THE SUMS OF PERIODIC SERIES. 577 



say with this understanding, that the result is to be substituted in (106); for it would be absurd to 

 speak, except by way of abbreviation, of a quantity which is zero except for particular values of 



X and «, for which it is infinite. The values of and will be obtained bv direct differ- 



^ djr df ^ 



entiation. We have therefore for the direct developement of y m in a triple series 



VM = 222 \ - (ju- + V- + ■m') J„„p + — — sin /xx sin ^y'^ sin nx sin vy sin iirx. 



But yw being equal to zero, each coefficient will have to zero, from whence we get A,„„p, and then 



8 „„„ 7<r . , . , . . . , . 



u = — — 2.2,Z -T, :; r Sin ux sin vv sin ux sin vv sin Tirz (107)- 



abc n'+v'+'sr' » " ^ 



One of the three summations, whichever we please, may be performed by means of the known 



formulae 



^ ■23- sin srz c e*''"" - e-"'"-'' 



2 — 7, — yr = - — re :rrc — > if2c>s;>0, (108), 



1 A:cos^« 6 e'"''-*-' + f-*<''-S'.' 

 ¥k^^ k^-.y—.^ ^^-e-"^ .if26>y,>0, (109), 



which may be obtained by developing the second members between the limits 2: = and x = c, 

 y — and y^ = b by the formulae (2), (22), and observing that the expansions hold good within 

 the limits written after the formula?, since e*''^"^' — e~ ''^~'* has the same magnitude and opposite 

 signs for values of « equidistant from c, and £"'"''■' + £-*('-'') jj^g the same magnitude and sign for 

 values of y^ equidistant from 6. If in equation (107) we perform the summation with respect to p, 

 by means of the formula (108), we get the equation (09) : if we perform the summation with respect 

 to n, by means of the formula (109), we get the equation (lOO). 



52. The following problem will illustrate some of the ideas contained in this paper, although, 

 in the mode of solution which will be adopted, the formulas given at the beginning of this Section 

 will not be required. 



A hollow conducting rectangular parallelepiped is in communication with the ground: required 

 to express the potential, at any point in the interior, due to a given interior electrical point and to 

 the electricity induced on the surface. 



Let the axes be taken as in Art. iS. Let x', y, z be the co-ordinates of the electrical point, m 

 the electrical mass, v the required potential. Then i' is determined fir.it by satisfying the equation 



in , 



y« = 0, secondly by being equal to zero at the surface, thirdly by being equal to — infinitely 



close to the electrical point, r being the distance of the points (x, y, z), (x, y', z'), and by being 

 finite and continuous at all other points within the parallelepiped. 



Let V = — + v„ so that v, is the potential due to the electricity induced on tlie surface, 

 r 

 Then «, is finite and continuous within the parallelepiped, and is determined by satisfying the 



general equation y?;, = 0, and by being equal to at the surface. Consequently f, can be 



r 



determined precisely as h in Arts. 48 or .ll. This separation however of « into two parts seems 



to introduce a degree of complexity not inherent in tlie problem ; for f itself vanisius at the 



surface; and it is when the function expanded vanishes at the limits that tlie application of the 



series (2) involves least complexity. On the other hand we cannot immediately expand w in u 



triple series of the form (10.'".), on account of its becoming iiidniti- at the point (x , y,if)- 



