1883.] under its own attraction, etc. 311 



If we write xa = r (x < 1) for the greatest, and a = yr (y < 1) for the 

 least, this equation becomes 



V(l-^ 3 ) ge l-*J(l-x 3 ) 



W- /to 8 ) (5). 



-JT/ V V y 



It Avill be found convenient in what follows to replace x, y by 

 u, v where 1 — x 3 = u, 1 — y 3 = v ; when the form never departs 

 very largely from the spherical form u, v are not large, in this case 

 we have very approximately 



2 

 rCl = 2 - t 



4 



> / 2 64 58 4 \ "] 

 _ 2 / , , 62 , , 55 A I ' 



whence neglecting cubes and higher powers of u, v 



1 



u o — v o ~ gg V o | 



■(6), 



1 2 



"o = w ° + 63 U ° 



.(7). 



Equation (5) can be put into a very elegant form, easily adapted 

 for the use of logarithmic tables, which give the logarithms of 

 secants and tangents, by introducing angles 0, <f> where 



cc = sin 3 #, y = cos 3 <£; 



when it will be found that (5) takes the form 



(p cos^</> cosec $ = sin 3 Q S ec log t . cot \0 



= 23025851 sin$0 sec log cot£0. 



Corresponding sets of values of 0, <f> are obtained with extreme 

 ease by means of this equation, by giving a series of values to 0, 

 finding the value of the logarithm of the right-hand member, and 

 taking as a rough approximation 



A closer approximation can then be obtained by proportional 

 parts. 



