200 ON THE CONSTANT TERM IN THE [25 



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



Moreover, all the arguments which occur in the value of z, and which 



are of the form 



2#;V(2fc + lh, 



will occur unchanged in the value of 1( but the coefficients of the cor- 

 responding terms will differ by quantities which involve either (? or y y, 

 as a factor. 



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



Also there will be additional terms in the value of z lt the arguments 

 of which are of the form 



where j does not vanish. The coefficients of these terms will all involve 

 ey, as a factor. 



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



Now the condition that g remains unchanged gives us the following 

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



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

 Hence, ultimately, ('' -**(?' 7i)- 



If this value of e~ be substituted for it, we see that every term of 

 the first class in the values of 



1 1 



--- , r r lt u u lt and v v 1 

 i\ r 



will be divisible by y 2 y~, and that every term of the second class in 

 the values of the same quantities will be divisible by e. Also every term 

 of the fii-st class in the value of z z l will be divisible by y y l ; and every 

 term of the second class in the value of the same quantity will be divisible 

 by ey,. 



Now in the formation of the quantities 



u ~ u > r ~ + (v ~ v ' )2 ~ (r ~ ri)2}> and (k -l) (z - ZiY ' 



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



