336 
ADDITION TO THE MEMOIR ON THE 
[692 
and thence 
- ^ (ca' - c'a) = u 12 + u* (60 - 1O0 3 + lid 4 - 6d 5 - 8d 8 + 10d 7 - 4d 8 ) 
Similarly we have 
+ v* (- 4 +lOd - 8d 2 - 6d 3 + lid 4 - 10d 5 + 6d 7 ) + v 12 . 
^ (be' - b'c) = u* 2 (5-50 + 4d 2 - 5d 3 + 2d 4 ) + u* (9d - 16d 2 + 0 3 + 10d 4 + d 8 - 16d 6 + 9d 7 ) 
+ v* (2 - 50 + 4d 2 - 50 s + 5d 4 ), 
^ (ah' - a'b) = u* (0 + 0 s - d 4 ) + v* (2 - 50 + 4d 2 + 3d 3 - 10d 4 + 3d 5 + 4d 8 - 5d 7 + 2d 8 ) 
+ w 12 (-l + d+d 3 ) ; 
say these values are 
u* 2 +pu* + qv* + v 12 , Xu 13 + /jlu* + vv*, pu 4 + cv* + tv* 2 . 
The required equation is thus 
0 = (u n + pu* + qv* + v 12 ) 2 — 4 (Am 12 + pu* + vv*) (pu* + <rv* + tv* 2 ), 
viz. the function is 
or say it is 
u** 
+ u 11 ' (2p — 4Xp) 
+ u 8 (2q0* +p 2 — 4Xcr0* — 4pp) 
+ (2d 12 + 2pq0* - 4\Td 12 - 4pa0* - 4z>p0*) 
+ v 8 (2\p0* 4- q~ — 4<pT0* — 4z/cr) 
+ w 16 (2^ — 4 z/t) 
+ v 24 , 
= (1, 6, c, d, e, f l\u 2 *, u* 8 , u 8 , 1, w 8 , w 1G , v 24 ). 
Supposing that this has a factor u 8 — ® + v 8 , the form is 
O 16 + Bu 8 + G+ Dv 8 + w 16 ) (u 8 — © + v 8 ) ; 
and comparing coefficients we have 
5-0 =b, 
(7-05 + d 8 — c, 
D0 8 — 0(7 + B0 8 — d, 
d 8 - ©i) + G =e, 
_© + 5 =/, 
where © has the before-mentioned value 
= (8, -28, +56, -70, +56, -28, + 8£d, d 2 , 0 s , 0*, 0\ 0\ d 7 ). 
From the first, second, and fifth equations, 5 = 6 + 0, (7 = c + ©5 —d 8 , T)=f+%\ and