221 
Urey = Az + bv, } ; 
Vx41 = Az ar bv, 
The solution is at once— 
Ue = Oky? + Che*, 
mae oe 
ante, (45 *\ r+ 0,( ) 7, 
1 
where 4, /, are the roots of the equation 
(— a,) (& — by) = bya. 
(2.) Let it be proposed to solve the system of three simultaneous 
equations— 
Ury = OV, + Cz | 
Von = Az a CW, Fe 
Wa = Agu, + 6,0, J 
The solution is at once 
Ux = Ck, + Ck + Cake >} 
0, = Dyk, + Dok? + Dake ‘, 
w,= Ek, + Ef? + Fyfe," 4 = 
where 
Ak, + Coll ask, + bs 
(oa ee YF 0 head ee 
ae kj? = ¢2b3” = hk? — 2b,’ 
“at Anh, + Cog s Askz + bye 
D, = C, Fe — Obs oe CaDs 5) =, = C, kee ¥ CDs ? 
Ankts + Colts _ py Gh + bate 
DOE Cal OG Rm ay 
and where &,, /,, ks are the roots of the cubic equation obtained by the 
elimination of a, #, v between the equations 
Ae + gv = ka) i ROM ee 
ba + bey = ip ‘, or b, > i he b; = 0, 
OA + ¢,pe=ke J) Cle Gs 
or, 
I3 — (by + €,43 + Cobg) i — (¢2b 143 + C,02b3) = 0. 
As regards the determination of the arbitrary constants C,, C2, C3, we 
are supposed to be given the values of w,, v,, W,, corresponding to a par- 
ticular value of x. Let it be supposed, for example, that the values of 
RI. A. PROC.—YOL. VII. 25 
