176 ME. L. N. G. FILOX OX THE ELASTIC EQUILIBEIUM OF 
lead to infinities, or at all events, to discontinuities in the stress. In other words, 
that, though we have chosen our constants so as to make (rr formally zero, yet 
the limit of ( rr ) as given by the series is not zero when r aj)proaches a. This 
would suggest that the series for o'v, considered as a series of I-functions, behaves at 
r = a in much the same way as a discontinuous Fourier series whose general term 
is sin nz behaves at z — tt. In fact, if we differentiate rr in the usual way with 
regard to r and then jjut r = o, we get a divergent series. 
It is easily seen, however, that no discontinuity really occurs except at the points 
where the shear is applied discontinuously. The general term in rr is of the form 
(dropping irrelevant factors) : 
cos 2n + lu ' T / w ^ 
1 (“) - rff( f)] 
-j 
I 'ya%^(a) — (1 + ya“)Ip(a) 
TT 
where v. = ~ rh ^ i ^)- 
Now, looking at the semi-convergent expansions for Iq and I^ we find, puttin^ 
a — d = r and p = a — 8 where 8 = jg small, 
_ -6 A I 1 ^ 
T / \ ^ I ^ r 2 
Jo(«) \ « 
1 ^ + terms of order S/cd and higher terms in 8/a 
and ^ , 
ll(«l 
-s 
1 + A 
1 — v; + higher terms 
where e = base of Napierian system of logarithms. 
The coefficient of cos [2n -fi l)w then reduces to the form 
1 
(2« -f 1) 
S / 8 8 
-h I,,(a)In (a)a X „ X e“^( 1 -j- terms in " + terms in — fi- &:c. -}- terms 
'2u- \ a u~ 
. 1 , \ 
y«i 
( 8 8 8 \ 
1 + + terms in and — + higher termsj 
(1 — y) Ii (a) Iq (a) [e“^8 (terms of order \ ] (a finite term)} 
— - yaff ^(a) |8e“^ (terms of order fi- e“^ (a finite term) 
-A(ya2V(a)-(H-ya-^)Ip(a)). 
f] 
