22 Mr. S. B. McLaren on Hamilton's Equations and the 



Suppose F = F 2 + Vf (24) 



and Div Fj = 0, . . . . . . (25) 



then Div F = V 2 <f 



or by (19) 



(27) shows that l<j>-\ -^-) is the potential due to a 



distribution of density p within the space. If we choose 

 tJt zero at the surface, then since (f) is also zero, it follows 



that <f>-\ -¥■ is the potential due to charge of density p and 



c at 



the charge it induces on the surface, also at the surface V 2 V" 



and ty are both zero, the former by (26), 



Again at the surface F is normal. (24) shows that F, is 

 also normal since yjr = 0. 



In view of (24) and the surface condition, Fj can be 

 expanded in a series of normal functions (Poincare, Am. 

 Journal of Maths,). 



r=< 



F^S^F,., (28) 



where 



(V 2 +OF r = -) 



Div F, = ! 



>,.... (29) 

 F r normal at surface j 



jF r W=47r J 



dY is an element of the volume enclosed, the numbers tc r are 

 arranged in an ascending series.. 



The last of (29) is required to determine ¥ r absolutely. 

 When r and 5 differ we have the normal property 



§F r ~F s dV=0 (30) 



;ases that /c r = k s , but (30) still holds. 

 l (28), (29), (30) that 



fF r FidV = 47r* r , (31) 



It may be in certain cases that K r = k s , but (30) still holds. 

 We can easily see from (2S), (29), (30) that 



