CHAP. VI.] VANISHING FORM. 307 



It is easy to see that the second member of this equation is 

 always nothing when the quantities m and n are selected from 

 those which we formerly denoted by m v m^ m 3 , &c. 



We have in fact 



W 



and hX= . 



comparing the values of /UT we see that the second member of the 

 equation (/) vanishes. 



It follows from this that after we have multiplied by adx the 

 two terms of the equation 



&amp;lt;#&amp;gt; (*0 = CW + a a w- a + o,w 8 + &c., 



and integrated each side from a? = to a; = X, in order that each of 

 the terms of the second member may vanish, it suffices to take 



for a the quantity xu or x^r [A-r-J . 



V V K J 



We must except only the case in which n = m, when the value 

 of laudx derived from the equation (/) is reduced to the form -, 

 and is determined by known rules. 



318. If A / -J- = /j, and A/ T = v, we have 



V A/ V A/ 1 



If the numerator and denominator of the second member are 

 separately differentiated with respect to v, the factor becomes, on 

 making fj, = v } 



We have on the other hand the equation 



d*u 1 du , it 



A+.-+-P l or ^4-- 



T 



202 



