209 



if we neglect small quantities which will be multiplied by B - A, &&}, 

 or observing that the latter factor is the same as D, it becomes simply 



N M X +M y 

 ~D AB ' 



and the term becomes 



M X +MA (a) 



B -A 3 1 + cos tl * /x 1 -AT 

 ABC 8 4 r 3 (n - n l ) n x D 



which is true for every value of ;. Suppose, for simplicity, that cos < 

 does not much differ from unity ; then on this supposition we should 

 have 



N ^ l lt^, if y = ^j^(^ + 3^)-^J&c. 



M x+ M y - Hsft + ^ - 3r,V) = (3 + ^ * ^ 



upon substituting these values for iV; and if, + M yi the expression a 

 becomes 



B-A 135 / M3 1 1 (SfcTfr + y^)-?) — U x 

 A£(7 64 V^ 3 J (n-n 1 )* 1 !) r 2 



This term, it will be observed, is multiplied by the cube of 

 whereas the terms which have been previously examined were only 



multiplied by the square. If now we resume the expression in the de- 

 velopment of N which is multiplied by 



we shall see that, on continuing the approximation, + B 2 ) and 

 (J.2 - Bi) will receive an increment such that when their new value is 

 substituted for them in the above expression, it will be multiplied by 



the cube of ^. The term which so arises will not destroy that pre- 

 viously found, but will be of the same order, and will modify it ; 

 therefore it must be sought out. To find it, 

 Let us return to the original equations for u> 2 , namely, 



da x 3 a C-B . (1+cos; — 



dt + ^2~r* T~ Sm< ( 2 cos4/-20-cos<cos0 



1 - cos < — ) 



- — - — cos 0 + 20) 



