Poisson's Equation for certain Volume Distributions. 139 

 Hence 



y(g, i?, g) _ _ f f 4rf 8tt^P 2 I <ft 16irrP a f _*_ 

 *• Jo I 3^ ** S^f 15 J rtloc l 



8 



167rrP 2l LW " r / 4tt 



log 



15 n , 1 . 



log: - ^ "■ loo- - 



a 



87rP 2 r ; r 5 



log 



where &x and & 2 are proper fractions dependent on r. 

 Therefore 



, 1 

 1 3V(fr *7, g) = 16ttP 2 [ ° g r ] ivh 8tt£ 2 P 2 



Wog --J 3 loo- - 51oo- 



From the above equation it follows at once that, since the 

 first differential coefficients of V are all zero at P ; 



3 2 V i aV(«,y + A,« ) 



oy~ *=u /' oy 



|^ = Lim i^f^±i) =+ae . 



Thus V 2 V has no meaning and, consequently, Poisson's 

 equation fails at P. 



3. Case II. * Let /o = cos ( log -). Then, by Newton's 

 theorem relating to the attraction of a spherical shell, at P 



Or 



and at Q . C 1 ' / 1 \ , 



^ TT 4tt1 £ 2 cos log - dt 



BV .) \ *t • n) 



or r 



* For Cases II. and III. see my paper " On the Second Derivates of 

 the Newtonian Potential due to a volume distribution having a dis- 

 continuity of the second kind " (Bulletin of the Calcutta Mathematical 

 Society, vol. vi., 1916), in which are studied, for the first time, volume 

 distributions for which lim p is non-existent. 



r=0 



