I 



THE SUMS OF PERIODIC SERIES. 583 



to the left of Oe and near O flowing in the direction A 0, while the fluid to the right is nearly at 

 rest. Of course, in the case of fluids such as they exist in nature, friction would prevent the 

 velocity in a direction tangential to Oe from altering abruptly as we pass from a particle on one side 

 of Oe to a particle on the other ; but I have all along been going on the supposition that the fluid 

 is perfectly smooth, as is usually supposed in Hydrodynamics. The extent of the surface of 

 discontinuity Oe will be the less the smaller be the motion of the cylinder ; and although the 

 ejysression (119) fails for points very near O, that does not prevent it from being sensibly correct 

 for the remainder of the fluid, so that we may calculate k'^ from (122) without committing a 

 sensible error. In fact, if y be the angle through which the cylinder oscillates, since the extent 

 of the surface of discontinuity depends upon the first power of y, the error we should commit 

 would depend upon y'. I expect, therefore, that the moment of inertia of the fluid which would 

 be determined by experiment would agree with theory nearly, if not quite, as well when a > tt as 

 when a < TT, care being taken that the oscillations of the cylinder be very small. 



As an instance of the employment of analytical expressions which give infinite values for 

 physical quantities, I may allude to the distribution of electricity on the surfaces of conducting 

 bodies which have sharp edges. 



-56. The preceding examples will be sufficient to show the utility of the methods contained in 

 this paper. It may be observed that in all cases in which an arbitrary function is expanded 

 between certain limits in a series of quantities whose form is determined by certain conditions to be 

 satisfied at the limits, the expansion can be performed whether the conditions at the limits be 

 satisfied or not, since the expanded function is supposed perfectly arbitrary. Analogy would lead 

 us to conclude that the derivatives of the expanded functions could not be found by direct differ- 

 entiation, but would have to be obtained from formulae answering to those at the beginning of this 

 Section. If such expansions should be found useful, the requisite formula^ would probably be 

 obtained without difficulty by integration by parts. This is in fact the case with the only 

 expansion of the kind which I have tried, which is that employed in Art. 45. By means of this 

 expansion and the corresponding formula; we might determine in a double series the permanent 

 temperature in a homogeneous rectangular parallelepiped which radiates into a medium whose 

 temperature varies in any given manner from point to point; or we might determine in a triple 

 series the variable temperature in such a solid, supposing the temperature of the medium to vary in 

 a given manner with the time as well as with the co-ordinates, and supposing the initial temperature 

 of the parallelepiped given as a function of the co-ordinates. This problem, made a little more 

 general by supposing the exterior conductivity different for the six faces, has been solved in 

 another manner by M. Duhamel in the Fourteenth Volume of the Journal de VEcole Poly technique. 

 Of course such a problem is interesting only as an exercise of analysis. 



G. G. STOKE& 



ADDITIONAL NOTE. 



If the series by which r' is multiplied in (II9) had been left without summation, the scries 

 which would have been obtained for //" would have been rather simpler in form than the series 

 in (122), although more slowly convergent. One of these series may of course be obtained from 

 the other by means of the developement of tan .r in a harmonic series. When s is an integer. 

 k'' can be expressed in finite terms. The result is 

 A:'-'' = 8«-'7r-Mog,2 + 8 «-'7r-"{2-' + 4-'... + («-!)-"( + 4-77-' {2"'' + 4-^. . + (*•- O'^j - /, , (« odd), 



k''= 8«-'7r-^ {l-' + .l-' ... + (*■- 1)-'} + 4 7r = {l"'' + .'5-^.. + (*' - l)""} "i- (« even). 

 Moreover when 2 * is an odd integer, or when a = 4,0", or = I.SS", &:c., Ic' can be expressed in 

 finite terms if the sum of the series 1 -' + rr' + 9'' + ••• l>e calculated, and then be regarded as 

 a known transcendental quantity. 



4F2 



