TRANSACTIOKS OF SECTION A. 557 



Now, by tbe theory of the potential of a surface density, as given in the ordinary 

 books on potential, 



ox d.v oy ay da oz 



where I, m, n are the direction cosines of the normal to the surface (drawn from 

 the side s<0 towards the side s>0). 



Here we may write 



/ = - {(IX + hy) + f J.;-, m = - {hx + bi/) + (j^r, n = ] + f - r, 

 and so we bare 



fx + xu,„ + yt(,r,j + ("r = 47r(s + xs^ + ys„ + €"r)[ - {ax + hy) + f j^r] 

 w^ + xu,nj + yuy,j + €',yr = 47r(s + .r*,. + ys^ + e"r)[ - {hx + *_y) + ej,r] 



M; + .VU:r: + yU^z + e'"r = 4cTt{s + .Ifo + yHy + f "?')[1 + f;rj 



As these hold for all values of x, y, for which r is less than some assignable 

 quantity, we have the results 



?^.„ = - iiras , ?/.,..- = iiTS,r 



Uyy = — irrbS , Uy: = -iTTSy 



u,.= +4iT{a + b)s , n„j= -4nhs 



where the value of ii.^ is determined by tbe fact that 



w.r,c + ><iiy + «.-; = (V'^'i — V '^' ) at the origin 

 = 



Since « + 6 = — + - - 



Pi Pi 



where p^, p., are the principal radii of curvature of the surface, it follows that 



VPi Pa / 



Potentialtheorie/ Bd.i. p. 50), and Poincare ('Potentiel Newtonien,'p. 251), when 

 allowance is made for tbe simplification introduced by using tbe axes selected 

 above. 



13. The Ajyplications of Fourier's Series to Mathematical Physics. 

 By H. S. Carslaw, D.Sc. 



In tbe problem of conduction of beat when tbe solution is given by the 

 infinite series 



V = 2«„ sin ??.ie'~'^'''', 



where 



2 C7 

 «« = - f {x') s'mni'fU', 

 77 Jo 



the presence of the factor e-^"-' preserves the convergency of the series when 

 differentiated term by term. 



In the problems of transverse vibrations of strings where the solution is 

 given by 



t^ = 2a„ sin nx cos nat 



