615] 
ON THE CONIC TORUS. 
521 
where 
H' — x?y 3 (1 — m 2 ) 
4- {(3 4- 8in- 4- m 4 ) z 2 + (— 3 + m 2 ) w 2 } 
4- xy {(3 + 11m 2 + 9m 4 + m 6 ) z 4 4- (— 6 — 12m 2 4- 6m 4 ) z 2 w 2 + (3 + m 2 ) tv 4 } 
4- (14- m 2 ) {(1 — m 2 ) z 2 — w 2 ) {(1 4- m 2 ) z 2 — w 2 } 2 , 
giving without much difficulty 
H' — z s (1 4- m 2 ) 3 (1 — m 2 ) 
+ 2Z 4 [(1 + 4m 2 -I- m 4 ) xy — (1 — m 4 ) w 2 ](1 + m 2 ) 
4- z 2 (xy — w 2 ) [(14- 12m 2 — m 4 ) xy — (1 — m 4 ) w 2 ] 
+ [(1 — m 2 ) xy — (1 + m 2 ) w 2 ] U; 
say this is 
where 
= z 2 H" + [(1 — m 2 ) xy — (1 + m 2 ) w 2 ] U, 
H" = z 4 (1 4- m 2 ) 3 (1 — m 2 ) 
4 2^ 2 (1 4- in 2 ) [(1 4- 4m 2 4 m 4 ) xy — (1 — m 4 ) w 2 ~\ 
4- (¿ry — w2 ) [(1 + 12??i 2 — m 4 ) xy — (1 — m 4 ) w 2 ], 
or, what is the same thing, 
H" = x 2 y 2 (1 + 12m 2 — m 4 ) 
+ 2xy [(1 + 4m 2 + m 4 ) (1 + m 2 ) z 2 4 (— 1 4 6m 2 ) w 2 ] 
+ (1 — m 4 ) {(1 + m 2 ) ^ 2 — tv 2 ] 2 . 
It consequently appears that the complete spinode curve or intersection of the 
quartic surface and its Hessian, being of order 4x8,= 32, breaks up into 
that is, 
and 
that is, 
U= 0, xy + (1 — m 2 ) z 2 — w 2 = 0, 
conic z — 0, xy — w 2 = 0 twice, order 4 
conic z + w = 0, xy — m 2 w 2 = 0, „ 2 
conic z — tv = 0, xy — m 2 w 2 = 0, „ 2 
H=0, z 2 H" = 0, 
U=0, z 2 — 0, conic 2 = 0, xy — w 2 = 0 four times, „ 8 
proper spinode curve U = 0, H" = 0, „ 16 
32; 
viz. the intersection is made up of the conic z = 0, xy — w 2 — 0 six times, the conics 
z±w = Q, xy — m 2 w 2 = 0 each twice, and the proper spinode curve of the order 16. 
C. IX. 66