THE SUMS OF PERIODIC SERIES. 581 



When s is an odd integer, the expression for k''^ takes the form 05 - os , and we shall easily find 



sV vr (n - s)n{n + s) 



where all odd values of n except s are to be taken. 



The quantity k' may be called the radius of gyration of the fluid about the axis. It would 

 be easy to prove from general dynamical principles, without calculation, that if k be the corre- 

 sponding quantity for a parallel axis passing through the centre of gravity of the fluid, h the 

 distance of the axes 



k'^ = fc- + h- (124), 



in fact, in considering the motion of the cylinder, which is supposed to take place in two dimen- 

 sions, the fluid may be replaced by a solid having the same mass and centre of gravity as the fluid, 

 but a moment of inertia about an axis passing through the centre of gravity and parallel to the 

 axis of the cylinder difl'erent from the moment of inertia of the fluid supposed to be solidified. 

 If K', K be the radii of gyration of the solidified fluid about the axis of the cylinder and a parallel 

 axis passing through the centre of gravity respectively, we shall have 



• 



•r'2 1 rr2 ,2 , * sin^a 8 . «7r 

 K^ = ^= K + h% h = 5_ = sm — (125). 



If we had restricted the application of the series and the integrals involving cosines to those 

 cases in which the derivative of the expanded function vanishes at the limits, we should have 



expanded <p in the definite integral / ^(fl, )3) cos ^\rf/3, and the equation (115) would have 





 given 



r(^>^) = a/3)e^'+x(/3)^-^ 



^, Y denoting arbitrary functions, which must be determined by the conditions (II6). We should 

 have obtained in this manner 



32 r^ l-e-^' dH 



^aJo 1-He-^"|3(/3S4)^ '^'~^>- 



It will be seen at once that k"' is expressed in a much better form for numerical computation by 

 [the series in (122) than by the integral in (12"). Although the nature of the problem restricts 

 [a to be at most equal to 27r, it will be observed that there is no such restriction in the analytical 



proof of the equivalence of the two expressions for (p, or for k''. 



In the following table the first column gives the angle of the cylindrical sector, the second 



I the square of the radius of gyration of the fluid about the axis of the cylin<ler, the radius of the 



cylinder being taken for the unit of lengtli, the third tlie square of the radius of gyration of the 



fluid about a parallel axis passing through the centre of gravity, the fourtii and fiftli the ratios of 



the quantities in the second and third to the corresponding quantities for the solidified fluid. 



Vol. VIII. 1'aut V. 4 1" 



