OF ARTS AND SCIENCES. 227 



Now 



X = X^ - X,3 = 2 QpQ,x,8 {a,\c,) 



and, since 



a^ — b"^ = const., a^ — c^ =: const., a^a = 6S5 = cSc, 



so that 



or, dropping subscripts, 



T=kQ,y, 



where 



and 



Q = r ^-i , 



o 



= r ^^ 



O 



Legendre has shown that 



and the same result may also be obtained as follows. By the theory 

 of potential, 



rfX ^ nTZ 



dx '^ dy ' dz ' 



for all points external to the attracting mass. Hence 



Now 



dx \ d(a2) "f" d(b-') ' (rfc2) I dx ~ did"-) '^^a I ^' 



and 



2xdx _ xMja'^) yV(52) zVfc^) _ d(a^) 



«- ^ ^0^ dx ' 



