Sir James Cockle on Criticoids. 205 



Hence, availing ourselves of the notation of art. 4, we have the 

 system 



Mj + o^A,, (9) 



u^-\-2a 1 u 1 + a 2 = \ 2i (10) 



w 3 + 3« 1 w 2 + 3« 2 w 1 H-flf 3 =A 3 (11) 



tt 4 + 4« 1 M 3 + 6ff 2 w 2 + 4a 3 tt 1 +0 4 =A 4 , . . . (12) 



whereof each member is capable of being represented by 



{u + a)™ = A m , (13) 



provided that after expansion we change exponents into suffixes. 

 6. All critical functions, under which term both invariants 

 and sernin variants may be included, are capable of being exhibited 

 as rational and entire functions of those critical functions which 

 I have denominated primary. In order to form a quarticriticoid, 

 I take a function corresponding to the quadrin variant of a quar- 

 tic, and I find that, on substitution and multiplication, many of 

 the terms cancel one another, and that on reduction we have the 

 following result, 



A 4 -4A 1 A s + 3AJ = fl 4 -4fl 1 ffl 3 + 3flJ + 12(fl a -a;)(tt 2 -tt5 



+ 3^-4m 1 m 3 + w 4 , (14) 



which, in virtue of (4), becomes 



A 4 — 4A 1 A 8 +3A» = fl 4 — 4a,fl 8 + 3flJ + 6^ — 12mJm 2 + 6m} 



d 3 v 



+ ^ l+12 ^-^-<)> • (15) 



whence, on reduction, we have 



d 3 u 

 A 4 -4A I A 8 + 3AJ = fl 4 -4fl 1 fl 8 + 3flJ+ -^ 



+ l2(n a -fl")K-i^)+6(« g -i4)» (16) 



Now, applying (2) and (9) successively, we find 



12(« 2 -a?)K-u?) + 6K-^)*=12(« 2 -«?)g 4- e(gi) 2 



^-4^-i) + K^-^ 



dA, 



dx dx J 



= qJ a a? da ^ | d K\( dA i da \ 

 L 2 l dx dx J \ dx dx) 



