32 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



whence on further differentiation (cf. equation (8)), 



_ 



dx' 2 " dx' 2 dx' 



so that equation (12) becomes 



From equations (8) and (ll), 



dx 



__ , , } 



' ' M ' y ' 



so that equation (13) may be written in either of the equivalent forms : 



'V , fd\'\ a \ ?'$' /1/A 



+ _ , .... (14) 



ax 7 " 



(15) 



in which V 2 stands for ^-^ + ^ + ^ , * for <f> (,r', ?/', z', X) and $' for 0> (a;', ?/', 2', X'). 



Thus if 3> satisfies either equation (14) or (15), and also equation (ll), then W, as 

 given by equation (7), satisfies all the conditions which have been seen to be necessary, 

 and will therefore give the true value of Vj V .* 



* Suppose there are, if possible, two solutions to the same problem, say "J? = $\ and 4> = "fv Since 

 W is determined when the problem is fixed, we must have 





so that 



where 



f\' rV 



SvZA. = <f>, f ?A, 

 o Jo 



0, or 



x (x', y', z, A') - x (f, V, s', 0) = (i) 



2 



for all values of x', y' t z. Thus, if x is such as to satisfy (i) we may add a term ^ to $ and still 

 obtain a solution of the same problem. A special case in which (i) is satisfied is when 



x (x', y', *, A') = /(*', y', z', A') {^ (A') - f (0)}, 

 where/ is any function which vanishes identically for the value of A' appropriate to the point x', y', zf, 



