CONTEMPORARY ADVANCES IN PHYSICS 297 



directions for these lines. Let the radius be taken as pointing out- 

 ward, and the normal as pointing inward towards the volume over 

 which we have integrated. Then the angle is greater than 90° and 

 not greater than 180°, its cosine is negative, and the negative sign 

 must be prefixed to the left-hand member of (20) that dS may be 

 positive. We make this transformation in (19), and the first of the 

 angle-integrals becomes : 



fM.inefd,(r'Ji-u)^=fdSco.(n.r)(^\'-^-^). (21) 



The second of the angle-integrals in (19) relates to the infinitesimal 

 sphere. We transform it as we did the first. Now, however, the 

 process is more simple, for the radius r is constant and equal to R, 

 and the angle (n, r) is zero; hence 



dS = E? sin dddd^ (22) 



and the second angle-integral becomes: 



/...n./..(.^-.)^ = /..(if-|). ,3) 



Here we meet the situation for which equation (12) was introduced — 

 the integration of a function over an infinitesimal sphere. Denote by 

 / the integrand in (23), viz., 



f=l^--^ (24) 



' RdR R^' ^ ' 



by/ the mean value of/ over the sphere of radius R, by Uq the value 

 of U at the centre of the sphere, which is the origin. As R approaches 

 zero, the surface-integral in (23) approaches a limit Aq which coincides 

 with the limit approached by ^ivR"}: 



A^ = Lim iwRidU/dR) - iirUo- (25) 



R=Q 



Unless the mean value of dU/dR should vary as the first or a higher 

 power of (1/i?) — a possibility which must be guarded against — the 

 first term on the right of (25) will vanish. Under this restriction, then, 

 Aq is equal to — 47rf/o- Now at the origin U is identical with s, by 

 definition (equation 3). Consequently Uo is identical with the value 

 of s at the origin at the moment / — the very thing which we set out to 

 calculate. For this — let it be called 5o — we have attained the follow- 

 ing equation : 



4x50 = f dWWdV -]- fdS cos (n, r)(dU/dr). (26) 



