156 
PROFESSOR CAYLEY’S SECOND MEMOIR ON THE 
115. Second equation : 
=0 = 
2(5) 
+( 4 , 1 ) 
+ {Supp. (4, 1)— *(4, 1)} 
+ (4, l)(6m + 5ra-3a) 
-(■ 1 , 3 ) 
— 30m — 30 w+a( 18) 
-8m 2 -20mft-8rc 2 +104m+104w+a( 6m+6rc-66) 
— 6m 2 — 3 mn + 18m + 9w+a( 3m — 9) 
+ 6m 2 +llmw + 5w 2 — 36m— 30w+a(— 3m— 3w+18) 
+ 8m 2 +12wm+3w 2 — 56m— 53w+a(— 6m— 3w+39). 
I stop for a moment to notice a very convenient verification of the term in { } ; putting 
therein a=3 n, the term is 
— 6m 2 — 3wm+18m+9w+(9mw— 27w) ; 
and if in this we write m=n=l, m 2 =w»=w 2 = 2, and when any higher terms enter 
m 3 =m 2 w=mw 2 =w 3 = 4, m 4 =m 3 « =mV=mw 3 =?i 4 = 8, &c., the value is —12 — 6 + 18 
+ 9+18 — 27, =0, viz. we should always obtain a sum =0. The reason is that the 
term in question should always admit of being expressed in the form ^+<^+rr + s/ ; 
the reduction to this form might be effected by the substitutions m=^(m+w)+- g-(* — /), 
w,=i(m+w)— jr(z— /), m 2 =2.-|(m+w)+2^+3^, n 2 =2.-|-(m+w) + 2r+3;, giving a result 
=A(m+w)+terms in (h, x, r, /), where A is a numerical coefficient calculable as above 
by simply writing m—n— 1, m 2 =mM=w 2 = 2, &c., and which is =0 when the term is of 
the proper form ph-\-qK-\-rr-{-SL. The complete reduction to the form in question is 
material in the sequel, but I advert to the point here only for the sake of the numerical 
verification. 
116. Third equation : 
3(5) 
+(3, 2) 
+ {Supp. (3,2)-*(3, 2)} 
+ (3, 2)(4m+3w— 2a) 
— 2 ( * 2 , 2 ) 
— 45m— 45 %+a( +27) 
+ 120m+120w+a( — 4m— 4w— 78)+3a 2 
+ 15m +a( n— 7) 
— 36m— 27w+a( 4m+3w + 18) — 2a 2 
— 54 m~ 48 w+a( +40)— a 2 . 
Verification is 15+3(1. 2 — 7)=0. 
117. Fourth equation : 
2(4, 1) 
+ (3, 1, 1)_ 
+ {Supp. (3, 1, 1)— *(3, 1, 1)} 
+(3, 1, l)(4m + 3w— 2a) 
-(. 2 , 1 , 1 ) 
— 16m 2 — 40mw— 1 6% 2 a) 
-|m 3 -10m 2 w-10mw 2 — |w 3 + i f%t 2 + 116m%+if% 2 ( 2 ) 
— ^m 3 + \m 2 n-\- 2mn 2 + ^m 2 + 1 ^mn— 7 n 1 o) 
2m 3 +^-m 2 w+ 8m% 2 + |w 3 — 26m 2 — ^mn— - 2 % 2 o) 
— 24m 2 — 3 Qmn— 12% 2 (5) 
