136 PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



Let us assume 



.. k cos y 



x -- at = - J i 



j > k cos & 

 = b= -. = 

 K sin y 



K sin y 

 Then when x, y, z is on the ellipsoid we have 



f r 



2 = c = - . 

 K sin 



Thus we may regard , >?, as the coordinates of a point on a sphere of unit 

 radius, or as direction cosines, if it is more convenient to do so. On substituting for 

 x, y, z their values in terms of 77, we find 



S, = (cosec 2 r ~ </ 2 ) ( 2 - ? 2 ) [- ? 2 2 - (1 - 2 9 2 h 2 + <m 



On performing the same operation to the points on the boundary of an element da 

 of surface of the ellipsoid, we find 



, 7 7 /j 3 cos B cos y , 



ndfr = cwcaa) = . ' _ --- ao>. 



sin 3 p 



where dJw is an element of the surface of the sphere of unit radius, or an element of 

 solid angle. 



Since on the surface of the ellipsoid ^., (v) cosec- y if, it follows that 



2 0*) 



Hence 



Tl \ _ ^ 3 COS /3 COS y / 2 _ ?\2 fr 2 /i 9 



It is easy to prove that 



cw 



_ f .3 3.3 , _ f > 2 o 7 



47T 



5 ' 



47T 



15' 



Therefore 



/.(cos) = 



_ 3 



On substituting for q* its value, viz., i (1 + r 2 - (1 - /cV 2 )*), and effecting 

 reductions we arrive at the result given in (16) above. 



