206 Sir James Cockle on Criticoids. 



But, by the characteristic property of the quadricriticoid, 



£-.£-*.-*m*-o. ^ 



hence we may write, and reduce, (17) as follows : 

 12(ff,-^)(u 2 -«f)+6(«,-^)' 



= 6{A 2 -A? + (a 2 -«f)}{A 2 -A?-(« 2 - fl ?)} 



= 6{(A 9 -A I )2-(« 2 -«?) 2 } (19) 



We also have, from (9), 



d 3 u l _ d 3 A l d 3 a x 



dec 3 dos 3 



(20) 



Consequently making in (16) the substitutions indicated by (19) 

 and (20), we have 



d 3 A d 3 a 

 A 4 -4A I A s +8A»=« 4 -4a 1 «3 + 3flJ+ -jj - -^ 



+ 6(A 2 -A^-6(« 2 -<) 2 , (21) 



which, after transpositions, becomes 



A 4 -4A 1 A 3 + 3A^-6(A 2 -A?) 2 



d 3 \ } 

 dx 3 



d 3 a 



= « 4 -4« 1 « 3 +3«l-6(« 2 -<) 2 -^-. . . (22) 



7. We may write (22) thus, 



d 3 A 

 A 4 -4A 1 A 3 -3A^ + 12A 1 A 2 -6At- *-£ 



d 3 a 



= tf 4 -4 V3 -34 + 12«X- 6 <- ^ph ■ ( 23 ^ 



and we see that the expression on the dexter of (22) or (23) re- 

 mains unchanged after the factorial substitution denoted by (6) , 

 and consequently that such expression (that is to say, 



d 3 a 

 a 4 -4a 1 a 3 -3al+l2a^-6a^- —±, 



which I shall represent by V) is a quarticriticoid. If we put 



