PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
281 
From (41) and (42) we have, if ft^ 2, 
n<t> n (aj) e (n_1>a| = <t>\(a j) —B 0 + 2 (<f>' l (a 1 ) — 2Ke~ a ’ sinh d x —B 0 )e 2?a ‘ 
g = l 
-2 V 2 h p e {2q - p)a ', 
Q= 1P =1 
from which we obtain 
ft sinh oti 0' n (aj) = (<p\ (d x ) — B 0 ) sinh ftdj —2K sinh (ft— l) sinh 
71—1 
— 2 2 sinh (ft— p) .(44) 
V = 1 
for ft ^ 2, while 0^ (a x ) is given by (43). 
Finally we have from (33), omitting the first equation of the series, if n ^ 2, 
ft (ft+1) (ft+ 2) e~ na ‘0„ +1 (aj)-(ft-l) (ft) (ft+1) e~ {n+lUl <j> n (a x ) 
= —2 (B u + K) e~ a ' sinh a. x 
n 
+ 2 2 p (c p + sinh <j> p (aj) e~ pa ’. 
p = 1 
By the aid of (36) we can reduce the right-hand side to 
ft (ft +1) (<p ' n+ 1 (ai)-e~V* ( a 0) e“' ia, + 2 2 p (c p + 6 p ) e"^ 1 , 
p = i 
from which it appears that 0 ra+1 (d x ) is finite and tends to zero as n increases, provided 
that the resultant of the applied forces over a = a x is zero. We then obtain for 
ft = 2, 
ft (ft 2 —l) sinh a x 0„ ( a 2 ) = (0'j (aj — By) {ft cosh ftdj — coth a 2 sinh ftd x | 
— K {(ft— l) sinh fta x —(ft+l) sinh (ft—2)dj} 
77 — 1 
+ 2 2 {(pc p +6 p coth «j) sinh (ft— p) d x 
p = i 
— (ft—p) cosh (ft—p) aj}, ... . (45) 
while 
20! (a x ) = 2c 0 + B 0 sinh 2a x + K(2 cosh 2 a —l).(46) 
It appears that equations (38), (40), (43), (44), (45) and (46) give the values of 
0„ (dj), \Js n (a t ), 0' B , (aj), i//„ (d x ) for ft ^ 1 in terms of B 0 , K and the given coefficients 
a n , &c., with the exception of \p- Y (d x ). 
Now we have only assumed a. x > 0 in order to establish the convergence of these 
functions, and hence the corresponding functions of a 2 will be given by the same 
formulae with a' n , b' n , c' n , d' n substituted for a n , b n , c n , d n , provided that the conditions 
for convergence are satisfied. It may be shown that the new conditions of con¬ 
vergence are identical with (30) and (31), or (30) and (32), according as a ? > or < 0, 
yol, ccxxi.— a, 2 R 
