - 2) (n - 3) = n 2 - n + (- 4&gt;n + b ) = a + 26 + 3c + (- 4n + 6) ; 

making this substitution in the equations (10), (11), (12), which contain (n - 2) (n 3), 
these become 

a (- 4&gt;n + 6) = 2 (8 - C) - a 2 - 4p - 9&lt;r - 2j - 3 X - low, 

6 (- 4w + 6) = 4& - 26 2 - 9/8 - 67 - 3i - 2p - j, 

c (- 4n + 6) = Qh - 3c" - 6/3 - 4&gt;y - 2i - So- - % - 3&&amp;gt;, 

which are the foregoing equations (6 ); adding to each equation four times the corre 
sponding equation with the factor (n 2), these become 

a 2 - 2a = 2 (8 - C) + 4 (K - B) - a - 2j - 3* - 3a&gt;, 
26 2 - 26 = 4& - /3 + 67 + 12 - 3i + 2p - j, 
3c 2 - 2c = Qh + 10/9 + 40 - Zi + 5a - x + a,. 

Writing in the first of these a 2 - 2a = n + 28 + 3/c a, and reducing the other two by 
means of the values of q, r, the equations become 

n - a = - 2(7- 



3r + c + 2i + ^ = oo- + /3 + 40 + w, 

which give at once the last three of the 8 2-equations. 
The reciprocal of the first of these is 

&lt;r = a-n + K r - 2j - 3% - 2C" - 4F - 3w , 
viz. wiiting herein 

a = n(n-l)-2b-3c and * = 3n(n - 2) - 66 -8c, 
this is 

o- = 4 (w - 2) - 86 - 1 Ic - 2j - 3 X - 2C" - 4&gt;B - 3o&gt; , 

giving the order of the spinode curve; viz. for a surface of the order n without 
singularities, this is =4n(w-2), the product of the orders of the surface and its 
Hessian. 



232 ON THE THEORY OF RECIPROCAL SURFACES. [750 

612. Instead of obtaining the second and third equations as above, we may to 
the value of 6 ( 4m + 6) add twice the value of 6 (n 2) ; and to twice the value of 
c ( 4&gt;n + 6) add three times the value of c (n 2), thus obtaining equations free from 
p and a respectively; these equations are 

b (- 2n + 2) = 4,k - 26 2 - 5 - 3i + 6t -j, 

c (- 5n + 6) = 12A - 6c- - 5y - 4t - 2 X + 30 - 3a&gt;, 

equations which, introducing therein the values of q and r, may also be written 

6(2n- 4) =2q+ 5/3 + 67 + Qt + 3i+j + 4/, 

c (5n - 12) + 30 = 6r + 18/3 + 5y + 4i + 2 X + 3o&gt;. 

Considering as given, ?& the order of the surface ; the nodal curve, with its singularities 
b, k, f, t\ the cuspidal curve, with its singularities c, h ; and the quantities /3, y, i 
which relate to the intersections of the nodal and cuspidal curves; the first of the 
two equations gives j, the number of pinch-points, being singularities of the nodal 
curve, quoad the surface ; and the second equation establishes a relation between 
0, %, &lt;u, the numbers of singular points of the cuspidal curve quoad the surface. 

In the case of a nodal curve only, if this be a complete intersection P = 0, Q = 0, 
the equation of the surface is (A, B, C%P, Q) 2 = 0, and the first equation is 

&(-2?i + 2) = 4&-2& 2 + Gt-j; 

or, assuming 2=0, say j= 2 (n 1)6 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, CQP, Q)- = 0, where A C - B 2 = MP + NQ ; and the 
second equation is 

c (- on + 6) = 12/i - 6c 2 - 2% + 3(9 - 3o&gt;, 

or, say 2^ + 3&lt;u = (5n 6)c-6c 2 + 12A + 30, which may also be verified. 

613. We may in the first instance out of the 46 quantities consider as given 
the 14 quantities 

: b, k, f,t : c, h, 0, X : 0, 7, i : C, B, 

then of the 26 relations, ]7 determine the 17 quantities 

a, 8, K, p, a- : j, q : r, a&gt; 

n : a , B , K : b ,f : c : i 

and there remain the 9 equations 

(18), (19), (20), (21), (22), (23), (24), (25), (28), 
connecting the 15 quantities 

p , a : k , t , f, q : h , , % , , / : , 7 : C", B . 



750] ON THE THEORY OF RECIPROCAL SURFACES. 233 

Taking then further as given the 5 quantities j , % , w, C , B , 

equations (18) and (21) give p, a- , 
equation (19) gives 2/3 + 87 + 3 ; 

(20) ,, 4/3 + 7+0 , 

(28) ff + W, 

so that, taking also t as given, these last three equations determine /3 , 7 , 6 ; and 
finally 

equation (22) gives k , 

(23) /, , 

(24) q, 

(25) r, 

viz. taking as given in all 20 quantities, the remaining 26 will be determined. 

614. In the case of the general surface of the order n, without singularities, we 
have as follow : 

n = n, 

a = n (n 1), 

8 = \n (n - 1) (n - 2) (n - 3), 

K = n (n 1) (n - 2), 

n = n (n I) 2 , 

a = n (n 1), 



K = Sn (n - 2), 

I = n (n - 1) (n - 2) (n 3 - ?i 2 + w - 12), 

k = in (n - 2) (n w - 6w 9 + 16w 8 - 54/i 7 + 164?i 6 - 288n 5 

+ 547w 4 - 1058n 3 + 1068n 2 - 1214?i + 1464), 

t = w (n - 2) (?i 7 - 4?i + 7?i 5 - 45?i 4 + 114n 3 - 111?? 2 + 548w - 960), 
q = n (n - 2) (re - 3) (n 2 + 2n - 4), 
p = ?^ (n 2) (?i 3 ?i 2 + n 12), 
c = 4?i (/i - 1) (?i - 2), 

A = i (n - 2) (16?i 4 - 64w 3 + 80w 2 - 108?z + 156), 
r = 2n (?i - 2) (3w - 4), 
a- = 4i (w - 2), 



7 = 4?z (71 - 2) (n - 3) (7i 3 - 3n + 16), 
the remaining quantities vanishing. 

c. XL 30 



234 ON THE THEORY OF RECIPROCAL SURFACES. [750 

615. The question of singularities has been considered under a more general 
point of view by Zeuthen, in the memoir " Recherche des singularity qui ont rapport 
a une droite multiple d une surface," Math. Annalen, t. iv. (1871), pp. 120. He 
attributes to the surface : 

A number of singular points, viz. points at any one of which the tangents form 
a cone of the order fi, and class v, with y + 77 double lines, of which y are tangents 
to branches of the nodal curve through the point, and z + stationary lines, whereof 
z are tangents to branches of the cuspidal curve through the point, and with u double 
planes and v stationary planes; moreover, these points have only the properties which 
are the most general in the case of a surface regarded as a locus of points; and 2 
denotes a sum extending to all such points. (The foregoing general definition includes 
the cnicnodes /* = i; = 2, y = rj = z = ^=ii = v = Q, and the binodes /* = 2, 77 = 1, 
i/ = y = &c. = 0.) 

And, further, a number of singular planes, viz. planes any one of which touches 
along a curve of the class p and order v, with y + r\ double tangents, of which y 
are generating lines of the node-couple torse, / + f stationary tangents, of which z 
are generating lines of the spinode torse, u double points and v cusps; it is, more 
over, supposed that these planes have only the properties which are the most general 
in the case of a surface regarded as an envelope of its tangent planes; and 2 denotes 
a sum extending to all such planes. (The definition includes the cnictropes /// = v = 2, 
y = ,/ = / = f = u = v = 0, and the bitropes // = 2, if = 1, v = y = &c. = 0.) 

616. This being so, and writing 

= v + 2i7 + 3 x = v + 27; + 3f, 

the equations (7), (8), (9), (10), (11), (12), contain, in respect of the new singularities 
additional terms, viz. these are 

a (n - 2) = ... + 2 [x (UL - 2) - 17 - 2], 



a (n - 2) (n - 3) = ... + 2 [x (- 4&gt;p + 7) + 277 + 4j, 

b (n - 2) (n - 3) = ... + 2 [y (- 4y*+ 8)] - 2 (4u + 3t&gt; ), 

c (n - 2) (n - 3) = ... + 2 [2 (- 4^ + 9)] - 2 (2w ), 

and there are of course the reciprocal terms in the reciprocal equations (18). (19), 
(20), (21), (22), (23). These formulae are given without demonstration in the memoir 
just referred to: the principal object of the memoir, as shown by its title, is the 
consideration not of such singular points and planes, but of the multiple right lines 
of a surface; and in regard to these, the memoir should be consulted. 



751] 235 



751. 



NOTE ON RIEMANN S PAPER "VERSUCH EINER ALLGEMEINEN 
AUFFASSUNG DER INTEGRATION UND DIFFERENTIATION* " 

[From the Mathematische Annalen, t. xvi. (1880), pp. 81, 82.] 

THE Editors of Riemann s works remark that the paper in question was contained 
in a MS. of his student time (dated 14 Jan. 1847) and was probably never intended 
for publication: indeed that he would not in later years have recognised the validity 
of the principles upon which it is founded. The idea is however a noticeable one: 
Riemann considers z x+h , a function of tc + h, expanded in a doubly infinite, necessarily 
divergent, series of integer or fractional powers of h, according to the law 

z x+h = V ~^kJd" x z.h", (2) 

i/= oo 

where the meaning is explained to be that the exponents differ from each other by 
integer values, in effect, that v has all the values a + p, a a given integer or fractional 
value, and p any integer number from -co to + oo , zero included. 

Riemann deduces a theory of fractional differentiation: but without considering 
the question which has always appeared to me to be the great difficulty in such a 
theory: what is the real meaning of a complementary function containing an infinity 
of arbitrary constants ? or, in other words, what is the arbitrariness of the complemen 
tary function of this nature which presents itself in the theory ? 

I wish to point out the relation between the paper referred to, and a short 
paper of my own "On a doubly infinite Series," Quart. Math. Journ. t. VI. (1851), 
pp. 4547, [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 series." And I then give the expansion 



where n is any integer or fractional number whatever, and the summation extends 
to all positive and negative integer values (zero included) of r. And I remark that, 
n being an integer, we have C n = T (n), and hence that assuming that this is so in 
general, or writing 

T (n) .e x = Z r [n- l] r as n - 1 - r , 

we have this equation as a definition of F (?i). The point of resemblance of course 
is that we have a doubly infinite expansion of e* in a series of integer or fractional 
powers of x, corresponding to Riemann s like expansion of z x+h in powers of h. 

Cambridge, 10 Sept. 1879. 



752] 237 



752. 

ON THE FINITE GROUPS OF LINEAR TRANSFORMATIONS OF 
A VARIABLE; WITH A CORRECTION. 

[From the Mathematische Annalen, t. xvi. (1880), pp. 260263; 439, 440.] 

IN the paper " Ueber endliche Gruppen linearer Transformationen einer Verander- 
lichen," Math. Ann. t. xn. (1877), pp. 23 46, Prof. Gordan gave in a very elegant form 

fl IT r \ / ) 

the groups of 12, 24 and 60 homographic transformations - - ,-. The groups of 12 

and 24 are in the like form, the group of 24 thus containing as part of itself the 
group of 12 ; but the group of 60 is in a different form, not containing as part of 
itself the group of 12. It is, I think, desirable to present the group of 60 in the 
form in which it contains as part of itself Gordan s group of 12 : and moreover to 
identify the group of 60 with the group of the 60 positive permutations of 5 letters: 
or (writing abc for the cyclical permutation a into b, b into c, c into a, and so in 
other cases) say with the group of the 60 positive permutations 1, abc, ab.cd and 
abode. 

Any two forms of a group are, it is well known, connected as follows, viz. if 
1, a, /3, ... are the functional symbols of the one form, then those of the other form 

are 1, ^a^" 1 , ^fi^ 1 , ... (where in the case in question ^ is a functional symbol of 
the like homographic form, SYB = ~ n ). But instead of obtaining the new form in 

\jffs "T~ _L// 

this manner, I found it easier to use the values of the rotation-symbol 

7T 7T 

cos h sin - (i cos X + j cos Y + k cos Z) 
q q^ 

for the axes of the icosahedron or dodecahedron, given in my paper " Notes on 
polyhedra," Quart. Math. Jour. t. vn. (1866), pp. 304316, [375]; viz. if for any axes, 

\ ft, v denote the parameters of rotation tan - cos X, tan - cos F, tan - cos Z, then, 

q q q 



238 



ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE. 



[752 



by a formula which is in fact equivalent to that given in my note " On the 
correspondence of Homographies and Rotations," Math. Annalen, t. XV. (1879), 
pp. 238 240, [660], the corresponding homographic function of x is 

( v i) ac + \ + ifi 
(\ ift) x + v i 
where i denotes V 1 as usual. 

The new formulae for the group of 60, or icosahedron group, of homographic 

OLX I Q 

functions - ^ are contained in the following table, where the four columns show 

the values of the coefficients a, /3, 7, 8 respectively : and where in the outside column, 
the substitution is represented as a permutation-symbol on the five letters abcde: 
moreover for shortness is written to denote Vo. 

THE GROUP OF 60. 



1 










1 


1 








1 


1 


2 


-1 








1 


ab . cd 


3 





1 


1 





ac . bd 


4 





-1 


1 





ad . be 


5 


2 


-3+e-ri( i-e) 


-3+0+j (-1 + 6) 


-2 


be . de 


6 


2 


-s+0+i(-i+0) 


- 3 + + 1 ( 1-6) 


-2 


ae . be 


7 


2 


3-e + i(-l + 0) 


3-0 + i( 1-6) 


-2 


ad . ce 


8 


2 


3-0 + /( 1-6) 


3-0 + f (-1 + 0) 


-2 


ad, be 


9 


2 


-i-e+i( 1-0) 


-i-e+z(-i+e) 


-2 


ae . cd 


10 


2 


-l-0 + i(-l + 0) 


-l-0 + i( 1-6) 


-2 


ab . de 


11 


2 


1 + + f (-1 + 0) 


i+e+?( i-e) 


-2 


be .cd 


12 


2 


i+e+?( i-e) 


l + 0+l(-l+0) 


-2 


ab . ce 


13 


2 


-l-0 + i (-3-0) 


-l-0 + f( 3 + 6) 


-2 


ac . be 


14 


2 


-l-0 + f( 3 + 6) 


-1-6 + ? (-3-6) 


-2 


bd . ce 


15 


2 


l + + f( 3 + 6) 


1 + + f (-3-0) 


-2 


ae . bd 


16 


2 


1 + 0+ j (-5-0) 


l + 6 + z( 3 + 6) 


-2 


ac . de 


17 


i 


i 


1 


1 


a be 


18 


-1 


i 


1 


i 


acb 


19 


1 


-i 


1 


i 


adc 


20 


- i 


-i 


1 


-1 


acd 


21 


i 


i 


1 


-1 


adb 


22 


1 


i 


1 


-i 


abd 


23 


-1 


-i 


1 


-i 


bed 


24 


i 


-i 


1 


1 


bdc 



752] 



ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE. 



239 



25 


-l-0 + i( 3 + 0) 


2 


-2 


-1-0 + M-3-0) 


aec 


26 


1 + + i ( 3 + 0) 


2 


-2 


1 + + / ( - 3 - 0) 


ace 


27 


l + e + i(-3-e) 


2 


-2 


1 + + i( 3+0) 


bed 


28 


_l_0 + ; ( _3-0) 


2 


-2 


-l-e + i( 3 + 0) 


Me 


29 


-3 + + /( 1-0) 


2 


2 


3-0 + -i( 1-0) ! bee 


30 


-8+6+&lt;&lt;-l+6) 


2 


2 


3-0 + i(-l + 0) 


bee 


31 


3-0 + i(_l + 0) 


2 


2 


-3 + 0+t(-l + 0) 


aed 


32 


3-0 + i( 1-0) 


2 


2 


-3 + + t( 1-0) 


ade 


33 


2 


-l-0 + i(-l + 0) 


l + 0+i( -1 + 0) 




cde 


34 


2 


l + + f( 1-0) 


- 1 - + i ( 1-0) 




ced 


35 


2 


-i-e+M 1-0) 


l+0 + z( 1-0) 




aeb 


36 


2 


i+e+i(-i+e) 


-l-0 + z(-l + 0) 




abe 


37 


-l-0 + t(-3-0) 


2 


2 


1 + + ^-3-0) 


abode 


38 


-l-0 + i( 1-0) 


2 


2 


l + + i( 1-0) 


accbd 


39 


_i_e+z(-i+0) 


2 


2 


l + + i(-l + 0) 


adbee 


40 


-l-0 + t( 3 + 0) 


2 


2 


l + 0+i( 3 + 0) 


aedcb 


41 


1 + + i ( 3 + 0) 


2 


2 


-l-0 + t( 3 + 0) 


adceb 


42 


i +0+ /(_i + e) 


2 


2 


-l-0 + i(-l + 0) acbde 


43 


l + 0+(( 1-0) 


2 


2 


-l-0 + z( 1-0) aedbc 


44 


1 + + i ( - 3 - 0) 


2 


2 


_l_e + i(-3-0) 


abecd 


45 


-1-0 + i (-1 + 0) 


2 


-2 


-l-0 + i( 1-0) 


acbed 


46 


_3 + e + i(-l + 0) 


2 


-2 


-3+0 + i( 1-0) 


abdce 


47 


3_0 + t(-l+0) 


2 


-2 


3-0 + i( 1-0) 


aecdb 


48 


i+e+f (-1+0) 


2 


-2 


l+0 + i( 1-0) 


adebc 


49 


l + + t( 1-0) 


2 


-2 


l + + i(-l + 0) 


aecbd 


50 


3-0 + i( 1-0) 


2 


-2 


3-0 + i(-l + 0) 


acdeb 


51 


-3 + + i( 1-0) 


2 


- 2 


-3 + + i(-l + 0) 


abedc 


52 


- 1 - + i ( 1-0) 


2 


-2 


_i_e+i(-i+0) 


adbce 


53 


2 


-3 + + ^-1 + 0) 


3-0 + i(-l + 0) 


2 


aebdc 


54 


2 


-l-0 + &lt;( 3 + 0) 


l + + t( 3 + 0) 


2 


abced 


55 


2 


l + + t(-3-0) 


_l_ e+ i(_3-0) 


2 


adecb 


56 


2 


3-0 + t( 1-0) 


-3 + + i( 1-0) 


2 


acdbe 


57 


2 


-3 + 0+t( 1-0) 


3-0 + i( 1-0) 


2 


abdec 


58 


2 


-l-e + t(-3-6) 


l + e + i(_3_e) 


2 


adcbe 


59 


2 


l + + t( 3 + 0) 


-l-0+i( 3 + 0) 


2 


aebcd 


60 


2 


3-0 + i(-l + 0) 




2 


acedb 



240 



ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE. 



[752 



This contains (as one of five groups of 12) the group of the positive permutations 
of abed ; and, completing this into a group of 24, we have 



GROUPS OF 12 AND 24. 

ft 7 



1 


1 








1 


1 


2 


-1 





1 


ab . cd 


3 





1 


1 





ac . bd 


4 





_ 1 


1 





ad . be 


5 


-i 


i 


1 


1 


abc 


6 


-1 


i 


1 


i 


neb 


7 


1 


-i 


1 


i 


adc 


8 


-i 


i 


1 


_ -^ 


acd 


9 


i 


i 


1 


-I 


adb 


10 


1 


i 


1 


- i 


abd 


11 


-1 


-i 


1 


- i 


bed 


12 


i 


-i 


1 


1 


bdc 


13 


i 








1 


adbc 


14 


- i 








1 


acbd 


15 





i 


1 





cd 


16 





i 


-1 





ab 


17 


1 


-1 


1 


1 


acdb 


. 18 


-i 


-1 


1 


i 


bd 


19 


i 


1 


1 


i 


abed 


20 


1 


1 


1 


-1 


be 


21 


-1 


_ ^ 


1 


-1 


abdc 


22 


t 


-1 


1 


- i 


ac 


23 


-t 


1 


1 


- i 


adcb 


24 


-1 


1 


1 


1 


ad 



The groups of 60 and 24 thus each of them contain the group of 12, 

x i 



1 . 1 x . 1 + x x + i 

as, - , 1= --- , t T &gt; + -- 

X \+X \X ~X l 



+ 

~ 



It may be remarked that, to verify the periodicities of the forms contained in the 
group of 60, we have as the conditions that 



.. may be periodic of the order 2, - 

"&gt; 

S&gt; 



= 0, that is, a + 8 = 0, 



= 1 



752] ON GROUPS OF LINEAR TRANSFORMATIONS OF A VARIABLE. 241 

For instance, in the form 



[_ i _ @ + j (_ 3 - 0)] x + 2 
2a?+[l + +t(-3-)] 

we have 

a8 = _ (l + 6) 2 -(3+@) 2 , =- 20-80, 

a+8 = 

and therefore 



as it should be. 

Cambridge, 11 Nov. 1879. 



CORRECTION*, pp. 439, 440. 

I erroneously assumed that the symbol adcb could be taken as corresponding 
to the linear transformation ix: but this was obviously wrong, for it gave bd as 
corresponding to the transformation ix, and these are not of the same order, but 
of the orders 4 and 2 respectively. The proper symbol is adbc, as given above, and 
the remaining eleven symbols are then at once obtained. 

Cambridge, 17 Feb. 1880. 

[* The correction in the Table of the Groups of 12 and 24 has been inserted in the Table as now 
printed on p. 240; it applies to the second half of the column of symbols on the extreme right-hand.] 



C. XL 31 



242 [753 



753. 

ON A THEOREM EELATING TO THE MULTIPLE 
THETA-FUNCTIONS. 

[From the Mathematische Annalen, t. XVII. (1880), pp. 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 Theorie der Abel schen Functionen von drei Variabeln, 
(Leipzig, 1880), which theorem is there presented in the form : 



,-,(...;, &gt; &gt; 6 fa + */,...; fi, lO-e-^- -ete,...; p + p! , v + ), (3 ) 
but which I write in the slightly different form 

exp. [- H(u ; //, i/)] . (u + 2r ; p,, v) = exp. [- Zirinv] .S(u; p, + p!, v + v ). 



I remark that the theorem is given in the preliminary paragraphs the contents 
of which are, as mentioned by the Author, derived from Herr Weierstrass : and 
that the form of the theta-function is a very general one, depending on the general 
quadric function 



of 2/&gt; variables, p being the number of the arguments u l} ..., u p (in fact, the periods 
are not reduced to the normal form, but are arbitrary) ; and the characters v l , ..., v p \ 
//,!, ...,/ip, 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 MJ, ..., u p , and so in other cases), is that the function 

@ (u ; A + //, v + v } 



753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 243 



with the new characters /* + // and v + v is, save as to an exponential factor, equal 
to the function (u + 2r ; p, v) with the original characters //,, i&gt;, but with the new 
arguments u + 2or . 

Notation. 

This is in some measure a development of the notation employed in my " Memoir 
on the Theory of Matrices," Phil. Trans, t. CXLVIII. (1858), pp. 17 37, [152] I use 
certain single letters u, etc. to denote sets or rows each of p letters, u = (u lt ... , u p ): 
or if, to fix the ideas p = 3, then u=(u- i , u 2 , u 3 \ and so in other cases. 

But I use certain other letters a, etc. to denote squares or matrices each of p z 
letters ; thus, if p = 3 as before, 

a= o u , a 13 , ttj 3 , 



and in any such case the transposed matrix is denoted by the same letter enclosed 
in parentheses 

(a) = elm ft 21 , 



The sum w + v of the row-letters u, =(u lt u 2 , u 3 ) and v, (v^ v 2 , v 3 ) denotes the 
row (MJ+VI, ?&lt; 2 + ^2 3 + v 3 ): and in like manner the sum a + b of the two matrices, 
or square-letters a and b, denotes the matrix 



and similarly for a sum of three or more terms. 

The product uv, =(wj, u. 2 , ii 3 )(v 1 , v. 2 , v 3 ), of the two row-letters u, v denotes the 
single term u^ + u 2 v z + u 3 v 3 . We have uv = vu. 



The product 



au, = a u , a 12 , a l 



l} u. 2 , u 3 ), 



of a preceding square-letter a and a succeeding row-letter u, denotes the set or row 

On, 12, ais)(tti, "2, MS). fa*, 0, O!B)(WI, ai W s ), (a si , a 32 , a tt ) (Wj, i&lt; 2 , W,) ; 

the notation ua is not employed. 

312 



244 ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 

The product 



[753 



auv = &lt; a n , a I2 , 



HI, u 2 , u 3 ) (v lt v. 2&gt; v 3 ), 



of a preceding square-letter a followed by the two row-letters u and v, denotes the 
single term 



(a n , a 12 , a 1 



(a 3l , a^, a 33 )(u 1 , u it u 3 )v 3 . 



Observe that auv is not in general = avu ; but it is easy to verify that auv = (a)vu , 
and hence if (a) = a, that is, if the matrix a be symmetrical, then auv = avu. 

A product of two matrices 

ab, = 



denotes a matrix 



(Ill, &lt;Z 12 , (if 13 




PHI &i2, ^13 


J 


# 21 , #22, #23 




^21 &gt; ^22 &gt; 0&lt;j3 




^*J1 y ^32 &gt; 33 




0-n , Oi9 . bvt 

Ol 7 Kfi J K&gt; 




6n, 621, b 31 ), (b^, 622, & 32 ), (b l3 , 633, 633) 



(a 2 i 



viz. the top-line of the compound matrix is 
(an, a 12 , a 13 )(6 u , b. 21 , & 31 ), (a n , a w 



^, 32 , a u , a 12 , 

and so for the other lines: or expressing this in words, we say that any line of 
the compound matrix is obtained by compounding the corresponding line of the first 
or further component matrix with the several columns of the second or nearer 
component matrix. 

Clearly ab is not in general =ba. We may easily verify that (ab) = (b)(a), that 
is, the transposed matrix (ab) is that obtained by the composition of the transposed 
matrix (6) as first or further matrix, with the transposed matrix (a) as second or 
nearer matrix. Even if a and b are each symmetrical, we do not in general have 
ab = ba, but only (ab) = ba, or what is the same thing, ab = (ba). 

In a symbol such as abuv, we first combine a, b into a single matrix ab, and 
then regard the expression as a combination such as auv : the expression denotes 
therefore a single term. The theory might be explained in greater detail; but 
the mode of working with row- and square-letters will be readily understood from 
what precedes. 

In all that follows, u, p, v, JJL, v, n, -or , % are row-letters; a, 6, h, &&amp;gt;, to , 77, 77 
are square-letters : a and b are symmetrical, viz. a = (a), b = (b). 



753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 245 

And I write 

(*)O, i/) 2 , =(a, h, &)(, i/) 2 
aii? + 2huv + bv- 

i n n (ti 11 11 V 

ni &gt; U i2 , tl i3 V "i &gt; "2 , "3/ 

21 22 



+ 2 A n , 7t 12 , A 13 

*^21 ) 22 ) 23 

A 31 , // 3 . 2 , ^33 



^21} ^22 &gt; 23 
^31, ^32, &33 

to denote the general quadric function of the 2p letters u, v, with 

&gt;, =/t&gt;(2p+l) 



coefficients. It is assumed that the determinant formed with the ^p (p + 1) coefficients 
6 is negative : this is the necessary and sufficient condition for the convergence of 
the series. 

Definition of (u ; ft, v}. 



B (u ; fj,, v}, the general theta-function with p arguments u, and 2p characters 
is the sum of a p-tuple series of exponentials 



v, 



(u ; /i, v) = 2 exp. [(*) (u, n + v) 2 + 



n + v)], 



where each of the letters n, =(ni, ..,,n p ), has all integer values (zero included) from 

x to + oo . 



The general theorem in regard to (u; n, v). 



This is 



exp. [- H (u ; 



(u 



/,, v) = exp. [- 



. (u ; 



establishing a relation between the function (M ; p + p , v + v), with arbitrary character- 
increments /A , i/ , and the function (u + 2ar ; p, v) with the original characters, but 
with new arguments w+2tr . Also #(w; ///, i/ 7 ) denotes a function, linear as regards the 
arguments u, but quadric as regards p and v ; Ziripv is a single term depending 
only on p and z/; and the theorem thus is that the two functions differ only by 
an exponential factor. The relations between the constants will be obtained in the 
course of the investigation. 



246 ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. [753 

Demonstration. 

The truth of the theorem depends on the equality of corresponding exponentials 
on the two sides of the equation : viz. substituting for the theta-functions their 
values, and comparing the exponents or arguments of the exponentials : writing also 
for convenience 

G (u + 2w , n + v), 

to denote the quadric function (*) (u + 2vr , n + v) 2 ; we ought to have 
- // (u ; fi, v } + G (u + 2/, n + v) + Strip (n + v} 

= 27rifj,v + G (u, n + v + v ) + fai (/j, + //) (n + v + v), 



or say 

H(u ; p!, v } = G(u + %& , n + v) G (u, n + v + v ) fai (n+v + v) /. 

In this equation, if true at all, the terms containing n must destroy each other; 
assuming that they do so, the equation becomes 



H (u fS, v }=G(u + 2vr , v)-G (u, v + v } - 2m (v + v } //. 
Consider first the terms in n: the right-hand side is 

= a (u + 2r ) 2 + 2/i (u + 2or ) (n +v) + b(n + i/) 2 

- an* - 2hu (n + v + v } - b (n + v + v ) 2 - fainp! ; 

and the terms herein which contain n thus are 

2A (u + 2-cr ) n + bn 2 + 2bnv 
2kun bn* 2bn (v + v ) - fain/* , 

= kh-at n 2bnv fainp!, 

which, b being symmetrical, may be written 

= 2 (2Aw - bv - irifjf) n, 
and these terms will vanish if, and only if 

2}n& bv Trip = 0, 
a system of p equations connecting or , //, v . 

Assuming them to be satisfied, the remaining relation, 

H (u ; fi, v ) = G (u + 2vr , v}-G (u, v+v)- fai (v + v) fj, , 
becomes 

H (u ; fi, v) = a (u + 2W- ) 2 + 2h (u + 2vr ) n + bi&gt;- 

- w 2 - 2hu (v + v }-b(v + vj - 2m (v + v } p . 

Here, a and b being symmetrical, we have 

a(u + %& )- = aii* + 4ara- w + 4aCT /2 , b (v + i/) 2 = bv- 



753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 247 

and the value therefore is 

= 4a (vr u + TO 2 ) + 2A (2srV - uv) - b (2i/V + i/ 2 ) - ZTTI (v + v) /* . 

On the right-hand side, putting the term in h under the form 

-2h(u+ TO ) v + 2/m (2;- + v }, = - 2 (h) v (u + TO ) + 2/tTO y (2i/ + i/), 
and the last term under the form 

Trip (2v + P) Trt /iV, 



the equation becomes 

//" (-M ; //, z/) = (4aTO- 2 (A) i/ ) (M + TO- ) irip v 

+ (2/iTO - 6i/ - irifi) (2i/ 4- i/ ), 

where the second line vanishes in virtue of the foregoing equation 

2/iTO- - bv - irifi = ; 



the equation thus is 

H(u; n , v ) = (4aw - 2 (h) v ) (u + TV ) - irip v, 

which equation, regarding therein TO- as a linear function of // and v, shows that 
H(u; p!, v) is a function linear as regards u (and containing this only through u + -& ), 
but quadric as regards /* , v. 

Introducing the new row-letter , we may write 

H (u ; fjt, , i/) = 2f (u + TO ) 7rt /nV, 

viz. the expression on the right-hand side is here assumed as the value of the 
function 

H (u ; /a , v \ = G (u + 2w , v) G (u, v + v ) 2?n (v + v ) p ; 

and the theorem then is 

exp. [- H (u ; p , v )] . B (u + Its ; p, v) = exp. [- 2?n&gt;z/] . (u ; /* + /& , v + v ), 
where, by what precedes, 

2A/BJ- &Z/ TTi /Lt = 0, 

- (A) v - = 0, 



2p equations for determining the 2p functions TO , as linear functions of p, v : 
which equations depend on the p(2p + l) constants a, b, h. 

Suppose that the resulting values of TO and are 

TO = &&amp;gt;// + co v , 

!/ / , / / 

f = T//U, + 77 1/ , 

where o&gt;, co , ?;, T; are square-letters ; then, regarding a, b, h as arbitrary, the 4p 2 
new constants &&amp;gt;, o&gt; , 77, ?; cannot be all of them arbitrary, but must be connected 
by 4/3 2 p (2p + l), =p(2p 1) equations. 



248 ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. [753 

We may regard a&gt;, o&gt; , 77, 77 as satisfying these p (2p 1) equations, but as being 
otherwise arbitrary ; the foregoing equations then are 

2/m bv Trip = 0, 
2avr -(h)v f -? =0, 

TX = (a/jf + (a v , 

j i 

r w* + *? " &gt; 

which lead to the equations connecting a, 6, A with a&gt;, CD , 77, 77 . 

The first and second equations, substituting for rar and their values, become 
(2A&&amp;gt; - iri) fi + (2ha&gt; -b)v = 0, 

(2ao&gt; - T; ) // + (Saw - (A) - 77 ) v = 0, 

or fji, v being arbitrary, we thus obtain the 4/&gt; 2 equations 

2aco 77 =0, 

2A&) iri = 0, 
2aw - 77 - (A) = 0, 
2Aw - b =0, 

which are the equations in question. It is to be observed that m is, like the other 
symbols, a matrix, viz. it is regarded as containing the matrix unity ; or, what is the 
same thing, it denotes 

1, 0, 0,... 

0, 1, 0, 



TTl 



We can eliminate a, b, h from these equations and thus obtain the p (2p 1) 
equations before referred to, which connect the 4p 2 constants ca, &&amp;gt; , 77, 77 . I give, but 
without a complete explanation, the steps of the elimination. 

The equation 2a&&amp;gt; 77 = 0, may be written in the form 

that is, 

2 () (a)- (77) = 0, 
or since (a) = a] this is 

from the original form, and the new form respectively, we find 

2 (&)) aw - (&)) 77 = 0, 2 () a (o&gt;) - (77) o&gt; = ; 
and comparing these 

(o&gt;) 77 (77) &) = 0, (first result). 
The equation 2a&&amp;gt; - 77 - (h) = 0, or say (A) = - 77 + 2aw , may be written in the form 



753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 249 

that is, since a = (a), 

A = -(77 ) + 2 (ft/) a; 
and we thence deduce 

Aft) = (77 ) &lt;w + 2 (a) } aw. 

But from the equation 2aa&gt; 77 = 0, we have 2 (&/) aw (&/) 77 = 0, and the equation 
thus becomes hco = (77 ) w + (&&amp;gt; ) 77 ; which, in virtue of 2/i&&amp;gt; 7n = 0, becomes 

\TTI = (77 ) &) + (&) ) 77, (second result). 

From the equation above obtained, h = (77 ) + 2 (a/) a, we have 

Aeo = _ (T/) a) + 2 (to ) aw ; 

in virtue of 2A&&amp;gt; 6 = 0, this becomes 2 (77 ) &/ + 4 (&&amp;gt; ) aa&gt; = 6 ; an equation which 
may also be written 2 ((77 ) &&amp;gt; ) + 4 ((&/) aw ) = (6), or, what is the same thing, 
2 (&&amp;gt; ) 77 + 4 (to ) (a) G) = (6) ; or since (a) = a and (6) = 6, this is 

- 2 (to ) 77 + 4 (&) ) ato = b: 
and comparing with the original equation 

-2 (77 ) to + 4 (w ) aw =6, 
we obtain 

(w ) 77 - (77 ) &&amp;gt; = 0, (third result). 

We have thus the three systems 

(&lt;o) 77 (77) w = , |p (p 1) equations, 

(&) ) 77 (if) to = i7rt, p 2 

(ft, ) 77 -(77 ) ft, = , ip(p-l) 

in all p(2p 1) equations. As to these systems, observe that (w)ij, (n)w, etc., are 
all of them matrices of p 2 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 2 to 
p (p 1); and similarly in the third system the number of equations is =|p(p 1): 
but for the second system the number of equations is really =p 2 . It is hardly 
necessary to remark that in this second system ^iri is as before regarded as a matrix. 

The foregoing three systems of equations are in fact the equations (6) p. 4 of 
Dr Schottky s work. 

Cambridge, 12 July, 1880. 



c. XT. 32 



250 [754 



754 



ON THE CONNEXION OF CERTAIN FORMULAE IN ELLIPTIC 

FUNCTIONS. 

[From the Messenger of Mathematics, vol. ix. (1880), pp. 23 25.] 

IN reference to a like question in the theory of the double ^--functions, it is 
interesting to show that (if not completely, at least very nearly) the single formula 

a , . , (u-a) 
II ft*, a) = u .~ - + i log s-7 - ; , 

a (w + a) 

that is, 

sn acnadn asn 2 w du a 



f k z sn ac 

I 

Jo !- 



.11 ___ i_ 1 IOCT v 

a g w 



leads not only to the relation 

r r 
_^ du dusrfu, 

Jo Jo 



between the functions , sn, but also to the addition-equation for the function sn. 

Writing in the equation a indefinitely small, and assuming only that sna, cna, 
dn a then become a, 1, 1, respectively, the equation is 



f a& Q 

sn 2 w du = u ^^ + \ log 
Jo 6 



"0 &u 
= ua -pr-_- a ^ - , 
@M 

that is, 

% @"0 f . 

75T = w "FT7T ^ 2 I &lt;WI Sn 2 U, 

u J 
or, integrating from u = 0, this is 

"o / r 

log Su=C + %u* n - A; 2 du du sn 2 it, 

v^U ./ n . n 



754] ON THE CONNEXION OF CERTAIN FORMULAE IN ELLIPTIC FUNCTIONS. 251 

which, except as regards the determination of the constants, is the required equation 
for log u. 

Next, differentiating twice the equation for II (u, a), and once the equation obtained 

for ^ , we have 

d 



1 / -i, __ ft ) _ 1 (71 -L. n\ 

and 

i* _ __ 1/& gfi-2 7/ 

@a (M)Q 

where, for shortness, ~ u is written to denote ^ , and the like in 

2 & 2 u 

the first equation ; the right-hand side of the first equation therefore is 

^ & 2 {sn 2 (u a) sn 2 (u + a)}, 
or the equation becomes 

, d sn 2 u 

2 sn a en a dn a -=- .,- j = sn- (u + a) sn- (u a). 

du 1 k- sn 2 tt sn 2 a . 

that is, 

4 sn u sn it sn a en a dn a 
T- Tn , = sn- (M + a) - sn 2 (ti - a). 

(1 k- sn 2 M 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 u were unknown) it would appear that sn u must 
be a mere constant multiple of en u dn u ; assuming, however, the actual value, 
sn u = en u dn u, the formula is 

4 sn u en u dn u sn a en a dn a 
(1 k 2 sn 2 u sn 2 of 

= sn 2 (u + a) sn 2 (u a) 

= {sn (u + a) + sn (u a)} {sn (u + a) sn (u a)}. 

The factor {sn (u + a) + sn (u a)} becomes = 2 sn u for a = 0, and this suggests that 
the factor snu on the left-hand side is a factor of {sn (u + a) + sn (u a)}. That cnu 
is not a factor hereof would follow from the properties of the period K ; viz. for 
u = K, en u = 0, but {sn (u + a) + sn (u a)}, = 2 sn (K + a) is not = ; and, similarly, that 
dn u is not a factor from the properties of the period iK ; hence, en u, dn u belong 
to the other factor {sn (u + a) sn(u a)}, and by symmetry en a, dn a belong to the 
first-mentioned factor. And we are thus led to assume 

sn (u + a) + sn (u a) = 2M sn u en a dn a, 

sn (u + a) sn (u a) = 2M sn a en u dn u, 
where 

denom. = l~k* sn 2 a sn 2 u, 

and MM = l. Some further investigation is wanting to show that M and M are 
constants, but assuming that they are so and each = 1, the formulas give at once the 
ordinary expression for sn (u + a) ; that is, we have the addition-equation for the 
function sn. 

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 

/"/ /T* L | , /I 

powers and roots of the function ^ is developed in a complete and satisfactory 

manner; the results in the main agreeing with those obtained in the original memoir, 
Babbage, " On Trigonometrical Series," Memoirs of the Analytical Society (1813), Note I. 
pp. 47 50, and which are to some extent reproduced in my " Memoir on the Theory 
of Matrices," Phil. Trans., t. cxLVin. (1858), pp. 1737, [152]. I had recently 
occasion to reconsider the question, and have obtained for the nth function &lt;fr n x, where 

$x = -j , a form which, although substantially identical with Babbage s, is a more 

CX -|- Ct 

compact and convenient one ; viz. taking A, to be determined by the quadric equation 

(A + l) 2 



the form is 



ad be 



(X n+1 - 



(\w-l)(cx+d)+(\ n -\)( ex -aY 

The question is, in effect, that of the determination of the nth power of the 
matrix ( a, b ); viz. in the notation of matrices 

c, d 



c, d 



755] 



ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 



UNIV] 

.CALIF 
253 



means the two equations x l = ax + by, y 1 = cx + dy, and then if x 2 , y 2 are derived in 
like manner from as 1} y i} that is, if x z =ax l + by l , y 2 =cx 1 + dy 1 , and so on, as 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 nib. 
power of ( a, b ) is, in fact, the matrix ( a n , b n ). 

) Cv 



Cjj j 



In particular, we have 



a, b ) 2 , =( a a , 6 a ), =( a 2 



C, C 

and hence the identity 

(a, b ) 2 (a 

c, d 
viz. this means that the matrix 

( a 2 (a + d) a + ad be, b 2 - (a + d) b 



a, b ) + (ad-bc)( 1, ) = 0; 



c, d 



0, 1 



) = ( 0, ), 
d-bc 0,0 

or, what is the same thing, that each term of the left-hand matrix is =0; which is 
at once verified by substituting for a 2 , b 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 Jike manner with a single quantity: 
the theory is more fully developed in my Memoir above referred to. I proceed 
with the solution in the algorithm of matrices. Writing for shortness M=( a, b ), 

! c, d 
the identity is 

M 2 - (a + d) M+ (ad - be) = 0, 

the matrix (1,0) being in the theory regarded as = 1 ; viz. M is determined by 

0, 1 

a quadric equation ; and we have consequently 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, /3, the roots 
of the equation 

v? (a + d) u + ad bc 0, 
viz. we have 

a. n - Aa + B = 0, 

/3 n - A/3 + B = 0; 

or more simply from these equations, and the equation for M n , eliminating a, /3, we 
have 

M n , M, 1 =0; 

a n , a, 1 



254 

that is, 



ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 



M n (a-/3)-M (&lt;x n - j3 n ) + a/3 (a n ~ l - /S 71 " 1 ) = 0. 



[755 



But instead of a, @, it is convenient to introduce the ratio X of the two roots, 
say we have a = X/3 ; we thence find 

(X + l)/3 = a + d, 

X/3 2 = ad be, 
giving 

(X+l) 2 = (a + d) 2 
X ad be 

for the determination of X, and then 

Q _a + d 
P*X+1 

(a + d) X 



a = 



X+l 



The equation thus becomes 
M n (\- 



or we have 



- X) +i = 0, 



M n = . 



It is convenient to multiply the numerator and denominator by X + l, viz. we 
thus have 



The exterior factor is here 



1 fa + d\ n ~ l 



X 2 - 1 VX + 1 
moreover (X + l)/3 is =a + c?: hence 

M=( a, b ), 
c, 



and 

the formula thus is 
#*- 



= ( a, b )-( a + d, ), =( -d, b ); 



c, d 






, a + d 



c , a 



a, 6 )+(x-X)( -rf, 6 ) 

, | 

c, d \ c , -a 



viz. we have thus the values of the several terms of the ?ith matrix 

M = ( 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 x + b n and c n x + d n , we then 
obtain 

(X - X) (- dx + b)}, 



- 

A/ ~~ JL 



and in dividing the first of these by the second, the exterior factor disappears. 

It is to be remarked that, if n = 0, the formulae become as they should do a^ + b = x, 
CoX + d =l; and if n = 1, they become a-jc + ^ = ax + b, c 1 x + d 1 = cx + d. 

If X M 1=0, where m, the least exponent for which this equation is satisfied, is 
for the moment taken to be greater than 2, the terms in { } are 

(X - 1) (ax + b ) + (1 - X) (- dx + b), 
and 

(X - 1) (ex + d) + (1 - X) ( ex -a); 

viz. these are (X l)(a + d)x, and (X l)(a + d), or if for (X l)(a + d) we write 

(X 2 l)r -., , the formulae become for n=m 
A. ~t~ J. 

i / CL ~T~ CL\ 

&m x + u m = - - - 1 X, 
\A,+ I/ 



viz. we have here 

n v I 

" W * I 



- j~ &gt; 

c m x + d m 

or the function is periodic of the with order. Writing for shortness ^= , s beino- 
any integer not = 0, and prime to n, we have X = cos 2S- + i sin 2^-, hence 

1 + X = 2 cos S- (cos + i sin &), 



n + x) 2 

or - =4cos 2 ^; consequently, in order to the function being periodic of the nth 

order, the relation between the coefficients is 

STT (a + d) 2 
4 cos 2 = v - 7 ,-. 
n ad be 

The formula extends to the case m = 2, viz. cos | (STT) = 0, or the condition is 
a + d = Q. But here X + 1 = 0, and the case requires to be separately verified. Recurring 
to the original expression for M 2 , we see that, for a + d = Q, this becomes 



that is, 



a 2 + be, 
, 



, =(a*+bc) 



1, 

0, 1 



ow ^ , = x &gt; 
or the result is thus verified. 



256 ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. [755 

But the case m = l is a very remarkable one; we have here X=l, and the 
relation between the coefficients is thus (a + d)" 2 = 4 (ad be), or what is the same thing 
(a d) 2 + 4&lt;bc = 0. And then determining the values for X = 1 of the vanishing fractions 
which enter into the formula?, we find 

a n x + 6 n = (a + d) 11 1 {(n + 1) (ax + b) + (n - 1) (- dot + &)}, 

c n x + d n = 2 - (a + d) n ~ l {(n + 1) (cos + d) + (n - 1) ( ex - a)\, 
or as these may also be written 

a n x + 6 n = (a + dy 1 1 {x [n (a - d) + (a + d)] + 2nb], 

c n x + d n = (a + d) n ~ l {x . 2nc + [- n (a -d) + a + d]}, 

which for n = 0, become as they should do a # + b = x, c x + d =l, and for n=l they 

become a 1 x + b 1 = ax + b, c-p + d l = ex + d. We thus do not have a * 7 * = x and the 

c^x + d 

function is not periodic of any order. This remarkable case is noticed by Mr Moulton 
in his edition (2nd edition, 1872) of Boole s Finite Differences. 

If to satisfy the given relation (a - d)- + 4&gt;bc = 0, we write 26 = k (a - d), 2c = - - (a - d). 

K 

then the function of x is 



and the formula for the ?ith function are 

a n x + l&gt; n = cp(a + dy 1 1 {(a + d) x + n (a - d) (x + &)], 

Cn x + d n = ^(a + d) n ~i {(a + d) -n(a- d) (j + 
^ ( \P 

which may be verified successively for the different values of n. 

Reverting to the general case, suppose n = oo , and let u be the value of ^ (x). 
Supposing that the modulus of X is not =1, we have \ n indefinitely large or 
indefinitely small. In the former case, we obtain 



_ 



X (ex + d) + ( ex -a) c(\,+ 
which, observing that the equation in X may be written 

\a-d _6(X + 1) 
cX+l~ \d-a 



755] ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 257 

is independent of x, and equal to either of these equal quantities ; and if from these 
two values of u we eliminate \, we obtain for u the quadric equation 

CM 2 (a d) u b = 0, 
that is, 

au + b 



u = 



cu + d 



as is, in fact, obvious from the consideration that n being indefinitely large the nth and 
O + l)th functions must be equal to each other. In the latter case, as \ n is indefinitely 
small, we have the like formulas, and we obtain for u the same quadric equation : 
the two values of u are however not the same, but (as is easily shown) their product 
is = 6-r-c; u is therefore the other root of the quadric equation. Hence, as n 
increases, the function &lt;f&gt; 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 + 4tbc = 0, which is 

1 26 
the relation existing in the above-mentioned special case ; and here u = (a d), = -, , 

c d d 

which result is also given by the formulae of the special case on writing therein n = oo . 



C. XL 



33 



258 [756 



756. 



A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY 

QUANTITIES. 

[From the Messenger of Mathematics, vol. x. (1881), pp. 1 3.] 

LET A, B, C be given imaginary quantities, and let it be required to construct 
the roots of the quadric equation 

I 



X-A^X-B X-C 
The equation is 



that is, 

and we have therefore 

3Z - (A + B + C) = J{(A + B+C)*-3(BC+CA + AB)}, 



or as this may be written 

X = (A +B+ C) V(i(4 + Bo&gt; + Ctf).%(A + Bo&gt;- + Co.)}, 
where w is an imaginary cube root of unity, 

= cos 120 + i sin 120 suppose. 

Taking an arbitrary point as the origin, let the imaginary quantity A, =a + a. i 
suppose, be represented by the point A, coordinates a and a ; and in like manner 
the imaginary quantities B and C by the points B and C respectively. 

Then Bat, Bw 2 are represented by points B lt B 2 , obtained by rotating the point 
B about the origin through angles of 120 and 240 respectively; (7&&amp;gt; 2 , Ceo are repre- 



756] A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY QUANTITIES. 259 

sented by points C 1} (7 2 obtained by rotating the point C about the origin through 
angles of 240 and 480 (= 120) respectively : and 

$(A + B + C), i(A+Ba,+ C&lt;o 2 ), $(A+Be&gt;*+Ca&gt;) 

are represented by the points G, G lt G 2 which are the C.G. s of the triangles ABC, 
AB 1 C 1 , AB 2 C 2 respectively. The formula therefore is 



where, if a, a are the coordinates of G, then OG is written to denote the imaginary 
quantity a + a t; and the like as regards OG t , OG 2 . Taking ^ (OG 1 .OG 2 ) = OH, we then 
have H a point such, that the distance OH from the origin is = geometric mean of 
the distances OG 1} OG 2 , and that the radial direction* of the distance OH bisects 
the radial directions of the distances OG 1} 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 l . GG 2 ), so that the points X , X" are in 
fact the point H, and the opposite point in regard to G. 

The theory of the more general equation 

P 9 r _ 

X-A^X-B X-C~ 

(p, q, r real) is somewhat similar, but the construction is less simple ; we have 
(p + q + r) X 2 - {(q + r) A + (r + p) B + (p + q} C } X + pBC + qCA + rAB = Q. 

Writing herein q + r, r+p, p + q = l, m, n, the equation becomes 

(I + TO + n) X 2 - 2 (IA + mB + nC) X + (- 1 + m + n) BC + (I - m + n) GA+(l + m- n) AB = 0,. 
that is, 

{(I + m + n } X -lA-mB- nC} 2 

= (IA + mB + nC) 2 + {I 2 - (m + w) 2 } BC + {m 2 - (n + I) 2 } CA + {n 2 -(1 + m) 2 } AB. 
Here the right-hand side is 
= 1 2 A 2 + m 2 & + n 2 C 2 + (I 2 - m 2 - n 2 ) BC + (-l 2 + m 2 - n 2 ) CA + (-l 2 - m 2 + n 2 ) AB, 

which is 

= - I 2 (C - A ) (A - B) - m 2 (A - B) (B - C) - n 2 (C - A) (A - B), 

and consequently is a product of two linear factors; these, in fact, are 

{PA + %(-l 2 -m 2 + n 2 VA) 5 + (- I 2 + m 2 - n 2 + VA) C}, 



* Radial direction is, I think, a convenient expression for the direction of a line considered as drawn as 
a radius of a circle from the centre, and not as a diameter in two opposite radial directions. 

332 



260 A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY QUANTITIES. [756 

where 

A = I 4 + m 4 + w 4 - 2ra 2 tt 2 - 2w 2 Z 2 - 2Z 2 m 2 . 

It is to be observed that A, = (I 2 - m 2 ft 2 ) 2 4ra 2 w 2 , is negative ; hence, calling 
the factors fA+gB + hC, f A+g B + h C respectively, the coefficients /, g, h, and 
/ , g , h are imaginary ; moreover f+ g + h = 0, / + g + h = 0. 

The values of X thus are 



and then passing to the geometrical representation, we have j - represented 

by the point which is the C.G. of weights I, m, n at the points A, B, C respectively; 
on account of the imaginary values of the coefficients the construction is not immedi 
ately applicable to the factors 

fA+gB + hC, f A+g B + h C; 
but a construction, such as was used for the factors 

A + wB + a&gt; -C, A + w 2 B + coC, 
might be found without difficulty. 



757] 261 



757. 

ON A SMITH S PRIZE QUESTION, RELATING TO POTENTIALS. 

[From the Messenger of Mathematics, vol. xi. (1882), pp. 15 18.] 

A SPHERICAL shell is divided by a plane into two segments A and B, one of them 
so small that it may be regarded as a plane disk: trace the curves which exhibit the 
potentials of the two segments and of the whole shell respectively, in regard to a point 
P moving along the axis of symmetry of the two segments. 

Criticise the following argument : 

The potential of the segment A in regard to a point P, coordinates (x, y, z), is 
one and the same function of (x, y, z) whatever be the position of P ; similarly the 
potential of the segment B in regard to the same point P is one and the same function 
of (x, y, z) whatever be the position of P: hence the potential of the whole shell in 
regard to the point P is one and the same function of (x, y, z) whatever be the 
position of P. 

The question is taken from my memoir "On Prepotentials," Phil. Trans, vol. 165 
(1875), pp. 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 1 &lt;- p 2 ), where p, p-^ , p 2 are the distances of 

the attracted point from the centre of the sphere and from the centre and the circum 
ference respectively of the segment. The segments might therefore just as well have 
been any two segments whatever, or (to take the most symmetrical case) they might 
have been hemispheres. 

As to the argument : the assertion in regard to the potential of the segment 

[* This Collection, vol. ix. p. 333.] 



262 ON A SMITH S PRIZE QUESTION, RELATING TO POTENTIALS. [757 

A is based upon the consideration of this segment alone ; and, on the ground that 
we can without crossing the segment pass from any one position of P to any other 
position of P, it is inferred that the potential is one and the same function of the 
coordinates, whatever be the position of P : it is therefore unassailable by any 
considerations in relation to the non-existent segment B. Similarly the assertion in 
regard to the potential of the segment B is based upon the consideration of this 
segment alone, and it is unassailable upon any considerations in regard to the non 
existent segment A : the potential of the whole sphere is certainly the sum of the 
potentials of the segments A and B: it is therefore altogether off the purpose to 
object that in the case of the whole sphere we cannot pass from a point outside 
the sphere to a point inside the sphere without crossing one or other of the segments 
A and B. I consider that the two assertions are each of them true, and that the 
conclusion is a legitimate one, but it is true only in the sense in which a + x + \/[(a a?) 2 ] 
is one and the same function of x whatever be the value of x : this is so, if 
&lt;v/[(& x} 2 ] denotes indifferently or successively the two functions (a x): but if, a 
and x being real, \/[(a x) 2 ] is taken to mean the positive value, then the function 
a + x + V[(a #) 2 ] i g = 2a or = 2x according as a x is positive or negative. 

Fig. 1. 



In further illustration, let the dark line of fig. 1 represent the intersection of 
an unclosed surface, or segment, by the plane of xz taken to be that of the paper, 
and consider the potential of the segment in regard to a point P in the plane of 
the paper, coordinates x, z. We have the potential V defined as a function of x, z 
by an equation V= a definite integral, depending on the parameters x, z, and being in 
general a transcendental function of (x, z); V is a real, one-valued, finite, continuous 
function of x, z : in particular, if the point P, moving in any manner, traverses the 
dark line, there is not any discontinuity in the value of V. There is however in 
this case a discontinuity in the differential coefficients of V: if to fix the ideas we 
imagine P moving parallel to the axis of x, so that z is taken to be constant and 
V a function of x only, then when the path of P crosses the black line there is 

dV 

in general an abrupt change of value in -=- . Taking V as a coordinate y at right 

dx 

angles to the plane of the paper, a section by any plane parallel to that of xy is 
(when the trace of the plane upon that of xz does not meet the dark line) a 
continuous curve ; but when the trace meets the dark line, then for this value of x 
there is an abrupt change of direction in the section. 



757] ON A SMITH S PRIZE QUESTION, RELATING TO POTENTIALS. 263 

If (as may very well happen in particular cases) V is algebraically determinate, 
then, qua one-valued function of (x, z}, V is not any root y at pleasure of an 
algebraical equation &lt;f&gt; (x, y, z} 0, but it is for any given values of (x, z}, some one 
determinate root y of this equation : and we thus see how in this case the before- 

dV 

mentioned discontinuity in the value of -7- must arise : viz. when the trace of the 

plane meets the dark line the section is a curve having a double point; and, for 
the positions of P on the two sides of the dark line, we have V the ordinate 
belonging to different branches of the curve of section. If the path of P passes 
through an extremity of the dark line, then the curve of section will, instead of a 
double point, have in general a cusp ; and when the path of P does not cross the 
dark line, then the curve of section is a continuous line without singularity. It may 
be added that the surface $ (x, y, z) = must have a nodal line which as to a certain 
finite portion thereof is crunodal, giving the before-mentioned double points of the 
sections, but as to the residue thereof is acnodal or isolated. 

It may happen that (the surface being algebraical) any particular section thereof, 
instead of being a single curve having a double point as above, breaks up into two 
distinct curves, so that for the two positions of P, we have V the ordinate of two 
distinct curves : and this is what really happens in the case of P a point on the 
axis of a circular disk or a spherical segment : thus in the case of the disk, taking 
c for the radius, and x for the distance from the centre of the disk, the formula 
is V 2-Tr {V(c 2 + tf 2 ) + x} ; or writing F-r-27r=;y, the section is made up of the two 
distinct hyperbolas y (y 2x) = c", and y (y 4- 2#) = c 2 . 

It may be remarked that in each case, it is only for P on the axis that the 
potential is algebraical. 

In the case of the hemispheres, drawing OM a radius at right angles to the 
axis, the formula for the potential of an axial point P is of the form 



or writing F=27ry we have for the hemisphere A, the curve (1) or (2) according 
as (x a) is positive or negative ; and for the hemisphere B the curve (3) or (4) 
according as x + a is positive or negative ; viz. the equations are 

\ / t/ yi*^ ^ y j 

x 

/ C\\ f // 9 i *) \ I / \1 

( 2 ) y = - w( a + x ~} + (* - *)}. 

# 

/ O \ f // 2 **\ / i \ I 



264 ON A SMITH S PRIZE QUESTION, RELATING TO POTENTIALS. [757 

being four cubic curves. The whole curve (1) is shown in fig. 2. and the others are 

Fig. 2. 

y 



equal or opposite curves: the rationalised equation of (1) is in fact 

2a 2 (y + a) 
(y + &lt;#-#&gt; 

and by writing a for a, and in each equation cc for x, we have the rational 
equations of the other three curves. 

But, drawing only the required portions of the curves, we have fig. 3 exhibiting 

Fig. 3. 




the potentials of the two hemispheres A and B ; and also the discontinuous potential 
of the whole shell, the ordinate for this last being the sum of the ordinates for the 
two hemispheres respectively. 



758] 



265 



758. 



SOLUTION OF A SENATE-HOUSE PROBLEM. 



[From the Messenger of Mathematics, vol. xi. (1882), pp. 23 25.] 

PROVE that, if a+b + c = and x + y + z = 0, then 
4 (ax + by + czf 

- 3 (ax + by + cz) (a- + ft 2 + c 2 ) (x 2 + y- + z 2 ) 



54iabcxyz = 0. 

I do not know the origin of this identity, nor do I see any very simple way 
of proving it: that which seems the most straightforward way is to transform the 
third line, which, omitting the factor 2, is 

1, 1, 1 .1 1, 1, 1 

a , b , c x , y , z 

a 2 , b 2 , c 2 x 2 , y 2 , z 2 

3, a +b +c , a- +b- +c- 

x + y + z , ax + by + cz , d-x + b-y + c-z 
x 2 + y 2 + z", ax- + by- + cz 2 , a-x- + b 2 y 2 + c-z 
and therefore when a + b + c = and x + y + z = 0, is 

= 3 (ax + by + cz ) (a-x 2 + b 2 y 2 + c 2 z 2 ) 
3 (a-x + b 2 y + c-z} (ax- + by- + cz- ) 

- (ax +by +cz) (a- + b 2 + c 2 ) (x 2 + y 2 + z 2 ) ; 
c. xi. 34 



266 SOLUTION OF A SENATE-HOUSE PROBLEM. [758 

or, as this may be written, 

= 6 (ax + by + cz ) (a 2 x* + fey + c 2 z 2 ) 

- (ax + by + cz) (or + b 2 + c 2 ) (a? + y 2 + z 2 ) 
-3 (ax + by + cz) (a 2 x 2 + fey + c 2 * 2 ) 

- 3 (a-x + b-y + c 2 z) (ax 2 + by 2 + cz 2 ). 
Here the third and fourth lines, omitting the factor - 3, are 

2 (a?a? + fey + c 3 z 3 ) + (ab 2 + a-b) (xy* + x 2 y) + (ac 2 + a*c) (as* + x-z) + (be- + b 2 c) (yz 2 + ifz\ 

where, in virtue of the two relations, each of the last three product-terms is = abcxyz, 
and the whole is thus 

= 2 (a 3 # 3 + fey + c 3 ^) 

+ Sabcxyz. 
The product of the two determinants is thus 

= 6 (ax + by + cz) (aW + fey + C 2 z-) 

- (ax + by + cz) (a 2 + b 2 + c 2 ) (x 2 + y 2 + z 2 ) 

- 6 (a 3 ^ 3 + &y + c 3 * 3 ) 
9 abcxyz ; 

and this being so the identity to be verified is 

4 (ax + by + cz) 3 

+ (- 3 + 2 =) - 1 (ax + by + cz) (a 2 + b 2 + c 2 ) 
- 12 (ax + by + cz) (aW + fey 



+ (18 - 54 =) - SQabcxyz = 0. 
We have here the terms 

12 (aW + fey + c s z 3 - Sabcxyz), 
= I2(ax + by + cz) (aW + fey + c 2 ^ 2 - bcyz - cazx - abxy), 

so that the left-hand side is now divisible by ax + by + cz, and throwing out this 
factor the equation becomes 

4 (ax + by + cz) 2 
- (a 2 + b 2 + c 2 ) (x 2 + y 2 + z 2 ) 



+ 12 (a 2 ^ 2 + fey + c 2 z 2 - bcyz - cazx - abxy) = ; 



758] SOLUTION or A SENATE-HOUSE PROBLEM. 267 

or, as this may be written, 

4 (a?a? + 6 2 t/ 2 + cV bcyz cazx abxy) 
- (a 2 + 6 2 + c 2 ) (x- + y- + z-} = 0, 

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 (a 2 *- 2 + 6 2 y 2 + c-z- bcyz cazx abxy} 
- 2 (a 2 + b- + c 2 ) O 2 + f + 2*} 

x( 3a 2 - 6 2 - c 2 4- 26c - 2ca - 2a6) 
= (x + y + z) + ?/ (- a 2 + 3fr- - c 2 - 26c + 2ca - 2a6) 
z - a 2 - 6 2 + 3c 2 - 26c - 2ca 



/ a ( 3x 2 y- 2 2 + 2yz 2zx Zxy) \ 
+ (a + b + c) J 4- 6 ( x- + 3y* - z 2 2yz + 2zx 2xy) L , 

4- c ( x- y- + 82- 2yz Zzx + 2xy) 
\ / 

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- ft + ft 2 y + y-ct = a given value k, then a, ft, 7 must denote the 
roots, taken not in any order whatever, nor yet in a uniquely determinate order, but 
with a certain restriction as to order, viz. if the roots in a certain order are a, b, c, 
these roots being such that a-b + Ire + c~a = k, then clearly the relation in question 
3? ft + /3 2 7 + 7 2 a = k, will be satisfied if a, ft, 7 = a, b, c, or = b, c, a, or = c, a, b 
(but not if a, ft, 7 = b, a, c, or = either of the remaining two arrangements) ; the 
relation thus allows a to be = a, or = b, or = c ; that is, a is = any one at 
pleasure of the roots of the cubic equation, and it is thus determined by the cubic 
equation, and not by any inferior equation ; but a being known, the other two roots 
ft and 7 will be uniquely, and therefore rationally, determined. 

It is worth while to see how the result works out ; suppose, for greater simplicity, 
the cubic equation is a? 7x + 6 = having roots (1, 2, 3), and that the given 
relation is a-ft + ft 2 y + 7-0. 1, then the cubic equation gives 

a + ft + 7 = 0, aft + 017 + 7 = - 7, afty = - 6, 
and we have, besides, the relation in question 

a.- ft + /3 2 7 + 7 2 a = - 1 ; 
eliminating 7 we have 

a 2 + a/3 + /3 2 =7, #( + ) = 6, a 3 + 3-/3 - /3 3 + 1 = 0; 
or, as it is convenient to write these equations, 

ft- + aft + 1? - 7 = 0, 

ft"-+ a/3-- =0, 

B 

ft 3 - 3a 2 /3 - a* - 1 = 0. 



759] ILLUSTRATION OF A THEOREM IN THE THEORY OF EQUATIONS. 269 

If from these equations we eliminate ft, we obtain two equations in a, which it 
might be supposed would determine a uniquely; but, by what precedes, a is any 
root at pleasure of the cubic equation and can thus be determined only by the 
cubic equation itself, and it follows that any equation obtained by the elimination 
of ft must contain as a factor the cubic function a 3 7 a + 6, and be thus of the form 
M(a? 7a+ 0) = 0, where M is a function of a; one result of the elimination is 
a 3 7a + 6 = 0, and every other result is of the form just referred to, M (a 3 7a + 6) = ; 
hence we have definitely a 3 7a+6 = 0, viz. the roots of the equation J/ = do not 
apply to the question. 

a 

In verification, observe that the first and second equations give a 2 7 = - , that 
is, a 2 6a + 7 = 0. To eliminate ft from the first and third equations we first find 

a/3 2 + (4a 2 - 7) ft + a 3 + 1 = 0, 
or say 

/ 7\ r 1 

ft- + 4a }ft + a 2 + - = 0, 

V a/ a 

and combining herewith the first equation 

we obtain 

/ 7\ i 

\ a.J a. 

that is, 



-3 2 +7 
substituting in the first equation, 

(7a + l) 3 
+ a(7a+l)(-3a 2 + 7) 

that is, 

49 14 1 

-21-3+49 +7 
9 0-105 +343 -343 



90-126-3+441 +21-342, 
or, dividing by 3, 

3a 6 - 42a 4 - a 3 + 147a 2 + 7a - 114 = 0, 

which, in fact, is 

(a? - 7 a + 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 

7ff -i- 1 

three values 1, 2. 3, but a being known we have 8 = rr =. , and thence 

3a 2 + 7 

-7a-l = 3a 3 -14a-l 
*" + -3a 2 +7 -3a 2 + 7 

in particular, as a = 1, then ft = 2 and 7 = 3. 




270 



[760 



760. 



REDUCTION OF _ TO ELLIPTIC INTEGRALS. 

[From the Messenger of Mathematics, vol. XL (1882), pp. 142, 143.] 

WEITIXG s, c, d for the sn, en, and dn of u to a modulus k, which will be deter 
mined, and denoting by 6 a constant which will also be determined, the formula of 
reduction is 

- 1 + Oscd 



oc ^ 



1 + Bscd 



To find from this the value of y, = ty(l - a?), putting for shortness X = 6scd, the 



formula is # = - and we thence have 
l + A 



where 

1 + 3Z 2 = 1 + 3(9V (1 - s 2 ) (1 - k 2 s 2 ), 

= 1 + SPs- -3^(1+ k 2 ) s 4 + 
may be put equal to (1 + # 2 s 2 ) 3 , that is, 

= 1 
viz. this will be the case if 

30 = 

that is, 

&lt;9 2 = -l-A; 2 , ^ = 3^ 2 ; 

these give 

&-&+ 1=0; 

that is, k z = w, if &&amp;gt; = |- + ^ ^3, an imaginary cube root of unity ; and then 

$ 2 = 1 + (a, =o)- (QJ- &lt;w), = io) 2 \/3 ; 



f dx 

7601 KEDUCTION OF 7- =*-* TO ELLIPTIC INTEGRALS. 271 

J J (1 #r) 3 



that is, 

e= + 



as may be verified by squaring. 

Hence finally, 6 and k denoting the values just obtained, 

_ - 1 + Oscd 
X 1 + Oscd 



l + 0scd 
or, writing as before, X = Oscd, we have 

dx . 

~ 

whence 



and then 

dX = jl - 2 (1 + #) s 2 + 3^5 4 } ^M, =0(1 + 8 s a ) 3 dw ; 
that is, 



or say 



the required formula. 



272 [76 



761. 



ON THE THEOREM OF THE FINITE NUMBER OF THE 
COVARIANTS OF A BINARY QUANTIC. 



[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvn. (1881 

pp. 137147.] 

GORDAX S proof, the only one hitherto given, is based upon the theory of derivative 
(Uebereinanderschiebungen). It is shown that the irreducible covariants of the binar 
quantic f are included in the series 

(f,fr.(f,fY, (/ h\ (/, h)*, ... 

of the derivatives of the quantic upon itself or upon some other covariant, and tha 
the number of the irreducible covariants thus obtained is finite. And not only sc 
but for the quintic and the sextic the complete systems were formed, and the number 
shown to be = 23 and 26 respectively. 

It would seem that there ought to be a more simple proof based upon the con 
sideration of the fundamental covariants : for the cubic (a, b, c, d$x, y) z , these an 
the cubic itself (a, ...$&gt;, y}*, the Hessian (ac- b 2 , ...$#, y) 2 , and the cubicovarian 
(a z d 3a6c + 26 3 , ...]#, y) 3 ; and so in general for the quantic (a, ...$#, y) n , we have i 
series of fundamental covariants the leading coefficients whereof are the seminvariants 

a, ac - b-, a-d - Sabc + 26 3 , a 3 e - 4a 2 6c + Qab 2 c - 36 4 , &c. 

It is known that every covariant can be expressed as a rational function of these, 01 
more precisely that every covariant multiplied by a positive integral power of the 
quantic itself can be expressed as a rational and integral function of the fundamenta 
covariants, and we may for the covariants substitute their leading coefficients, or saj 
the seminvariants ; hence, every seminvariant is a rational function of the fundamenta 
seminvariants, and more precisely, every seminvariant multiplied by a positive integral 



761] ON THE THEOREM OF THE FINITE NUMBER OF CO VARIANTS. 273 

power of the first coefficient a is a rational function of the fundamental seminvariants. 
Thus, in the case of the cubic, we have the discriminant V, 

= a*d- - Qabcd + 4ac 3 + 4&gt;b 3 d - 36 2 c 2 , 
obtained from 

a, ac-b 2 , a-d - Sabc + 26 3 , 
by the formula 

2 v = (a?d - Sabc + 26 3 ) 2 + 4 (ac - 6 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 3abc + 26 3 , 

and V. It appears a truism, and it might be thought that it would be, if not easy, 
at least practicable, to show for a quantic of any given finite order n, that we can 
in this manner, as rational functions of the n 1 seminvariants, obtain only a finite 
number of new seminvariants, so that all the seminvariants would be expressible as 
rational and integral functions of a finite number of seminvariants; and, consequently, 
all the covariants be expressible as rational and integral functions of a finite number 
of irreducible covariants. But the large number, 23, of the covariants of the quintic 
is enough to show that the proof, even if it could be carried out, would involve 
algebraical operations of great complexity. 

The theory may be considered from a different point of view, in connexion with 
the root-form a (x ay) (x fty) . . . , or say (ac a.) (x 0) . . . of the quantic ; we have 
here what may be called the monomial form of covariant, viz. the general monomial 
form is 

(a - /3) m (a - y) n (/3 - 7)? ...(as- a)3(x-/3) r ..., 

where in all the factors (whether a /3 or x a) which contain a, in all the factors 
which contain /3, . . . , and so for each root in succession, the sum of the indices has 
one and the same value, = suppose. Thus, for the cubic 



we have the monomial covariants 

&lt;-)( -7) ( -7). 



and so for the quartic 

(x -a)(x- /3) (x -y)(x- B), 

we have the monomial invariants 



Observe that the monomial form is considered as essential ; a syzygetic function of 
c. XL 35 



274 ON THE THEOREM OF THE FINITE NUMBER [761 

two or more monomials is not a monomial, and we are thus in no wise concerned 
with identities such as 



or 

(a - 8) (/3 - 7) - ( - 8) (a - 7) + (7 - 8) (a - /3) = ; 

notwithstanding these syzygies respectively, 

(/3 7) (x a), (a 7) (as /9), and (a /3) (# 7 ) 
are regarded as independent covariants of the cubic, and 

( a _S) (/ 3_ 7 ), (/3-S)( a - 7 ), and ( 7 - 8) (_), 
as independent invariants of the quartic. 

It is only when a monomial covariant is equal to a power or product of simple 
or other powers of lower monomial covariants that it is regarded as a function of 
these lower monomial covariants and therefore as not irreducible. Thus 



is a reducible monomial covariant, expressible in terms of the lower irreducible 
monomial covariants 

-S and a - 



The theorem of the finite number of the irreducible monomial covariants (as just 
explained) of the root-quantic is a question of the same kind as, but entirely distinct 
from, that of the finite number of the covariants of the quantic in the ordinary form ; 
and there are thus the two questions ; (A), that of the finite number of the irreducible 
monomial covariants of the root-quantic; and (C), that of the finite number of the 
irreducible covariants of the ordinary quantic. 

But we can pass from (A) to (C) by means of a lemma (B), which I have not 
proved, but which seems highly probable, and which I enunciate as follows : (B) The 
infinite system of terms X, rational and integral functions of a finite set of letters 
(a, b, c, ...) which remain unaltered by all the substitutions of a certain group 
(r(a, b, c, ...) of substitutions upon these letters, includes always a finite set of terms P 
such that every term X whatever is a rational and integral function of these terms P. 

In explanation of this lemma, observe that, if G (a, b, c, ...) denotes the entire 
group of substitutions upon these letters, so that the functions which remain unaltered 
by the substitutions of the group are in fact the symmetrical functions of (a, b, c, ...), 
then the theorem is " The infinite system of rational and integral symmetrical 
functions of (a, b, c, ...) includes always a finite set of terms P such that every such 
rational and integral symmetrical function is a rational and integral function of the 
terms P, viz. the terms P are here the several symmetrical functions 

a + b + c + . . . , ab + ac + be + . . . , abc + . . . , &c." ; 



761] 



OF THE COVARIANTS OF A BINARY QUANTIC. 



275 



and so, if G (a, b, c, ...) is the group of all the positive substitutions, then we have 

the like theorem for the rational and integral two-valued functions of (a, b, c, ...), 

viz. the terms P are here the two- valued function (a b)(a c)(b c) ..., and the 
symmetrical functions 

a + b + c + ..., ab + ac + bc + ..., abc + ..., &c., 
as before. 

I return to the theorem (A), but instead of the covariants of a root-quantic of 
any order, I consider first the invariants of a root-quantic of any even order. The 
general form is 

(-) w (a-7)G8-7)P..., 

where in all the factors which contain a, in all the factors which contain /3, and so 
for each root in succession, the sum of the indices has one and the same value = 6. 
Writing 12 for the index of a (3, 13 for that of a 7, and so in other cases, then 
assuming always 12 = 21, 13 = 31, &c., the indices, taken each twice, form the square 



21 



31 



12 



13 



23 



the order of which, or number of its rows or columns, is equal to the order of the 
quantic; the terms of the dexter diagonal are each =0, and the square is sym 
metrical in regard to this dexter diagonal. Moreover, the square is such, that the 
sum of the terms in each row (or column) has one and the same value = 6 ; and 
conversely, every such square, say R 6 , represents an invariant. 

Thus, for the quartic (as - a) (x - /3) (as - 7) (as - B), the square E 9 is a square of 
four rows (or columns) representing the invariant 

(a-y3) 12 (a-7) 13 (a-S) 14 , 



in which 



(7-S) 34 , 

12 + 13 + 14 = (9, 
21 + 23 + 24 = 0, 

31+32 + 34 = 0, 

41 + 42 + 43 = 0. 



352 



276 ON THE THEOREM OF THE FINITE NUMBER 

There are three squares R l} viz. these are the squares 



[761 



1 



representing the before-mentioned invariants (a - /3) (y - 8), (a - y) (/3 - 8), (a - 8) (/3 - 7) 
respectively: say these are a, b, c, and every other invariant is a rational and 
integral function of these; in fact, the ^-equations give easily 12 = 34, 13 = 24, 14 = 23, 
so that the general form of the invariant is = a 12 b 13 c 14 , 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 e (0=12 + 13 + 14) is a sum 

= 12. ^+13. E/ + 14. RS, 

with positive integer coefficients 12, 13, 14, say for shortness it is a sum of squares 
jRj. And so any like expression with a negative coefficient or coefficients may, for 
shortness, be called a difference of squares R lf 

Observe that, in general, two squares R e , R^ are added together by adding their 
corresponding terms, the result being a square R e +^ , 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 e , giving a square R e -^,. 

In the case of the sextic 



there are fifteen squares R 1} which may be represented as follows: 

12.34.56 
12.35.46 
12.36.45 
13.24.56 



13.25.46 


2/2 


13.26.45 


* 


14.23.56 


#3 


14.25.36 


2/3 


14.26.35 


Z 3 


15.23.46 


at t 


15.24.36 


2/4 


15.26.34 


4 


16.23.45 


# 


16.24.35 


2/5 


16.25.34 


5 



761] 



OF THE COVARIANTS OF A BINARY QUANTIC. 



277 



viz. 12.34.56 here represents the square 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 ) 56 . But 
it is not true that every square R e is a sum of squares R l ; this is not the case, 

for the square R 2 , 

= 12.13.23.45.46.56, 

representing the invariant 

(a - /3) 12 (a - 7) 13 (0 - 7) 23 (S - e) 45 (S - ) 46 (e - ) 56 &gt; 
is not a sum of squares RI. 

But the square last referred to is a difference of squares R^. it is in fact 
= 12 . 36 . 45 + 13 . 25 . 46 + 14 . 23 . 56 - 14 . 25 . 36, 

or, what is the same thing, the corresponding invariant is the product of the 
invariants 12.36.45, 13.25.46, 14.23.56, divided by the invariant 14.25.36; viz. 
it is a rational function of invariants RI. 

It is required to show, first, that every square R e is a difference of squares J2 a ; 
and thence, secondly, that it is a sum of a finite number of squares Rk (being, in 
fact, squares R! and R 2 ). 

For the first theorem we equate the general expression of R 6 with the assumed 

value 

x l . 12 . 34 . 56 + 2/1 . 12 . 35 . 46 + z l . 12 . 36 . 45 + ... + z s . 16 . 25 . 34. 

We thus obtain 

fifteen equations satisfied by 

x l = 34 - 26 + r + 8 - t, 

#3 = 14 p S , 



12 = 2/1 + #1 + ^1 

13 = # 2 + 2/2 + ^2 

14 = x s + 2/3 + z 3 



15 =#4 + ; 2/4 + ^4 

16 = x 5 + 2/5 + z 5 

^jO = #3 -J- X T $/j 

24 = x 2 + 2/4 + y s 

25 = 2/2 + 2/3 + * 5 

26 = z 2 + z 3 + z 4 

34 = X-L + z + z 5 

35 = 2/1 + 2/5 + ^3 

36 = 2/3 + 2/4 + ^1 

45 = x s + z l + z 2 

46 = ar 4 + 2/1 + 2/ 2 
56 = x l + # 2 + #s 



= 15-26-3 

= 45 q r , 

= 12-34 + 26 -q-r-s + t, 
= 25 p , 

P 



2/3 = 

2/4 = 36 

y, = 16-45 



-p- 



-t, 



. q 



= 26 



r 

s 
r s 



278 OX THE THEOREM OF THE FINITE NUMBER [761 

connecting x lt y lt z lt ...,z s with the terms 12, 13, etc. of R e (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 l , y lt z 1} ..., z s , these 
values satisfy identically the relations which ought to be satisfied by the terms 12, 
13, etc., viz. the equations obtained by the elimination of 6 from the equations 

12 + 13 + 14 + 15 + 16 = 0, 

12 + 23 + 24 + 25 + 26 = 6, 

16 + 26 + 36 + 46 + 56 =6. 

The equations are thus insufficient to determine the values of a\, y lt z l} ..., z s , 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. 2 , z a , z s 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 oc 1 , y 1} z 1} ...,z s , 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&gt; 

= x l . 12 . 34 . 56 + 2/1 . 12 . 35 . 46 + z l . 12 . 36 . 45 + . . . + z s . 16 . 25 . 34, 
equal in general to a difference of squares R l . 

Suppose in such difference of squares J^ we have any term, say 12 . 34 . 56, 
occurring with the coefficient 1. Since the expression represents a square 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 - 12 . 34 . 56, 
12 . 36 . 45 + 15 . 26 . 34 + 14 . 23 . 56 - 12 . 34 . 56, 

12 . 36 . 45 + 16 . 25 . 34 + 13 . 24 . 56 - 12 . 34 . 56, 
12 . 36 . 45 + 16 . 25 . 34 + 14 . 23 . 56 - 12 . 34 . 56. 

The first of these is 35.46.15.26.13.24, viz. it is 13.15.35.24.26.46 which is a 
square R z (of the form mentioned above); the second is 35.46.15.26.14.23, which 
is 15 . 23 . 46 + 14 . 26 . 35, a sum of squares ^ ; and similarly each of the other 
combinations is either a square R 2 or a sum of squares R lt We have thus got rid 
of the negative term 12.34.56, and in like manner if the negative term had been 

- m . 12 . 34 . 56, = - 12 . 34 . 56 - 12 . 34 . 56 - &c., 



761] OF THE CO VARIANTS OF A BINARY QU ANTIC. 279 

or, whatever the negative terms may be, we get rid one by one of each negative 
term ; and thus ultimately express R e as a sum of squares R^ and R z . Or, what 
is the same thing, the invariant R originally expressed as a rational function of 
invariants R lt is finally expressed as a rational and integral function of invariants 
R! and R 2 . 

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 1} and then as a sum of squares 
R lt -R 2 , or it may be higher squares R s , &c., but certainly as a sum of a finite 
number of squares Rk. For a root-quantic of any odd order n, the investigation 
would be of a somewhat different form, since here there are no squares R 1 , 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 Rg is a sum of a finite number of 
squares R k . And a like reasoning would apply to co variants instead of invariants : 
viz. the reasoning (although for simplicity it has been given for a very particular 
and special case) does, I think, really establish the theorem (A) in its generality, 
viz. the theorem that for a root-quantic of any given finite order, the number of 
irreducible monomial covariants is finite. 

From any monomial covariant of the root-quantic, by taking the sum of the 
forms belonging to the different roots, so as to obtain a symmetrical function of the 
roots, that is, a rational and integral function of the coefficients, we obtain a covariant 
of the quartic in its ordinary form (a, ...$#, y) n . Consider for a moment the before- 
mentioned case of the invariants of the root-quartic 

(x - ay) (x - fty) (x - 7 y) (x - By), 
now put 

= -(a, b, c, d, e$x, 2/) 4 ; 

tl* 

and to make the reasoning clearer, take a, b, c, f, g, h = (a 8) (ft 7), (ft - 8) (7 a), 
(7 8) (a ft), (a 8) (7 ft), (ft 8) (a 7), (y B)(ft a) respectively, these being, 
with the signs +, the before-mentioned three monomial invariants. In the root-theory, 
every monomial invariant is a rational and integral function of a, b, c, f, g, h. Every 
invariant of (a, ...$#, y)*, qua rational and integral function of the coefficients, is, 
when expressed in terms of the roots, a rational and integral function of the roots, 
and then qua invariant is a sum of monomial invariants, and as such a rational and 
integral function of a, b, c, f, g, h. But every such rational and integral function 
of a, b, c, f, g, h is not a symmetrical function of a, ft, 7, B, and consequently not 
in the present theory an invariant of (a, ...]#, y) 4 ; the invariants are those rational 
and integral functions of a, b, c, f, g, h which are symmetrical functions of (a, ft, 7, B), 
that is, which remain unaltered by every substitution whatever upon the roots 
(a, ft, 7, B). Now each such substitution gives a substitution upon a, b, c, f, g, h, 
and the 24 substitutions upon a, ft, 7, B give a group of 6, = . 24 substitutions 
upon (a, b, c, f, g, h); the invariants are thus the rational and integral functions of 
(a, b, c, f, g, h) which are unaltered by each of the substitutions of a certain group 
G(a, b, c, d, e, f) of 6 substitutions. Theorem (B) asserts that, among the terms in 



280 ON THE THEOREM OF THE FINITE NUMBER OF CO VARIANTS. [761 

question, that is, among such rational and integral functions of (a, b, c, f, g, h), we 
have a finite number of terms P, such that every one of the terms is a rational 
and integral function of the terms P ; and recollecting that a + b + c = 0, these terms 
P are in fact two terms be + ca -f- ab and (b c) (c a) (a b) ; the conclusion being, 
that the invariants of the quartic (a, b, c, d, e\x, y}* are all of them rational and 
integral functions of the last-mentioned two functions, that is, of 

/, = ae - 4&gt;bd + 3c 2 , and J, = ace - ad z - b*e + 2bcd - c 3 . 

As regards the group G (a, b, c, f, g, h) of 6 substitutions upon a, b, c, f, g, h, 
observe that the 24 substitutions of (a, /3, 7, 8} operating upon a, b, c, f, g, h give 6 
substitutions taken each four times; for instance, the substitutions 1, 0/3.78, ay . (3&, 
aS . j3y leave each of them a, b, c, f, g, h unaltered, that is, they each give the 
substitution 1. And we thus find for the group Cr(a, b, c, f, g, h) the 6 substitutions 

1, 

abc . fgh, 
acb . fhg, 
af . bh . eg, 
ah . bg . cf, 
ag . bf . ch. 

For the functions of a, b, c, f, g, h, which remain unaltered by the substitution of 
this group, observe that we have f, g, h = a, b, c ; so that any function of 
the six letters may be represented as a function of a, b, c. An odd symmetrical 
function, for instance abc, does not remain unaltered, for it is by any one of the last 
three substitutions changed into fgh, that is, into abc; on the other hand, the 
two-valued function (b c) (c a) (a b) does remain unaltered : the functions which 
remain unaltered are therefore the even symmetrical functions of a, b, c (that is, the 
symmetric functions 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 + ca + ab and (b c) (c a) (a b). 

The proof applies to the general case of the theorem (C ), viz. taking the theorem 
(A) to be proved, and putting the root-quantic 

O - ay) (ac - @y) . . . = - (a, ...$#, y) n , 

Cv 

then we have a, b, c, d, ... a system of monomial covariants of the root-quantic; 
and all the covariants of (a, ...]#, y) are rational and integral functions of (a, b, c, d, ...) 
which remain unaltered by the substitutions of a certain group G (a, b, c, d, ...); hence, 
assuming the theorem (B), they are rational and integral functions of a finite number 
of irreducible covariants. And the demonstration thus depends upon that of the 
theorem (B). 



762] 281 



762. 



ON SCHUBERT S METHOD FOR THE CONTACTS OF A LINE 

WITH A SURFACE. 



[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvn. (1881), 

pp. 244258.] 

I WISH to reproduce in part 33, "Coincidenz von Schnittpunkten einer Geraden 
mit einer Flache " of Schubert s very interesting work Calcul der abzahlende Geometric, 
Leipzig, 1879, explaining in the first instance (but not altogether in the manner or from 
the point of view of the author) the general principles of the theory. 

We have to do with conditions relating to a subject; the subject is a geometrical 
form or entity of any kind depending upon a certain number of constants; and the 
condition is onefold, twofold, &c., according as it imposes a onefold, twofold, &c., 
relation upon these constants. The number of constants is the Postulandum of the 
subject, and the manifoldness of the condition is called also its Postulation. A condition 
is incomplete when its postulation is less than postulandum of subject, complete when 
its postulation is equal to postulandum of subject; two or more incomplete conditions, 
making up a complete condition, are supplementary to each other. The case where 
the postulation exceeds the postulandum, or say that of a more than complete 
condition, is not in general considered ; it may however sometimes present itself. 
For instance, the subject may be a line with n points upon it ; the number of 
constants is here = n + 4. A condition that the line shall meet a given line, or that 
a certain one of the n points shall lie on a given plane, is a onefold condition; 
the condition that such point shall lie upon a given line is a twofold condition: and 
so in other cases. 

Conditions are denoted by letters, and simultaneous conditions by a product; for 

instance, the subject is a line carrying the n points 1, 2, ..., n; g is the condition 

that the line meets a given line; p 1 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, 

fj e the condition that it lies in a given plane, 
g p the condition that it passes through a given point, 

then g e +g p 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 2 =p g . 

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 e the condition that it lies in a given plane, 
g p the condition that it passes through a given point, 
then g 2 =g e 



This equation has (so far) no numerical signification, and I regard it as having 
no logical signification. Schubert, however, gives it a logical signification by means of 
his " Princip der speciellen Lage " (Principle of Special Situation), viz. the condition 
of the line meeting each of two given lines is, in the particular case where the 
two given lines meet, equivalent to the condition, that the line shall either lie in 
the plane of the two given lines or else pass through their point of intersection. 



762] CONTACTS OF A LINE WITH A SURFACE. 283 

Third example. Subject a line bearing upon it the points 1 and 2, 
e the condition of the coincidence of the two points, 
p that the point 1 shall He on a given plane, 

q " 

g that the line shall meet a given line, 

then e = p + q g. 

This equation has (so far) no numerical signification, and it does not appear to 
have any logical signification. In fact, in the actual form of the equation we have 
a sign which has not had given to it any logical interpretation ; and if we write 
the equation in the form e + g = p + q, there seems to be no logical signification in 
the assertion, the condition that either the points shall coincide, or else the line 
meet a given line, is equivalent to the condition that either the first point, or else 
the second point, shall lie in a given plane. 

Any equation connecting complete conditions is a numerical equation ; and to 
render a condition complete, we have only to join to it a supplementary condition X 
of the proper postulation. Thus, in the last example the postulandum is = 6 ; e, p, 
q, g are onefold conditions, and joining to each of them one and the same fivefold 
condition X, we have Xe = Xp + Xq Xg. And, taking X to be an arbitrary fivefold 
condition, the original equation e=p+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 S 1} S 2 respectively. The subject 
is the line of the given regulus with its two points ; and the meaning of the equation 
is that the number of subjects with two coincident points is equal to the number 
of subjects with the point 1 on a given plane, phis the number of subjects with the 
point 2 on a given plane, minus the number of subjects for which the line meets 
a given line. Although for the moment concerned only with the meaning of the 
theorem, not with its truth, I stop to show d posteriori that the theorem is in fact 
true: take k for the order of the regulus; m,, ra 2 for the orders of the surfaces 
$! , $ 2 respectively ; then it is to be shown that X e, Xp, Xq, Xg are each = km^m^ 
(values which satisfy the equation). First X e : the points 1 and 2 here coincide at 
a point of the curve of the order m^n^, which is the intersection of S 1 and S. 2 ; 
the regulus meets this curve in km-^m^ points, and through each of these we have a 
line of the regulus having upon it the two coincident points; that is, Xe = km l m z . 
Next Xp: the point 1 is here on the plane curve of the order m lt which is the 
intersection of S l with the corresponding given plane ; the regulus meets this plane 
curve in km l points; through each of these we have a line of the regulus intersecting 
$ 2 in m 2 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 2 .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 $ x in m^ 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 

Xg, = k . m 1 . m 2 . 

The general theorem Xe = Xp + Xq Xg is proved by means of Chasles theorem 
of united points as follows: the subject is a line, or say, for convenience, an axis , 
bearing upon it the two points 1 and 2 ; we consider in conjunction therefore a given 
line X, and through this draw the planes P lt P 2 passing through the points 1 and 
2 respectively. 

Suppose that when 2 lies in a given plane there are a positions of the axis, 
and on each of these /3 positions of the point 1 ; and, similarly, that when 1 lies 
on a given plane there are a positions of the axis, and on each of these /3 positions 
of the point 2 ; then, 1 lying in a given plane, the number of subjects is a/3, or 
we have Xp = a/3 ; and, similarly, Xq = cc /3 . Take now for the point P! an arbitrary 
plane through X; then, 1 lying on this plane, the number of the points 2 is =a/3, 
or, since each of these determines with X a position of the plane P 2 , the number 
of these planes is = ay9, that is, it is = Xp ; and, similarly, taking P 2 an arbitrary 
plane through X, the number of the planes Pj is a /3 , that is, it is = Xq ; viz. the 
two planes P a , 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 | 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 + qg as a numerical equation as follows, viz. taking for the subject 
a line with two points, which line and points are regarded as satisfying a given fivefold 
condition, then 

e is the (additional onefold) condition that the two points shall coincide, 

p that the point 1 shall lie in given plane, 

q that the point 2 shall lie in given plane, 

g that line shall meet given line. 

The conditions e, p, q, g are thus in effect complete conditions, having values which may 
be connected by an equation ; there, in fact, exists between them the relation 



762] CONTACTS OF A LINE WITH A SURFACE. 285 

The like remarks would apply to the before-mentioned equation (subject a point) 
P*~Pg either adding to it a onefold condition X, and so taking it in the form 
Xp* = Xp g , or understanding it in its original form p t =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+qg) is in its actual form an 
equation without logical or numerical signification ; but if we adjoin to it a supplementary 
condition K, such that postulation of K + do. of Jf=do. of X (or, what is the same 
thing, that the condition KM shall be supplementary to the several conditions contained 
in the original equation e = p + qg), then the equation in question, Me Mp + Mq Mg, 
is to be interpreted as meaning 

KM e = KMp + KMq - KMg, 
that is, 

Xe = Xp + Xq - Xg, 

which is numerically true. We thus see that the original equation e=p + q g implies 
the new equation 

Me = Mp + Mq - Mg, 

which is its algebraical consequence. And if we regard, for instance, A + B as the 
condition that either the condition A shall be satisfied or else the condition B shall 
be satisfied, then A + B is a condition, and as such we have 



It is going a step further to say that if we have, for instance, an equation 
M = A + B C between conditions M, A, B, C, then that, instead of 

Me = Mp + Mq - Me, 
we may write 

(A + B - C ) e = ( A + B - C) p + ( A + B - C) q - (A + B - C) e ; 

this is, in fact, treating A + B C as being to all intents and purposes a condition such 
as M, or an alternative condition A + B. It is, in fact, assumed that the step is per 
missible ; and we thus make such deductions as 

(e + p + q - g) (e - p - q + g) = ; 



286 ON SCHUBERT S METHOD FOR THE [762 

that is, 

e*-(p + q-g)* = 0, 
or 

e 2 = (p + q - g)-, =p* + 2pq + q 2 - 2pg - 2qg + g" ; 

viz. this is an equation such as the original equation e = p + q g, acquiring a numerical 
signification when we adjoin to it a supplementary condition X of the proper postulation. 

The section above referred to deals with the question to determine the number of 
lines which satisfy the several relations of contact in regard to a given surface F of the 
order ??, without point-singularities, that is, the surface represented by the general 
equation (*$#, y, z, w) n = 0. 

The chief results are contained in the following table, the notation of which will be 
explained : 

1. 2 g s = n(n-l), 

2. e s bg e = n, 

3. e 3 g e = 3w(w-2), 

4. s g p = n(n-l)(n-2), 

5. e s & 3 2 = 2n, 

6. .g e = in(?i-2)(w-3)(n + 3), 

7. g p = iw(n-l)(n-2)(n-3), 

8. e 2 -A 2 = n (n - 3) (n + 2), 

9. e^Co = tt(n 3 -2n 2 + 2?i-6), 

10. e 4 g = 2?i(n-3)(3ro-2), 

11. e 4 6 4 = ??(ll?i-24), 

12. e 32 g = n (n - 3) (n - 4) (if- + 6n - 4), 

13. 3,63 = n (n - 4) (3rc 2 + 5?i - 24), 

14. 6 3 .,6 2 = ? ? ( ?l _2)(?i-4)(?i 2 + 2n + 12), 

15. e^g = ^n(n-3)(n-4)(n~5)(n+3n-2), 

16. 622363 = M( w ,-2)(?i-4)O-5)0 2 + 5/i+ 12), 

17. e 5 = 5w(w-4)(7w-12), 

18. e 42 = 2n (n - 4) (w - 5) (n + 6) (3n - 5), 

19. e ;53 = i (n - 4) (n - 5) (n 3 4- 3/i 2 + 29n - 60), 

20. 63,2 = %n(n-4&gt;)(n-o)(n-6)(n 3 +9n- + 20n-60), 

21. eoooo = T Vrc(ri-4)(w-o)(H-6)(?i-7)(y + 6n 2 + 7n-30), 

22. 622-6! = J(n-4)(n-5)(n-6)(n+8n*-2ii-12X 

23. 636^ = ( n -3)(n 2 +2), 

24. eo^Cj^ = ?i 2 (w - 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 2 denotes a 
2-pointic intersection, e 32 a 3-pointic and a 2-pointic intersection. The letters b, c, d refer 
to the points of contact or intersection, thus e 32 6 3 , b 3 is the point of 3-pointic intersection ; 
222^1, &i is one of the points of simple intersection; 6j is also the condition that the point 
in question lies on a given plane ; g, g s , g e , g p have their ordinary signification explained 
a little further on. Thus (15) 222*7 denotes the number of triple tangents which can be 
drawn to meet a given line; or, what is the same thing, it is the order of the regulus 
formed by the triple tangents. 

The following are elementary formulas used in the investigation of the foregoing 
results. 

Subject a line having upon it a point, 

Postal. 

p the condition that point is in a given plane 1 

p g 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 s 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. 
p g = p 2 2 (logical) 

Pg =P 2 + 9e 2 

g s = gg e 3 (logical) 

9s = gg p 3 (logical) 

Pffp = P 3 + 9s 3 (demons, infra) 

p 4 = 4 

g e g p = 4 

g e 2 = G 4 

.) s~y A 

^Jf) ^Jf J& 

p 3 g = p 2 g e 4 (demons, infra) 



Pffs = p*g e +G 4 

P 3 ffe =0 5 

p 3 g p =pG 5 

p*g s = pG O 



288 ox SCHUBERT S METHOD FOR THE [762 

P s + 9s , we have Q=g e +p z pg, 0=g e +g p g-, and thence 

o = (p + g) [g e +p(p- g)} -p [g e + g p - g 2 } 

= P9e + 99 e + P 3 ~ Pg* 

-P9e-P9 P 



P 3 g = p-g e - from pg=p*+g e , we have p 2 g = p* + p*g e = p*g e , since p* = 0, 
Pffs = p 2 g e + 0,, g s =gg e pg e =pgg e = (p- + ge)ge=p"g e + G, 

Pffs = p*g p g s = gg P pg* = pgg p = (P- + g e )g p = p*g p , since g e g p = o ; 

and in a similar manner we prove the last three equations. 

For the demonstration of the formulae of the table we take the subject to be a line 
bearing upon it the points 1, 2, ..., n, which are its intersections with a given surface of 
the order n. The symbols p lt p, ... refer to these points respectively; thus, p l is the 
condition that the point 1 may lie on a given plane ; and then, writing 



=pi+p 3 -g, 



it appears that e will denote the condition of the coincidence of the points 1 and 2 ; e 
that of the points 1 and 3, &c. Hence also, ee will denote the twofold condition of the 
coincidence of the points 1, 2, 3 ; and so in other cases. But, according to the notation 
above explained, e is also denoted by e 2 ee by e 3 , ee" by e 22 , &c. 

We thus have 

* 2 = pi + p-2-g, 

e a =(PI+ p-2 - g} (PI +PS- g\ 

2*22 = (PI + p., - g) (p 3 + p,- g), 

4 = (PI +p-2-g) (PI + p 3 -g) (PI +p t - g), 

**&gt; = (P! +p-2-g) (PI +p s -g) (P* +PS- g), 

Bezs = (p, +p 2 -g) (p 3 +p^-g) (p 5 +P S - g), 

e 5 = (PI +p 2 -g) (pi+pt-ff) (Pi+p* - g) (PI + p 5 - g), 

e 42 = (PI + p-2 - g} (PI + p 3 -g) (PI +p*-g) (p s + p s - g}, 



2e 322 = (p, + p, - g) (p, +p.-g) (p, +p 5 -g) (p s +p 7 - g), 

7 +p s -g}- 



J.1C1 J I ,t 
*~S 



762] CONTACTS OF A LINE WITH A SURFACE. 289 

We can now, by a mere analytical process of development and reduction, express 
each of the foregoing values as a linear function of 

Pi 2 p 2 2 , p*p a p, PiP&spi, and G. 
(Schubert says, as a linear function of these four symbols and pip. 2 g e ; but in fact Pip. 2 g e 



s = 



Observe, first, that we may, p. 287, in all the general equations instead of p write 
p 1} p 2 , &c. ; and, further, that any symbol containing for instance p^ is =0. For the 
symbols now belong to the intersections of the line with a given surface ; pf is the 
condition that a certain one of these intersections shall lie in three given planes, that is, 
that it shall coincide with a given arbitrary point; this cannot be the case, for the 
arbitrary point is not on the surface F; and therefore p 1 a = Q. 

We thus have Pig=pi 2 + g e , thence Pi*g=pi*+Pig e , that is, pfg=p 1 g e ; and thence 
further Pi s g=pi z g e , that is, p 1 2 g e = Q. 



Again, from p 2 g=p. 2 2 + g e , Pi9=p* + g e &gt; we have 

Pi*(p*+ge)=pip*(pi&gt; 

which, in virtue of p?g e = and pfp^ = 0, becomes 



As a simple instance of the reductions, take 

%ff,, = (pi+p-2 

Here 

Pigs, =p 2 g s , = pi*g e + G, = G, since p l *ff e = , 
and 

99 B = g*g e = (g e + g p ) g e = g&lt;? + g e g p = G, since g/ = 0, g e g p = G ; 

whence the value is 

e 2 g s =G + G-G, = G. 

As a more complicated example, take 



Observe that, after the multiplication is effected we may, in any way we please, 
interchange the suffixes, p l 2 p 3 p i =p 1 2 p 2 p s , p^pf^p^pj, &c. ; the suffixes serve only to 
distinguish from each other symbols in the same product (thus pf is different from 
PipzpspJ, but there is nothing to distinguish one point of intersection from another. 
Thus the foregoing expression containing the terms (p, +p 3 +p 4 + p 5 ) (p^ -g) 3 , these may 
be combined into the single term 4&gt;p 2 (p 1 g) 3 ; expanding in powers of p l -g and 
reducing in this manner, the value of e 5 is, in fact, found to be 

= (Pi ~ g}" + 4^ 2 (p 1 - g) 3 + 6p 2 p 3 ( Pl - &lt;7) 2 + 4&gt;p,p 3 p. ( Pl -g)+ Pl p a p 3 p t . 



Developing this in powers of g, omitting the terms containing p.? which vanish, and 
further reducing, the value is 



t) - Sp.g 3 
c. XL 37 



290 



ON SCHUBERT S METHOD FOR THE 



[762 



We have 

g 4 = 2G, p^g 3 =pig* =pi9e + G, =G. 

Next for the terms in g* t from p\gp\-\-ge. we have 

Pi 9= Piffe, 



and thence 



or, since p^g s = G as before, the whole term is = 18pfp 2 g + 24&gt;G. The terms in g thus 
become = g(6p 1 2 p 2 IQp^ps), and from the same equation Pig = p^ + g e we find 



P*P*ff =PiP 
The value is&gt; thus finally found to be 



The whole series of like results is 



7&gt;iV 



1- 2 9s 








+ 1 


2. ,, b,,ff e 


+ 1 






1 


3- e 3 #e 


3 






- 3 


4- g p 








+ 1 


5. &, 


2 


+ 1 




+ 1 


6. 2 ffl &lt;7 e 


+ 4 






- 3 


7 o 
^ &gt;? i/j) 








+ 1 


8. &, 


- 3 


+ 2 




+ 1 


9. e, 6 2 c 2 


2 




+ 1 


+ 1 


10. C 4 


- 2 


+ 4 




- 2 


11. b, 


- 6 




+ 1 


+ 4 


12. 30 # 


3 


+ 6 




- 2 


13. b s 


7 


1 


+ 2 


+ 4 


14. 6 2 


6 


- 3 


+ 3 


+ 4 


15. 6e 222 &lt;jr 


4 


+ 8 




- 2 


16. 2,, b. 


7 


4 


+ 4 


+ 4 


17. 5 


- 10 


- 10 


+ 5 


+ 10 


18. 


- 10 


- 16 


+ 8 


+ 10 


19. 2e 33 


9 


- 18 


+ 9 


+ 10 


20. 2 C322 


9 


- 24 


+ 12 


+ 10 


21. 24C0222 


- 8 


-32 


+ 16 


+ 10 


22. ee^ 6j 


- 6 


- 12 


+ 8 


+ 4 


^O 7 2 


3 


+ 3 




+ 1 


24. c 2 ftjCjC?! 


- 1 


- 1 


+ 2 





762] CONTACTS OF A LINE WITH A SURFACE. 291 



But in these formulae pip%, Pi 2 p 2 p 3 , Pip^psjh, G have numerical values which are 
different according to the number of points of intersection presenting themselves in 
the several formulae; viz. this number being called i, we have for the formula? in 



6 2 3 33 4 e 32 e 222 e 5 ^42 ^33 322 e 2222 ^222^1 e 3^1 

i= 23445656678 7 4 5, 
and the values of the symbols are 

pfp* =ri&gt;(n- 2) (n - 3) ...(n-i + 1), 

= n(TO-l)(TO-3) ...(n-i + 1), 

* = n* (2n* - 6n + 3) (TO - 4). . .(n - i + 1), 
G = n(n-l)(n-2) ...(n-i+1). 

Thus, suppose i = 4, the subject is a line bearing the points 1, 2, 3, 4, which are 
intersections of the line with the surface F; we have then G as the condition in 
order that this line (or, say, the line of the subject) may coincide with a given line, 
which given line intersects the surface in n points; any four of these (their order 
being attended to) may be regarded as being the points 1, 2, 3, 4 ; or there are 
(n l)(n 2)(n 8) subjects satisfying the prescribed condition (that the line of the 
subject may coincide with the given line). Hence here G = n (n 1) (n 2) (TO -3); 
and so in general G = n(n l)(n 2). . .(n - i + 1). 

Next, for pfpf. Here pf is the condition that the point 1 shall lie in each 
of two given planes, that is, in a given line, say L^; and, similarly, pf is the condition 
that 2 may lie in a given line L 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, 
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 x ; 
we therefore take for 1 any one of the n intersections of L^ with F; p s is the condition 
that 2 may lie in a given plane P 2 , 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 &lt;r, and the two cones, 
vertex 1, which stand upon the plane curves respectively, intersect in the n lines 
joining 1 with the n points &lt;r, and in TO S -TO other lines. The line of the subject is 
then any one of these n*-n lines, or, since the vertex is any one of n points, the 
line is any one of TO(TO S -TO), = ri&gt;(n-l) lines; the remaining points 4, 5, ..., i are 
any i-3 of the remaining TO -3 intersections of the line with F ; hence the formula 

Pi*p 2 p 3 = n* (n -l)(n- 3) (n - 4). ..(n - i + I). 

For p!p 9 p a 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 SCHUBERT S METHOD FOB THE [762 

intersections with F of the line of intersection of their planes, the order of the scroll 
generated by the lines which meet three of the sections is 2n 3 3w 2 ; this scroll meets 
the fourth section in n (2n 3 3w 2 ), = 2w 4 3n 3 points ; or we have this number of lines 
meeting each of the four sections. But among these are included 3% 2 (n 1) lines 
which have to be rejected, viz. the sections 1 and 4 meet in n points, each of which is 
the vertex of cones through the sections 1 and 2 respectively; these cones meet in n 
lines, which are to be disregarded, and in n 2 n other lines, and we have thus n (n 2 n), 
= n" (n 1) lines ; and similarly from the intersections of 2 and 4, and from the inter 
sections of 3 and 4, n z (n 1) and n z (n 1) lines, in all 3?i 2 (?i 1) lines. Hence the 
number of lines meeting the four sections is 

2n 4 - 3w 3 - 3n 3 + 3w 2 , = 2n* - Qn 3 + 3n? ; 

taking any one of these for the line of the subject, the remaining points 5, 6, ..., i are 
any i 4 of the remaining n 4 intersections, or we have the required formula 

* = n 2 (2rc 2 - Qn + 3) (n - 4). . .(n -i + l). 



The four numbers p^p/, Pi 2 p 2 p 3 , Pip 2 p 3 p 4 , G for any line of the table being now 
known, we can at once calculate the required values e 2 g s , &c., as the case may be ; for 
instance, 



- lOrc 2 (TO - 1) (n - 3) (n - 4) 
+ on a (2n 2 - Qn + 3) (n - 4) 



= on (-4)(7w-12). 

In fact, throwing out n(n 4), the remaining terms give 

-10n 3 + 50n 2 - 60n, 
-10n 3 +40?i 2 - 30n 
+ 10w 3 - 30w a + I5n 
+ Wn 3 - 60w 2 + 11 On - 60 



35n-60, = 5(7n-12). 
And we obtain in like manner the other formulae of the table. 

The remainder of 33 contains investigations of less systematically connected 
theorems, and I quote the results only. 

25. If on the surface 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; editor, 
class of torse is =r(n 1). 

In particular, for curve of 4-pointic contact, r = n(lln 24), class of torse is 



No. of tangent planes through line, or class of surface, =n(n-\)-. 



762] CONTACTS OF A LINE WITH A SURFACE. 293 

26. e 3 b 3 g = e 3 b 3 2 + 3 g e = 2n + 3n (n - 2), = n (3n - 4). 

3^3 ff, = n (3w 4), is the order of curve of contact of the 3-pointic (chief) tangents 
which meet a given line. 

Parabolic tangents are coincident chief tangents. 

No. of 4-pointic parabolic tangents = 2n (n 2) (lln 24). 

27. Order of parabolic curve = 4n (n 2). 

Order of regulus formed by parabolic tangents 

= 2n(n-2)(3w-4). 
The parabolic curve and curve of contacts of an e 4 tangent meet in 

4(*-2)(lln-24) 
points, i.e., they touch in 2n (n 2) (lln 24) points. 

28. Umbilici. No. is =2w(5n a - 14m + 11). 

29. No. of points at which the chief tangents being distinct are each of them 

4-pointic, or, what is the same thing, No. of actual double points of 
curve e 4 , 

= 5n (7w 2 - 28w + 30), 

n = 3, No. is 15 (63 - 84 + 30), = 135, viz. this is the number of points of 
intersection of two of the 27 lines ; or, what is the same thing, the number 
of triple tangent planes is =45. 

30. No. of parabolic tangents which have besides a 2-pointic contact is 

= 2n (n - 2) (n - 4) (3w 2 + on - 24). 

31. No. of double tangent planes such that line through points of contact is at one 

of these points 3-pointic 

= n (n - 2) (n - 4) (n 3 + 3?i 2 + 13n - 48). 

32. No. of points where one chief tangent is 4-pointic, the other 3-pointic and (at 

another point of the surface) 2-pointic is 

= n (n - 4) (27w 3 - 13w 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) (4w 5 - 4% 4 - 95w 3 + 99n a + 544ft - 840). 

34. The curve of contacts b 3 of an e 32 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 6 4 of 
an e 42 tangent, 2 at the points where the chief tangents are e 4 and e 32 . 



294 [763 



763. 

ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 

[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvn. (1881), 

pp. 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 



For the 4 squares (for which the nature of the theorem is better seen) it is 

(x? + x?- + ac 3 2 + x?) (y? + y 2 - + y s * + y?} = (x^ + x%* + x s y s + x t y t ) 2 



- # 4 2/i + #22/3 - 



763] ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 295 

or, as this may be written, 

(X 2 + x 2 2 + # 3 2 + x/} (y-c + y 2 2 + yf + ?/ 4 2 ) - (x^ + x 2 y 2 + x 3 y 3 + x.y,)- 

= (12 + 34) 2 
+ (13 - 24) 2 
+ (14 + 23) 2 ; 

where 12 is used to denote x^j* x^, &c., and the truth of the theorem depends 
on the identity 12.34-13.24 + 14.23=0. Clearly, the first step for forming the 
equation is to arrange the duads in a synthematic form 

12.34 
13 . 24 
14.23, 

and then to determine the signs : such an arrangement exists in the case of 8, and 
the signs can be determined; it exists also in the case of 16, but the signs cannot 
be determined to satisfy all the necessary relations. 

In the case of 8, we have the synthematic arrangement 

12.34.56.78 
13.24.57.68 
14.23.58.67 
15.26.37.48 
16.25.38.47 
17.28.35.46 
18.27.36.45, 

being the only type of synthematic arrangement. This is, in fact, important as regards the 
16-question, and it will appear that the case is so ; but in the 8-question, starting from 
this arrangement, we have to show that there exists an equation which, for convenience, 
I write as follows : 



(x* + . . . + x&lt;?) (y? + . . . + */ 8 2 ) - (awi + - - - + fftfs) 8 
= (12 + 34 + 56 + 78) 2 
+ (13 + 24 + 57 + 68) 3 
+ (14 + 23 + 58 + 67) s 
+ (15 + 26 + 37 + 48) 2 
+ (16 + 25 + 38 -4- 47) 2 



296 ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. [763 

but in which it is to be understood that each duad is affected by a factor + 1 
which is to be determined; say the factor of 12 is e 12 , that of 34, e^; and so in 
other cases. It is however assumed that 6 12&gt; e^, e x , e 78 ; eu, e u , e l5 , e 16 , e 17 , e 18 are 
each =+1. 

We have then on the right-hand side triads of terms such as, 2 into 

6,3634 12 . 34 4- 6 13 e 24 13 . 24 + 14 . 23, 
which triad ought to vanish identically, as reducing itself to a multiple of 

12.34- 13.24 + 14.23; 
viz. we ought to have 

12 e 34 == ^13 e 24 = e 14^23 j 

or, using now and henceforward when occasion requires, 12, 34, &c. to denote e 12 , 634, &c. 
respectively, we have 

12.34-+*, 

13 . 24 = - k, 



where k, = + 1, has to be determined (in the actual case we have 12 = 4-1, 34 = 4-1, 
13 = 1, 14=1; and therefore the first equation gives k=l, and the other two then give 
24 = - 1, 23 = 4- 1). 

We have in this way triads of values corresponding to the different tetrads 

1234 
. 1256 
1278 
1357 
1368 
1458 
1467 
2358 
2367 
2457 
2468 
3456 
3478 
5678, 

which can be formed with the several lines of the formula. Thus we have from the 
first line 1234, 1256, 1278; then from the second line (not 1324 which in the form 
1234 has been taken already) 1357, 1368, ...; and finally from the last line 5678. 



763] 



ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 



297 



We might consider each line as giving 6 tetrads, but the tetrads would then be 
obtained 3 times over ; the number of tetrads is thus 6 x 7 -f- 3, =14 as above. 
And observe, that the systems of values for the coefficients e = 1 are obtained 
directly from the tetrads, without the employment of any other formula. 

We thus obtain the system of signs as follows : 



12 


+ 1 


13 


+ 1 


14 


+ 1 


15 


+ 1 


16 


+ 1 


17 


+ 1 


18 


+ 1 


23 


+ 1 


24 


-1 


25 


+ 1 


26 


-1 


27 


+ 1 


28 


-1 


34 


+ 1 




35 


a 


-e 


36 


b 


e 


37 


a 


e 


38 


-b 


-e 


45 


c 


e 


46 


d 


e 


47 


-d 


-e 


48 


c 


-e 


56 


+ 1 




57 


a 


-e 


58 


c 


e 


67 


d 


e 


68 


b 


e 


78 


+ 1 





C. XI. 



38 



298 ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. [763 

viz. the original assumptions 12 = + 1, &c., and the tetrads 1234, 1256, 1278 give all 
the signs +1 up to 34 = + 1 ; from the tetrad 1357 we have 

13.57 + 1 a, 

15.37 - 1 a, 
17.35 + 1 a, 

that is, 35 = a, 37 = a, 57= a, where a, = 1, is still undetermined; and similarly, 
the tetrads 1368, 1458, 1467 give the remaining signs b, c, d. The tetrad 2358 then 
gives 

23.58 + 1 c, 

25.38 - 1-6, 
28.35 +-1 a, 

that is, a = b = c ; and similarly the tetrads 2367, 2457, 2468 give a = b = d, 
a = c = d, b = c = d respectively ; the four tetrads thus give a = b = c = d, say each 
of these = 6. But retaining for the moment a, 6, c, d, the tetrad 3456 then gives 

34.56 +11, 
35.46 - a d, 
36.45 + b c, 

that is, 1 = ad = be, and similarly the last two tetrads 3478 and 5678 give 
1 = ac = bd and 1 = ab = cd respectively ; substituting the values in terms of 6, 
the several equations give only &* 1, that is, 6 = I at pleasure ; and the series of 
signs for the 8-formula, containing this one arbitrary sign #= + 1, is thus determined. 

Passing to the case of 16, we have in like manner to form a synthematic arrange 
ment of the numbers 1, 2, ..., 16 in 15 lines, each containing the 16 numbers in 8 duads 
(no duad twice repeated), and this containing all the 120 duads. And, using for the 
moment letters instead of numbers, the necessary condition is, that ab . cd occurring in one 
line, ac . bd must occur in another line, and ad .be in a third line. Observe that as well 
the order of the letters in a duad as the order of the duads is thus far immaterial; so 
that a line containing bd . ca may be considered as containing ac . bd. 

Considering any such combination ab . cd, the line which contains it may be 
taken to be the first line ; and the line which contains ac . bd may be taken to be 
the second line. And then writing 1, 2, 3, 4 in place of a, b, c, d respectively, the 
first line will contain 12.34, and the second line will contain 13.24. Let e be any 
other symbol occurring in the first line, say in the duad ef, and in the second line 
say in the duad eg; then g must occur in the first line in some duad gh, or the 
first line will contain ef.gfi, and then the second line as containing eg will contain 



763] 



ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 



299 



also fh, that is, it will contain eg . fh. And then writing 5, 6, 7, 8 in place of e, f, 
g, h respectively, the first line will contain 56 . 78 and the second line will contain 
57 . 68. And continuing the like reasoning, it appears that the first line and the second 
line may be taken to be 



and 



12.34.56.78.9 10. 11 12. 13 14. 15 16, 
1 3. 2 4. 5 7. 6 8. 9 11. 10 12. 13 15. 14 16, 



respectively. There will then be a line containing 1 4 which may be taken for the 
third line, a line containing 1 5 which may be taken for the fourth line, and so on ; 
viz. the successive lines may be taken to begin with 1 2, 1 3, 1 4, ..., 1 16 respectively. 

Proceeding to form the synthematic arrangement, and starting with the first and 
second lines and first column as above, it appears that in each of the remaining 
lines there are three duads which occur of necessity, and putting these in the second, 
third, and fourth places (the order of the duads in any line being immaterial), it is 
seen that the second, third, and fourth columns can be filled up in one, and only 
one way ; see the annexed first-half : 

First-half common to all. 



1 2 


3 4 


5 6 


7 8 


1 3 


2 4 


5 7 


6 8 


1 4 


2 3 


5 8 


6 7 


1 5 


2 6 


3 7 


4 8 


1 6 


2 5 


3 8 


4 7 


1 7 


2 8 


3 5 


4 6 


1 8 


2 7 


3 6 


4 5 


1 9 


2 10 


3 11 


4 12 


1 10 


2 9 


3 12 


4 11 


1 11 


2 12 


3 9 


4 10 


1 12 


2 11 


3 10 


4 9 


1 13 


2 14 


3 15 


4 16 


1 14 


2 13 


3 16 


4 15 


1 15 


2 16 


3 13 


4 14 


1 16 


2 15 


3 14 


4 13 



382 



300 



ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 

Four forms of second-half. 



[763 



I. 



9 10 


11 12 


13 14 


15 16 


9 11 


10 12 


13 15 


14 16 


9 12 


10 11 


13 16 


14 15 


9 13 


10 14 


11 15 


12 16 


9 14 


10 13 


11 16 


12 15 


9 15 


10 16 


11 13 


12 14 


9 16 


10 15 


11 14 


12 13 


5 13 


6 14 


7 15 


8 16 


5 14 


6 13 


7 16 


8 15 


5 15 


6 16 


7 13 


8 14 


5 16 


6 15 


7 14 


8 13 


5 9 


6 10 


7 11 


8 12 


5 10 


6 9 


7 12 


8 11 


5 11 


6 12 


7 9 


8 10 


5 12 


6 11 


7 10 


8 9 



III. 



9 10 


11 12 


13 14 


15 16 


9 11 


10 12 


13 15 


14 16 


9 12 


10 11 


13 16 


14 15 


9 15 


10 16 


11 13 


12 14 


9 16 


10 15 


11 14 


12 13 


9 13 


10 14 


11 15 


12 16 


9 14 


10 13 


11 16 


12 15 


5 15 


6 16 


7 13 


8 14 


5 16 


6 15 


7 14 


8 13 


5 13 


6 14 


7 15 


8 16 


5 14 


6 13 


7 16 


8 15 


5 11 


6 12 


7 9 


8 10 


5 12 


6 11 


7 10 


8 9 


5 9 


6 10 


7 11 


8 12 


5 10 


6 9 


7 12 


8 11 



II. 



9 10 


11 12 


13 14 


15 16 


9 11 


10 12 


13 15 


14 16 


9 12 


10 11 


13 16 


14 15 


9 14 


10 13 


11 16 


12 15 


9 13 


10 14 


11 15 


12 16 


9 16 


10 15 


11 14 


12 13 


9 15 


10 16 


11 13 


12 14 


5 14 


6 13 


7 16 


8 15 


5 13 


6 14 


7 15 


8 16 


5 16 


6 15 


7 14 


8 13 


5 15 


6 16 


7 13 


8 14 


5 10 


6 9 


7 12 


8 11 


5 9 


6 10 


7 11 


8 12 


5 12 


6 11 


7 10 


8 9 


5 11 


6 12 


7 9 


8 10 



IV. 



9 10 


11 12 


13 14 


15 16 


9 11 


10 12 


13 15 


14 16 


9 12 


10 11 


13 16 


14 15 


9 16 


10 15 


11 14 


12 13 


9 15 


10 16 


11 13 


12 14 


9 14 


10 13 


11 16 


12 15 


9 13 


10 14 


11 15 


12 16 


5 16 


6 15 


7 14 


8 13 


5 15 


6 16 


7 13 


8 14 


5 14 


6 13 


7 16 


8 15 


5 13 


6 14 


7 15 


8 16 


5 12 


6 11 


7 10 


8 9 


5 11 


6 12 


7 9 


8 10 


5 10 


6 9 


7 12 


8 11 


5 9 


6 10 


7 11 


8 12 



And it is to be noticed that in this first-half the upper part, or first seven 
lines, give in fact the synthematic arrangement for the 8-question ; so that (as 
remarked above) in this 8-question there is but one form of synthematic arrangement. 

Proceeding to fill up the remaining columns, the duad 59 cannot be placed in 
any line which contains a 5 or a 9 ; that is, it must be placed in some one of the 



763] 



ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 



301 



last 4 lines; and placing it successively in each of these, it appears that the columns 
can be filled up in one, and only one, way ; we have thus the above " four forms 
of second-half," each of which, taken in conjunction with the common first-half, gives 
a synthematic arrangement of the 16 numbers. 

Each of these synthematic arrangements may be converted into a square, the 
first line of which is formed with the numbers 1 to 16 in order, and the other 
fifteen lines of which are derived from the fifteen lines of the synthematic arrange 
ment respectively : thus the line 

1 2. 3 4. 5 6. 7 8. 9 10. 11 12. 13 14. 15 16 
gives the second line of 

1 2. 3 4. 5 6. 7 8. 9 10. 11 12. 13 14. 15 16, 

2 1. 4 3. 6 5. 8 7. 10 9. 12 11 . 14 13. 16 15, 

and so in other cases. And conversely, by comparing with the first line of the 
square each of the other fifteen lines respectively, we have the fifteen lines of the 
synthematic arrangement ; we thus obtain the four squares presently given. These 
squares are not required in the sequel, but they serve to put in a clearer light 
the construction of the synthematic arrangements ; by converting in like manner into 
a square the formula p. 332 of Young s paper, it appears that his arrangement is in 
fact the first of the foregoing four arrangements. The squares are 



1234 
2143 
3412 
4321 


5678 
6587 
7856 
8765 


9 10 11 12 
10 9 12 11 
11 12 9 10 
12 11 10 9 


13 14 15 16 
14 13 16 15 
15 16 13 14 
16 15 14 13 


5678 
6587 
7856 
8765 


1234 
2143 
3412 
4321 


13 14 15 16 
14 13 16 15 
15 16 13 14 
16 15 14 13 


9 10 11 12 
10 9 12 11 
11 12 9 10 
12 11 10 9 


9 10 11 12 
10 9 12 11 
11 12 9 10 
12 11 10 9 


13 14 15 16 
14 13 16 15 
15 16 13 14 
16 15 14 13 


1234 
2143 
3412 
4321 


5678 
6587 
7856 
8765 


13 14 15 16 
14 13 16 15 
15 16 13 14 
16 15 14 13 


9 10 11 12 
10 9 12 11 
11 12 9 10 
12 11 10 9 


5678 
6587 
7856 
8765 


1234 
2143 
3412 
4321 



302 



ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 

II. 



12345 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 16 


21436 


5 


8 


7 10 


9 


12 


11 


14 


13 


16 15 


3412 


7 


8 


5 


6 11 


12 


9 


10 


15 


16 


13 14 


4321 


8 


7 


6 


5 12 


11 


10 


9 


16 


15 


14 13 


5678 


1 


2 


3 


4 14 


13 


16 


15 


10 


9 


12 11 


6587 


2 


1 


4 


3 13 


14 


15 


16 


9 


10 


11 12 


7856 


3 


4 


1 


2 16 


15 


14 


13 


12 


11 


10 9 


8765 


4 


3 


2 


1 


15 


16 


13 


14 


11 


12 


9 10 


9 10 11 12 


14 


13 


16 


15 


1 


2 


3 


4 


6 


5 


8 7 


10 9 12 11 ; 13 


14 


15 


16 


2 


1 


4 


3 


5 


6 


7 8 


11 12 9 10 16 


15 


14 


13 


3 


4 


1 


2 


8 


7 


6 5 


12 11 10 9 


15 


16 


13 


14 4 


3 


2 


1 


7 


8 


5 6 


13 14 15 16 


10 


9 


12 


11 


6 


5 


8 


7 


1 


2 


3 4 


14 13 16 15 


9 


10 


11 


12 5 


6 


7 


8 


2 


1 


4 3 


15 16 13 14 


12 


11 


10 


9 ! 8 


7 


6 


5 


3 


4 


1 2 


16 15 14 13 


11 


12 


9 


10 


7 


8 


5 


6 


4 


3 


2 1 



III. 



1234 
2143 
3412 
4321 


5678 
6587 
7856 
8765 


9 10 11 12 
10 9 12 11 
11 12 9 10 
12 11 10 9 


13 14 15 16 
14 13 16 15 
15 16 13 14 
16 15 14 13 


5678 
6587 
7856 
8765 


1234 
2143 
3412 
4321 


15 16 13 14 
16 15 14 13 
13 14 15 16 
14 13 16 15 


11 12 9 10 
12 11 10 9 
9 10 11 12 
10 9 12 11 


9 10 11 12 
10 9 12 11 
11 12 9 10 
12 11 10 9 


15 16 13 14 
16 15 14 13 
13 14 15 16 
14 13 16 15 


1234 
2143 
3412 
4321 


7856 
8765 
5678 
6587 


13 14 15 16 
14 13 16 15 
15 16 13 14 
16 15 14 13 


11 12 9 10 
12 11 10 9 
9 10 11 12 
10 9 12 11 


7856 
8765 
5678 
6587 


1234 
2143 
3412 
4321 



763] 



ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 

IV. 



303 



1234 
2143 
3412 
4321 


5678 
6587 
7856 
8765 


9 10 11 12 
10 9 12 11 
11 12 9 10 
12 11 10 9 


13 14 15 16 
14 13 16 15 
15 16 13 14 
16 15 14 13 


5678 
6587 
7856 
8765 


1234 
2143 
3412 
4321 


16 15 14 13 
15 16 13 14 
14 13 16 15 
13 14 15 16 


12 11 10 9 
11 12 9 10 
10 9 12 11 
9 10 11 12 


9 10 11 12 
10 9 12 11 
11 12 9 10 
12 11 10 9 


16 15 14 13 
15 16 13 14 
14 13 16 15 
13 14 15 16 


1234 
2143 
3412 
4321 


8765 
7856 
6587 
5678 


13 14 15 16 
14 13 16 15 
15 16 13 14 
16 15 14 13 


12 11 10 9 
11 12 9 10 
10 9 12 11 
9 10 11 12 


8765 
7856 
6587 
5678 


1234 
2143 
3412 
4321 



The foregoing investigation of the synthematic arrangements is exhaustive : it 
thereby appears that there are at most four types, viz. that every synthematic 
arrangement is of the type of one or other of the four arra