1885.] K', E', J', G' in ascending powers of the modulus. 245 

 In the same way the third relation, viz. Kfi 2 — K 2 G t = 1, gives 



_ l 2 . 3 2 . 5 2 . T _ l 2 . 3 2 . 5 2 



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



l 2 3 2 l 2 



+ «2 J. fi X Ha ( ad( l) 



2 2 . 4 2 . 6 



These three equations suffice to exemplify the curious kind of 

 arithmetical formula? to which the 21 relations give rise. I have 

 not examined in detail the complete system of results. 



§ 45. It may be remarked that if the first term of the right- 

 hand member of the first two of the three equations be trans- 

 posed to the other side of the equation, the left-hand member 

 becomes the same in each case, and, supposing n = 4 as before, we 

 find 



1 2 .3 2 .5 2 .7 2 1 2 .3 2 .5 2 .7 _ 1 2 .3 2 .5 £ 2 



2 2 . 4 3 . 6*. 8 2 2 2 . 4*. 6". 8 ^ ' ~ 2 2 . 4 2 . 6 2 X 2 2 ^ ^ 



1 2 .3 1".3" , x 



1 1 2 .3 2 .5 2 

 + 22 x £2 42 (-2 (aaq) 



_ I 2 . 3 2 . 5 l 2 , 



~ 2 2 . 4 2 . 6 X 2 2 ( q) 



1 2 .3 1 2 .3 2 , N 

 + 2 2 74 X 2 2 74 2 ^ q) 



1 l 2 3 2 5 2 



+ 2 X 2 2 74 a f6 2(adtl) ' 



