198 ON THE CONSTANT TERM IN THE [25 



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

 and which are of the form 



2^y</>/<', 



will occur unchanged in the values of , r lt u lt and v lt provided that <f), 



TI 



and therefore also j- or c, remains unchanged, but the coefficients of the 



corresponding terms will differ by quantities which involve either e e t or y 2 

 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 -, ? u 1} and v lt 



r i 

 the arguments of which are of the form 



where k does not vanish. The coefficients of these terms will all contain 

 y 2 as a factor. 



Call the terms with these arguments terms of the second class. 

 And (z z l )' = z 1 ", which contains y" as a factor in every term. 



Now the condition that c remains unchanged gives us the following 

 relation between e", e*, and y": 



He" = He? + Ky, 

 taking into account only the terms of lowest order in e 2 , e, 2 , and y 2 . 



Hence, ultimately, 



,_H.._ n 



If this value of y 2 be substituted for it, we see that every term in the 

 values of , r r lt u u 1} v v lt and (z z,) 2 will be divisible by e e l . 



Hence the constant part of will be divisible by (e e^f, and 



therefore also by (e 2 e^) 2 , since this constant part involves only even powers 

 of e 2 and e?. 



