Sir James Cockle on Criticoids. 



309 



a quantoid. For, let 





j\x + a l )=a {l, a lf « 2 , ..alfcv, l) n ; . 



. . (42) 



then we have 





f(x) =fl (l, a l9 fl a> . . a n Jx-a v l) n , . 



• • (43) 



or, which is the same thing, 





/(#)=«o(l,0, B 2 , B 8> ..B B X*,1)*, • ■ 



, . (44) 



provided only that 





B 9 =fl a — «;, 



, . (45) 



B s =flr 3— 3«j« 2 + 2flJ, 



, . (46) 



&c. = &c. ; 



and B 2 , B 3 , &c. (B, vanishes identically) are the primary critical 

 functions. That they are critical in this sense, viz. that they 

 remain unaltered under the linear substitution, appears from the 

 circumstance that 



/(# + A 1 )= ao (l,A 1 ,A s ,,..A„X*, 1)" • • (47) 

 leads to 



/(*) = « (1,0 ) B 2 ,B S) ..B,J5, 1)", . . . (48) 



which is identical with (44). We have in fact a set of relations, 



B S =^-««=A S -AJ, (49) 



&c. = &c. = &c, 



which show that the transformation of /(#) by the linear substi- 

 tution leaves the critical functions unaltered. 

 10. We may treat the quantoid 



f{eA^y) = ef^(l,a„a i ,..a n X^,\fy . . (50) 



in an analogous manner, remembering that a factorial now re- 

 places the linear substitution. Thus we have 



f(y)=e/^(l, a v a 2 , . . aj±, l)"(-J^). . . (51) 



Now 



Phil. Mag. S. 4. Vol. 39. No. 260. March 1870. 



(52) 



