196 ON THE CONSTANT TERM IN THE [25 



Then all the arguments which occur in the values of - , r, u, and v 



will also occur in those of - , ?\, w,, and v lt but the coefficients of the 



TI 



corresponding terms will differ by a quantity which contains e* as a factor. 

 Let the terms with these arguments be called terms of the first class. 



Also there will be additional terms in the values of - , r lf i^, and v lt 



**i 



with arguments of the form 



*%j+W, 



where j does not vanish, and the coefficients of these terms will contain e 

 as a factor. 



Let the terms with these arguments be called terms of the second class. 

 Now, in the formation of the quantities 



- - -Y and t- - -} {(u - Ui y + (v- 7. )' -(r- r,)'} 

 ' i ' / 



terms with the argument zero can only arise by multiplying together three 

 terms of the first class, one term of the first and two of the second class, 

 or three terms of the second class, one of which at least involves e 2 as a 

 factor. Such a term formed in the first of these ways would be of the 

 order of e s at least, while one formed in the second or third of these ways 

 would be of the order of e 4 at least. Hence, by the principle before proved, 



the value of can contain no constant term of the order of e 2 . 



r, r 



Hence B generally, and as this holds good for every value of e', 

 we must have 



B = 0, , = 0, B, = Q, &c. 



CASE II. 



In the next place, let the values x, y, z, as before, belong to the 

 solution in which e and y vanish, and let the values x 1} y lt z, belong to 

 the solution in which e is still equal to 0, but y has a finite value, while 

 nt + e, and therefore also n, retains the same value as before. 



