Schlomilctis Iheorem in BesseVs Functions. 569 



Equation ((5) is the result of averaging F (£) over all 

 directions indifferently in the osy plane. Let us abandon 

 this restriction and take the average when f is indifferently 

 distributed in all directions whatever. The result now be- 

 comes a function only of R, the radius vector in space. If 

 be the angle between R and one direction of f, ^ — H cos 0, 

 and we obtain as the mean 



c 



F(Rcos(9)sin(9^=i-{F 1 (R)-F 1 (0)}, . (7) 



where F/ = F. 



This result is obtained by a direct integration of F(f) over 

 all directions in space. It may also be arrived at indirectly 

 from (6). In the latter f(r) represents the averaging of 

 F(f) for all directions in a certain plane, the result being- 

 independent of the coordinate perpendicular to the plane. 

 If we take the average again for all possible positions of this 

 plane, we must recover (7). Now if be the angle between 

 the. normal to this plane and the radius vector R, r= Rsin 0, 

 and the mean is 



Jo 



sin (9) $mdd0. .-■ .-■•..: . (8) 

 Jo 

 We conclude that 



R /(R sin 0) sin JO = Fj(R) - F^O), . . (9) 



which may be considered as expressing F in terms of /'. 

 If in (6), (9) we take F(R) = cos R, we find * 



f^JofRsin 0) sin d0 = U~ 1 sin R. 

 Jo 

 Differentiating (9), we get 



F(R) = f 2 /(R sin <9)sin0 dO + Rf * /'(Rsin 0) (I- cos 2 0) d0. 

 : : J° J° .... (10) 



Now 



R p^cos 2 0f'(Ji sin 0) d9= ( cos . <//<R sin (9) 



= -/TO) + f a? /(R sin 0) sin tf<7. 



* £?ic. j?rfc Art. Wave Theory, 1888; Scientific Tapers, vol. iii.p.98. 

 PML Moo. S. 6. Vol. 21. No. 124. 4pn7 1911. '2 P 



