PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 83 



Of this a particular case is 



(it + S)fff&k = -ff&.pUvds, 



which suggests many curious theorems. 



27. As a verification of it, let the closed surface 2 which determines the 



limits of the integrations be itself 



f=C, 



which, of course, subjects the form of £ to further limitations. 

 The right hand member is obviously equal to 



3C x vol. of 2, 



because — S.pUv is the perpendicular from the origin to the tangent plane at p 

 to the element ds. The left hand side may be broken up into a set of shells 

 bounded by surfaces whose equations are 



where e varies from to 1. [This follows from the assumption that £ is homo- 

 geneous.] The volume of the surface corresponding to any value of e is obviously 



e 3 x vol. of 2 . 

 Hence 



d<; — 3e 2 de x vol. of 2 , 



so that the left-hand member of the equation above becomes 



(ft + 3) /* dCeT+'de x vol. of 2 = 3C x vol. of 2 , 



• J 



and the proposition is proved. 



28. A very interesting case is when 



g Tp 3 ' 

 in which case n = — 3, and our equation appears to become 



(3 - SW^p = = - CTS.^TJvds. 



It is obvious, however, that there is an infinite element on the left hand, 

 when Tp = 0, i.e., when the origin lies inside 2 ; and it is easy to see that the 

 correct result is a simple case of the well-known equation of § 4. In fact, the 

 expression on the right denotes, as is evident, the whole spherical opening sub- 

 tended at the origin by 2. Its value is therefore if the origin be without 2, 

 and 477 if within — 2 being supposed to be simply-connected. 



29. As a final example let us suppose in § 26 that f is a Spherical Harmonic. 



