CIRCULAr. CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 177 
Now, if we remember that [ct) — (!-}- ya})l^{a) is of order yal^{a), we see 
that the successive terms in the coefficient of cos 2n-\-lu are of the orders 
Se * 
Se 
ge-' 
(2)i+lb’ (1% + !)’ (2M+ld’ (27i + l>'5’ (2;t+iy5’ (2;^ + 1)^ ’ 
respectively. Also in considering discontinuities, we need only consider the terms 
towards infinity, for the terms at the beginning can introduce no discontinuity. 
But clearlv the series 
, cos 1 n, 
(2» + !)•’ ' 
Se 
are of the order d multiplied by a series, which is finite and continuous up to and 
including the value d = 0. They tend therefore to the limit 0 with d, and can 
introduce no discontinuity in the stress. 
Tlie same will be seen to hold of the series 
- cos 2w+l u, provided if 0. 
{2n + Id 
If however u ~ 0, we have to deal with the series 
ird ^ 
1 
TT(l 
- (-2'! + 1) -5; 
2c (2» + I) 
The series under the sign of summation is divergent if d = 0. If, however, d is 
small, ljut still finite, the series can be summed, and we have the expression equal to 
7rd /I + e 
This tends to zero when d is small, provided 
'n-d , . . ird ^rd 
-log(l - e 2,.) , -loo- 
& 9 
ic 
tend to zer( 
which is known to be the case, 
continuity in the stress. 
Now consider the series 
Hence this series again can never introduce a dis- 
—cos '2n + lu = Se~^cos 2n + Do 
2n + 1 
sc 
This is not of the same form. Tlie series under tlie S is sometimes oscillatory, and 
sometimes divergent, but is never convergent, if d is put equal to zero, 
VOL. CXCVTIT.—A, 2 A 
