280 
DR. G. B. JEFFERY ON PLANE STRESS AND 
which, on effecting the summation with regard to q, leads at once to 
for n — 1. 
n —1 
n\fr' n («i) = 2 cosech a x 2 a p sinh ( n—p) a x 
p = 0 
Treating (34) in a similar way we have 
(38) 
n{n+ 1) (n + 2) -</r n+1 (aj) —(n— l) (n) (n+l) \fs n (a x ) e 
= 2e no ‘ 2 p [c^ + sinh at x \f/ p (oq)] e~ pa '. . . (39) 
p = i 
We can readily show from (35) that 
sinh a x 2 pe- pa '\!s' p (a } ) = \n (n + 1) {yjr' n+l (ci. l )-e- a ^ f n (a 1 )}e“ na '- 2 pa p e- pa \ 
p = i p = i 
and hence (39) may be written 
n (n + 1) (n + 2) ^ n+1 (a,)-(n-1) ( n ) (ra + l) W, (a,) e“ a ' = n (n + l) [i/4+i (a 1 ) + <?~ a, '/4 1 ( a i)] 
+ 2e na ‘ 2 p (d p —a p ) e~ pa \ 
p = i 
As in the case of \jy' n (o^), we can show that the right-hand side tends to zero as n 
increases if conditions (30) are fulfilled. Hence \p- n (a x ) is finite for all values of n and 
tends to zero as n increases, and we have 
(n—l)(n)(n+l)\fr n (a l )e (n ~ l)a ' = 2 2 2 p (d p —a p ) e {2s ~ p)a ' 
Q= 1 P =1 
+ 2 g(g+l)e ?ai [^ / ?+1 (a 1 )-e“ a, V4(a 1 )], 
9 = 1 
which, on reduction, leads to 
n—1 
n (n 2 — 1) \fy n (<Xj) = 2 cosech aj 2 {(n—p)a p cosh (n—p) a x 
p = o 
+ (pd p —a p coth on) sinh (n—p) ol x ] . . (40) 
for n ^ 2. Equations (34) do not determine (o^). 
From (36) we have 
2(p' 2 (aj) e~ ai — <p\ (a 2 ) e~ 2a ' = <p\ (a 1 ) — 2e~ a ' (K sinh ctj + By cosh a l ) — 2b 1 e~ a ', . (41) 
and if n^2 
(n + l) <p' n+x (ot, l )e~ na '—7i(j)' n (a x ) e~ (n+1)ai = <p\ (a x ) — 2Ke _a ’ sinh a x — B 0 
n 
-2 2 b p e- pa \ ...... (42) 
p = i 
and hence, if the sequence </>„ (a x ) is to converge for large values of n, we must have 
(p' l (a 1 ) = B 0 + 2Ke _a ' sinh a x -{2 2 b p e~ pa> .(43) 
p = i 
