WHICH SATISFY GIVEN CONDITIONS. 
155 
Tenth equation : 
3{(3, 2)— 2(1, 2, 2) — (3, 2)(2m-6)} 
+ {(2, 2, l)-(r, 2, 2)(2m-7)-(I, 2, 2)(2m-7)} 
+ Supp. (I, 2, 2) =(I, 2, 2)2D. 
Eleventh equation : 
3{(3, 1, 1)— (T, 1, 1, 2)— (3, 1, l)(2m— 7)} 
+ 2{2(2, 2, 1)-2(1, 1, 1, 2)— (2, 2, l)(2m-7)} 
+ {3(2, 1, 1, 1)-(I, 1, 1, 2)(2m— 8)— (I, 1, 1, 2)(2m-8)} 
+ Supp. (I, 1, 1, 2) =(1, 1, 1, 2)2D. 
Twelfth equation : 
2{(2, 1, 1, 1)-4(T, 1, 1, 1, 1) — (2, 1, 1, l)(2m— 8)} 
+ {5(1, 1, 1, 1, 1)-(1, 1, 1, 1, l)(2m— 9) — (I, 1, 1, 1, l)(2m— 9)} 
+ Supp. (I, 1, 1, 1, 1) =(T, 1, 1, 1, 1)2D. 
113. I alter the forms of these equations by substituting for 2D its value 
=n — 2m + 2+a, and by writing for the expressions with (1) their values, 
(1, 4)=( • 4) — 5(5), &c., 
and except in the terms {Supp. (5)— «(5)}, &c., by writing for k its value —3 w+a 
The resulting equations, if the Supplements were known, would serve to determine the 
values of (5), (4, 1), &c. ; but I assume instead that the last-mentioned expressions are 
known (First Memoir, No. 50), and use the equations to determine the Supplements, or, 
what comes to the same thing, the values of the terms in { } which contain these Supple- 
ments. We have thus the twelve reduced equations, with resulting values of the sup- 
plements. 
114. First equation : 
= 0 = 
( 5 ) 
+ { Supp. (5)— *(5)} 
+(5)(8m+7w— 4a) 
-(.4) 
(that is, we have 
Supp. (5)-*(5)= 
— 15m — 15w + 9a 
— 3m + a 
8m+ In — 4a 
+10?w+ 8n— 6a 
3 m+a, 
and so in the subsequent cases, the equation gives the value of the term in { } which 
contains the Supplement). 
mdccclxviii. z 
