68 PROFESSOR J. J. SYLVESTER A.ND MR. J. HAMMOND 



while at the same time, since the order of r'ot does not exceed , we have 



and in like manner 



< 



Thus equation (5) becomes 

 Order 1 



= r l , and so on. 



1 s 4 + 



where 



order 

 Let 



then 



H 



a = 



-i 



' = -3r - - 



a = 



+|:Dr 



+ HE * 



A . _ JLI 



8 > 16 



where terms as far as, but not beyond, ^ (which is the order of s~ 3 w 5 ) have been 

 retained. 

 Now 



p consists of terms whose orders are 2, f , 1, f, f , \, . . . 

 q 1 ; l ; | ( _ft tf> ( _ ;\ 



a )> 4> 2' 8"> T6' 4> 



Thus the order of a' is , and in the above expression all terms of a superior to 



^ have been retained, and consequently (rejecting the square of a' whose order is 



^) in the first line of (6) we may write 



- r-V - 



+ 



+ ^ r- 



