480 ON CERTAIN APPROXIMATE FORMULA FOR [59 



Considering -^- 2 - i , -, - -^- , and a-/3 to be small quantities of 



the first order, the above expressions for -- - , X, Y, and T are true to 

 the fourth order. 



The quantity (77) which occurs as a factor in some of the terms 



\UCL(f)J 



of the third order may be put under a very convenient form in the following 

 manner. 



We have, by Taylor's , theorem, 



du 



In this make <o = -(a ft) and -(a /3) successively; therefore 



and 



Hence we have to the first order of small quantities 



p-q_/du 



a - 



and 



and therefore ( - -. ) = -. , ^ ~ q ' n -, to the first order. 



Making this substitution for (-4^) the expressions for X, Y, and T 



\ua<p/ 



become 



