THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 189 
Ree PO SOT TSO! (Zs 
where 
A=11P?+ 2PS — Q’, 
B= — PE — Q(16P°+3PS — Q’), 
and C=P°QE + P?(6PQ’+S8). 
By (6, 7); 
(a — AP)u? + (b — BP)w+ (¢e - CP) =0, 
or 7) on gh OO a ee ee (=) 3 
where 
A’=11P*S — PQ?+8?, 
B’= — PSE - QS(5P?+8), 
and C’= — P*?(P?Q’+ PS? - Q’S). 
Combining (7) and (8), we find 
(A’ B- AB’)u+ (A’ C— AC’) =0, 
or P(mE+n)u — P(pE+q)=0, 
ote ya PEt 
mE+n’ 
where 
m= P?Q?+ PS? - Q’S, 
n= — Q(121P*S — 16P°Q’ + 28P?S? — 9PQ’S + Q*t+83), 
p= — PQ(11P?S — PQ?+8?), and 
g= — P(11 P°Q?+ 11 P*S?+ 57P°Q’S — 7P°Q*+ 13P*S* 
+2PQ°S’ + Q'S + §%). ; 
Moreover, multiplying (1) into PS, and (2) into Q, sub- 
tracting the former from the latter result, and dividing 
the difference by w, we obtain 
Quw?—-PSu+P(P?Q-S8v)=0 . . (9). 
Next, multiplying (1) into Q, and subtracting (9) from 
the result, we have 
(PS — Q?+ Qu)v — P(P?Q — Su) =0, 
o = E(PQ-Su) _P(pE+g) | 
PS — Q?+ Qu mE+n 
where 
m = —8(LOP*Q? — P*S?+-3PQ*S — Q‘), 
n' = — Q(11P°Q? + 132 P°S? — 80PQ’S + 9P*Q!‘ + 41 P88 
— 35P°Q’S? + 11PQ‘S + 2PS* — Q*- Q’S'), 
