334 
PROFESSOR C. NIVEN ON THE INDUCTION OF ELECTRIC 
S„ +1 (Xa) = AT +1 sin (\a+^ V ) +A/ +1 cos Vj 
T„ +1 (Xa)=A 1 ” +1 cos (Xa+^V)-A.T +1 sin (xa+^Tr), 
S / ,_ 1 (\b)= Bp -1 sin ^Xb + tt j + Bf' -1 cos ^Xb+ — — tt ) 
T„_j(Xb) = Bp -1 cos (Xb + —~i rj — Bp -1 sin ^Xb +' 1 tt ) 
Substituting in (62), 
(A 2 ” +1 BT _1 —A 1 ” +1 B 3 ,l-:i ) cos (X(b— a)—7 t) —(A 3 ” +1 B 2 “ -1 + A 1 s+1 B 1 “ -1 ) sin (X(b — a)—7r)=0 
or more simply 
(A 1 ” +1 B 1 " -1 +A 2 “ +1 B 2 " -1 ) sin.X(b—a) — (A 2 ” +1 B 1 “ -1 — A 1 " +1 Bv -1 ) cos.X(b—a) = 0. 
This equation may be further simplified ; for, putting 
X . n A g B— i 
tan S-'=-A, tan A T , tan 8'= * ■ 
n 9 \ tt+I 7 1 H n —1 
V 
BT 
(63) 
it becomes 
whence 
sin {X.b — a — a! —/T} = 0, 
X.b — a=a'— /3' -{-nr 
(64) 
here a' and /T are the smallest positive angles which satisfy the above equations ; and 
3 may, when 2x>n(n+1), be expanded in ascending powers of — by Gregory’s 
series. 
tt 1 
U p to —, 
, J= ^ + fa- 1 >yt 1)(, ‘ +2) 4-AA ) [ 3 .or+») ; -8(» 5 +«)+i-4A • • • <65> 
2.4 
2.4.6 
The values of <x and (3‘ in terms of Xa and Xb may be similarly found, and when 
substituted in equation (64) will give the means of finding all the values of X. These 
being found, the complete expression for P may be written down and the values of the 
constants therein determined from the initial circumstances by known methods; we 
shall therefore proceed to find the value of P at a point outside the shell. 
