WHICH SATISFY GIVEN CONDITIONS. 
115 
the other lines are of course expressible in terras of (a, (3, y, S), but as the law of their 
formation would then be hidden, I abstain from completing the reduction. 
82. The series of formulae is 
[a] = ( a-\-l)rm 
+(a+l)aA 
[a, b~\=-(a-{- l)(6+l)( a +l) . . rm 
-(a+l)(ft+l)f «(«+ 1)1 A 
1 - 
+ r 2%+i)(j+i)p, 
— ab 
where a=a-\-b, (3 —ab; and coeff. of z expressed in terms of a , (3 is=a(l -\-u-\-(3)— /3. 
[«,M]= (a+l)(&+l)(c+lX*+l)(«+2) .‘.m 
+(a+l(«+l)(c+l)[ *(*+l)(«+2) | A 
(3(a+2) 
+ f-X c(a+l)(b+l)(c+l)(a+2) 
J+^+c+2)(a+l) 
1 
where a=a-\-b-{-c, (3 =ab-{-ac-j-bc, y=abc ; and the coefficient of z expressed in term& 
of a, (3, y is = — a 3 — a 2 /3 — a 2 y— 3a 2 — a/3 — 2a -j- 2(3 -j-y. 
0, b, c, d]=-(a+l)(b+l)(c+l)(d + l) (a + l)(a+2)(a+3) ..rm 
-(fl+l)(i+l)(c+l)(^+l) r «(«+l)(«+2)(« + 3)l A 
— (3 (a+2)(a+3) 
— 7 («+3) 
+ f + t d(a+l)(»+l)(c+l)(i+l)(*+2)(«+3)J *, 
- S cd(c+d+ 2) (a+l)(b+l) (a+3) I 
+22 bcd(b+c+d-\-3)(a + l) 
— 6 abed 
where a =a+5+c+d, . . }>=abcd. 
MDCCCLXVIIX. 
