1885.] K\ E', J', 0' in ascending powers of the modulus. 243 



By equating the coefficients of k~ n in this identity we find that, 

 if n > 1, 



If we define P x Q (adq) to denote Px Q (p — q), i.e. the con- 

 tinued product of P and Q and the adjunct of the quotient of P 

 divided by Q, we may write the last equation in the form 



Q>^-« )-^(/3p»-&) 



= P n _ x x Q l (adq) + P„_ 2 x Q 2 (adq) + P n _ 3 x Q 3 (adq) . . . 



+ ^x Q n _, (adq). 

 The left-hand member of this equation 



= a{Q„(ad)}- i 8{P ?i (ad)}-« Q re + /3. P w . 



When (3 = 1, the equation may be written 



«{«,(ad)}-a& + £.P. 



= P n (ad) -(- P^ x Q x (adq) + P„_ 2 x Q % (adq) . . . + P 1 x Q^ (adq), 



and, when a = 1, it may be written 

 /3{P>d)}-/3 P K+ « Q M 



= Q„ (ad) + (?„_, x P x (adq) + Q n _ 2 x P 2 (adq) . . . + Q t x P„_ t (adq). 



§ 43. Taking for example the first relation K 1 E 2 — K 2 E t = 1, 

 which was written out at full length, in terms of the series, at the 

 end of the last section, and putting for simplicity n = 4 (the law of 

 the formation of the terms being as clearly seen in this particular 

 case as in the general formula) we obtain the arithmetical theorem : 



1 2 .3 2 .5 2 .7 2 1 2 .3 2 .5 2 .7 2 1 2 .3 2 .5 2 .7 , 



2 2 .4 2 . 6 2 .8 2 2- . 4 2 . 6 2 . 8 2 ^ ' ~ 2 2 . 4 2 . 6* . 8 2 ^ ' 



1 2 .3 2 .5 1* , N 



+ 2^4 2 76^ X 2 5(adq) 



1 2 .3 1 2 .3 2 



+ 22 42 x 2~2~^ 2 ' ad( lJ 



, 1 r. 3 2 . 5 2 



+ 2^ x 2 2 4 2 6 2 ^ ^ 



