478 REPORT— 1880. 



but substituting for the sn's by (1), and equating tlie coefScient of k^ to zero, we 

 have 



sin ^ sin y sin (/3 — y) [cos -/3 + cos -y + cos ^(^ — y) ] 



+ sin y sin a sin (y — a) [cos -y + cos "a + cos ~(y — a) ] 

 + sin a sin /3 sin (a — ^) [cos -a + cos -/3 + cos ''(a — /3) ] 

 + sin (/3 - y) sin (y - a) sin (a -/3) [cos^(/3 - y) + cos'^(y-a) + cos^(a - /3) ] = 0, 

 ■wbich, in virtue of (2), may be written 



sin j3 sin y sin O — y) [siQ-/3 + sin^y + sin^O — y)] 

 + sin y sin a sin (y - a) [sin ^y + sin ^a + sin '^(y — a)] 

 + sin a sin /3 sin (a — (3) [sin ^a + sin -(3 + sin*(a — /3)] 

 + sin (/3 — y) sin (y - a) sin (a - /3) [sin ^((3 - y) + sin^(y — a) + ain*(a - ^)] 



= 0...(3) 



Similarly from 



sn a sn (/3 — y) + sn ^ sn (y — a) + sn y sn (a — 0) 



+ A;^ sn a sn /3 sn 7 sn (/3 — y) sn (y — a) sn (a — /3) = 



we have, by putting k = 0, 



sin a sin O - y) + sin /3 sin (y — a) + sin y sin (a — ^) = 0, 



and, by equating to zero the coeiBcient of k", 



sin a sin (/3 — y) [sin ^a + sin ^(/3 — y) ] 



+ sin /3 sin (y — a) [sin -/3 + sin -(y - a) ] 



+ sin y sin (a — /3) [sin "y + sin -(a — /3) ] 



- 4 sin a sin /3 sin y sin (/3 - y) sin (y — a) sin (a — /3) = 



If the object be, not to deduce trigonometrical formulae from elliptic function 

 formulae, but to verify the latter, the formulae deduced by equating to zero the 

 coefficient of k'' obtained by means of (1), generally afford a much better verifica- 

 tion than is obtained by merely putting k = 0. 



It may be mentioned that (3) may be easily verified by use of (2) and of the 

 formula 



sin'.r + sin"y.+ sin^(.r — y) = 2 - 2 cos .r cos «/ cos (i' — y). 



10. On Plane and S^lierical Curves of the Fourth Glass with Quadruple Foci. 

 By Henry M. Jeffery, F.B.8. 



I. On Plane Olass-Quaetics. 



1. All quartics with quadruple foci may be expressed by the geometrical rela- 

 tion 



Kp* = qr + \ 



if the line-coordinates p, q, r denote the quadruple focus P, and Q, R the foci of 

 the satellite-conic. 



It is proposed to examine every possible quartic in a group, in which P, Q, R 

 remain unaltered, while the parameters, k, X, vary indefinitely. 



2. When there are critical bitangential quartics in a group, the mutual relation 

 of K, X will be exhibited in a plane curve, of which they are the coordinates. 



This locus will be hereinafter designated the bounding curve, by which plane 

 space will be divided into regions. In some regions no quartic is possible, and if 

 K, X represent points on the bounding curve, critical quartics exist, with real or 

 imaginary bi-tangents. If two branches intersect in a node or imite in a cusp, two 

 bi-tangents will unite to form some higher singularity. In the remaining regions 

 quartics will occur, which alter their character as k or X becomes zero, i.e. as the 



