166 
PEOFESSOE CAYLEY’S SECOND MEMOIE ON THE 
*( 2 , 1 , 1 , 1 )= 
1 
M *- 1 
T 
3 mV— 3 mw 3 — |^ 4 +^m 2 w4- 24wm 2 + i ^m 
‘ — 37mw— 
37rc 2 
1 
Os|hf>. 
> 
II 
1 
■^m 4 
4 - fm 3 — %m 2 n 
+ fm 2 4 - fwm— 
¥^ 2 
— fm 3 w 
+ ^m 3 4 - ^m 2 n — ^rnn 2 
— ^m 2 4-^m^4- 
^ 2 
II 
Q 
•^|w 
1 
— 
\m?n 2 — fm 3 4 - \rrfn-\- %mn 2 
4 - 2 4m 2 — Smn— 
-2D = 
3m 3 
-21 m 2 
- D' = 
fm 
1 
nf ^ 2 
— j-m 4 — ^m a n — 2 £m 2 n 2 — 3wm 3 — \it + 4m 3 + ~^m 2 n + ym» 2 + 9n 3 — ^m 2 — 1 4mn — -f %z 2 
+ 2 2 5 w + a(fm 3 + m 2 w + mn 2 + — fm 2 — fnrn + 2 n 2 + -n — 7 5 ) 4- a 2 ( — f m — f n 4 - - 2 -) 
fm4~ |w + a( w 2 — m+ 8w— l) + a 2 ( — f) 
(3) + ^f%+a( 
<4) — Y|pm4- ^^4-a( 
(■>) 4 - 36m 4-«( 
( 6 ) 4- 18w4-«( 
4-fmw 
—10m— 10^4-40) 
— 9m 4-20) 
4 - 7m -12) 
+ 
6 ) 
— i 3 % 4 - 1 f lM +“(i m 3 + TO!!}l 4 -» 2 + 6 fi3- f w& 2 — ww4-f^ 2 — fw— 34)4-a 2 (— fm— f 4-13) 
(twelfth equation) 
129. We have consequently, by means of the results just obtained, 
Supp. (5) = a(5) 
+N 
Supp. (4", 1) 
= *(4,1) 
4-2J 4-R . . . 
Supp. (3, 2) 
= *(3,- 2) 
4-6K4-L4-3N+20 
Supp. (3, 1, 1) 
= *(3,1,1) 
4-D+E4-F+2G4-3J4-J' . . . 
Supp. (2, 3) 
= *(2,3) 
4-Q • 
Supp. (2, 2, 1) 
= *( 2 , 2 , 1 ) 
+ 3G4_I4_4J + 2J' 
Supp. ( 2 , 1, 1, 1) = <2, 1,1,1) 
-4-B4-4C+4D4-D' ....... (seventh equation) 
