THE SUMS OF PERIODIC SERIES. 579 



axis, both ends being closed: required to determine the effect of the inertia of the fluid on the 

 motion of the cylinder. 



If there be more than one partition, it will evidently be sufficient to consider one of the sectors 

 into which the cylinder is divided, since the solution obtained may be applied to the others. In 

 the present case the motion is such that udw + vdy + wdz, (according to the usual notation,) 

 is an exact differential d(p. The motion considered is in two dimensions, taking place in planes 

 perpendicular to the axis of the cylinder. Let the fluid be referred to polar co-ordinates r, ^ in a 

 plane perpendicular to the axis, r being measured from the axis, and from one of the bounding 

 partitions of the sector considered, being reckoned positive when measured inwards. Let the radius 

 of the cylinder be taken for the unit of length, and let a be the angle of the sector, and a. the 

 angular velocity of the cylinder at the instant considered. It will be observed that a = Sir 

 corresponds to the case of a single partition. Then to determine <p we have the general equation 

 d^d) 1 d(h 1 d^<h 



-T^. + -7^ + -3Z = (112), 



dr r dr r dd 



with the conditions 



1 dd) , „ 



-— ^ = eur, when 6 = or - a (113), 



r do 



d(h 



—i-=0, when r = 1 (114), 



dr • V /' 



and, that <p shall not become infinite when r vanishes. 



Let r = e"*, and take 0, \ for the independent variables; then (112), (113), (lU) become 

 (F(f) d-(b , ^ 



— = we''"", when = or = a (II6), 



-2 = 0, when X = (117). 



d\ 



Let (j) be expanded between the limits = and Q = a m a. series of cosines, so that 



d) = Ao + 2A„cos (118), 



a 



Aq, A„ being functions of X. Then we have by the formula (D) and the condition (116) applied 

 to the general equation (115) 



rf'Ao 



0. 

 dX' 



d'A„ 

 d\' 



/nTr\- 2(1) , , , ,1 



— A„-— 1 -(-1)" e-=* = 0; 

 \ a J a 



Iwhence Ad = J„\ + fi„, 



A ><=?.»-"•? 2a,a{l -(-1)"} ,, 



Since is not to be infinite when r vanishes, that is when X becomes infinite, wo have in the 

 ffirst place A,, = 0, A, = 0. We have then by the condition (117) 



