342 
PROFESSOR A. R, FORSYTH ON 
Writing s, c, cl respectively for smq cn u, (Imp the values of A, B and C are given 
by the equations 
A.q + Btqd-Oc^^ 1 
As 2 +Bc 2 +Cc4= 1 
A s s +Bc 3 -j-Cc/ 3 = 1 
or writing 1 
O 
we have 
and then 
C'oClo -CgC/o - CT 
cl. 2 S 3 — c/gSofi- • ■ • =K 
S-fio — A3C0 -fi • . • = S 
C x C.l ^ 
s. ? Co do 
O'! C. 
do 
= A 
AA = 
cr 
AB = k AC = S 
E(Wi)+E(w 2 )+E(w 3 )— Efai+Wa+Mg) 
8A: 2 cr/cSA 
= ~ (ff 8 4-* s -A s 8*) 3 + 4&VS 2 ’ 
As a verification of our formula assume 
U, — Uo=Uo 
ry » ■ — /y> ■ /y* 
so that 
Then since the equation 
A.s‘ + Bc + Cc/= 1 
has three equal roots, the values of A, B and C for our case will obviously be given bv 
A-q+Bq —)~ Oc/ 3 =1 
A -BA_CFA —0 
0 A 
B-i+a£=o : 
c i “1 
for if we write .q + f for q in the first, the coefiicient of £ must vajiish, which condition 
gives the second equation ; and similarly for the third. These last two equations 
give 
B 
C 
Jc~k'\ S -k\* d y' 
■-H 
