424 Mr green, ON THE DETERMINATION OF THE 



which agrees with the ordinary formula, since the mass of the ellipsoid 



47r«'6'c' 



IS 



3 



and «^ = d^ + h\ 



Examination of a j)articular Case of the General Theory exposed in the 



former Part of this Paper. 



14. There is a particular case of the general theory first considered, 



which merits notice, in consequence of the simplicity of the results to 



which it leads. The case in question is that where we have generally 



Avhatever /• may be 



a/ = a. 



Then the equation (19) which serves to determine 0, becomes by 

 supposing kn = k . a"' 



= il-'2r'^r')^r'^, + (.v-»-l)2/*'?.^-A<^ (39). 



If now we employ a transformation similar to that used in obtaining 

 the formula (14), No 6, by making 



^i = P cos 9i, ^2 = p sin 9i cos On, ^3 = p sin 0, sin 9.^ cos 63, &c. 



and then conceive the equation (39) deduced from the condition that 



'"^'"i- "f-d-sf =)'^ {^■*' (f )' - r^l 



must be a minimum (vide No 8), we shall have 



rf^,rf^2 c?f,, = p'-' sin0/-^ sina/-^ sin 9^-, dp de,d9, «?0,_„ 



\d^rl \dp) p" ' sin0,^sin0/ sine^,-,' 



and 1 - 2^,= = 1 - p\ 



Proceeding now in the manner before explained, (No 8), we obtain 

 for the equivalent of (39) by reduction 



d''(j) , ( _ _ , V cos g,. d(f> 

 d'<p s-l-np' dcl> 1 d9;''^^^~'' Um9rd9,. k 



dp' ^ p{l-p') 'dp p' ' sm9,'sm9^' sin0^,., 1 _^2 9-V*U). 



