110 APPLICATION OF THE PRELIMINARY RESULTS 



The equation (c), by making r = 0, becomes 



or by substituting for Y 



an equation which ought to hold good, for every value of #, from 

 # = X to a; = + X. 



In those cases to which our theory will be applied, 1 - g is 

 a small quantity of the same order as a?A, and thus the three 

 terms of the first line of our equation will be of the order a* AX; 



dX'" 

 making now x = + X, f g , a is shown to be of the order 



a* AX'", and therefore^-:- X'" is a small quantity of the 

 ax 



order aA ; but for any other value of x the function multiplying 



J~V~i" 



7 becomes of the order a 2 , and therefore we may without 

 ax 



sensible error neglect the term containing it, and likewise 

 suppose 



dx 



In the same way by making x = X, it may be shown that the 



J'V" 



term containing j is negligible, and 



dx 

 Thus our equation reduces itself to 





