422 
PROFESSOR A. CAYLEY ON THE 
say these values are 
u 1 ~-\-pu i '-\-qv 4 '-\-v l ' :i , Xtd 3 + yid + w^, pii 4 + crtd+ry 13 . 
The required equation is thus 
0 = (w 1 3 +y>ii 4 + qy 4 + v u )~ — 4 (Aid 3 + /rii 4 + v y 4 ) (p a 4, + cry 4 + ry 13 ) , 
viz., the function is 
R 34 
or say it is 
+'ii 16 (2p — 4Xp) 
+ n 8 (2q9 l '-\-p 2 — 4Xcrd 4 — 4/qo) 
+ (2# 13 + 2 pqO 1 ' — 4Xr+ 3 — 4pcr6 i — 4vp9 { ) 
+ y 8 (2 p9 l -\-q 2 — 4 pr9 [ ' — 4ycr) 
-{-v 10 (2q — 4 vt) 
+ v°\ 
= {l,b, c, cl, e,f IX# 4 w 16 , u\ 1, v s , v l() , y 34 ). 
Supposing that this has a factor ?i 8 — © + y 8 , the form is 
(w 16 +Bw s +C+ Dy 8 + y 16 ) (n 8 — © 4- v 8 ) ; 
and comparing coefficients we have 
B — © = b, 
C— ©B+ 9 s =c, 
D d s — © C + B 9 8 = d, 
9 s — ©D + C —e, 
-© +d =/; 
where © has the before-mentioned value 
= (8, -28, +56, -70, +56, -28, + 8X#, 0\ 9\ 9\ 9\ 9 Cl , 9 7 ); 
from the first, second, and fifth equations, B=6 + @, C = c + @B — 9 s , D=++@; and 
the third and fourth equations should then be verified identically. Writing down the 
coefficients of the different powers of 9 we find 
2p = 0+ 12 0 — 20 + 22 — 12 — 16 + 20 — 8(0° . . . 9 s ) 
— 4Xp=0 — 20 + 20 — 36 + 60 — 44 + 36 — 28 + 8 
b = 0— 8 + 20 — 56 + 82 — 56 + 20— 8 0 
0 = 0+ 8-28 + 56-70 + 56-28+ 8 0 
• \B=0 0— 8 0 + 12 0- 8 0 0 
