Sir James Cockle on Criticoids. 207 



u=a *-<- d i£> • (34) 



A = « 3 -3«,« 2 + 2a?-^', (25) 



rl 3 n 



V = fl 4 -4a 1 flr 8 -3flJ+12fl;fl a — GflJ-.^ 1 . . (26) 



I call □ , A? and V basic criticoids. There are no quadricriti- 

 coids which are not functions or multiples of □ , and no cubicri- 

 ticoids which are not functions or multiples of A ; but all values 

 of the expression 



V + aD 2 , 



wherein the factor X is arbitrary or indeterminate, are quarticri- 

 ticoids. The degrees of the foregoing criticoids are the greatest 

 suffixes which occur in them respectively. I call those criticoids 

 basic on account of their simplicity. In each only one single 

 differential coefficient occurs; and into each the coefficient of 

 greatest suffix and the differential coefficient enter only linearly, 

 neither being multiplied into the other or into any differential 

 or other non-numerical coefficient. 



8. Secondly, in the equations (9), (10), (11), and (12) put 



m 1 + « ] =0, (27) 



and let the result of this hypothesis be to change A,, A 2 , A 8 , and 

 A 4 into B x , B 2 , B 3 , and B 4 respectively. Then we have, Bj va- 

 nishing, 



= B,j (28) 



and, replacing a { by ( — w,) in (10), (11), and (12), we have also 



M 9 -2Mj + flr 2 = B a , (29) 



u 3~ •^u l u 2 + Sa q u i +« 3 =B 3 , (30) 



u 4 — 4^3 + 6<7 2 w 2 + 4ff3W x -M 4 = B 4 ; . . (31) 



which again, in virtue of (2), (3), and (4), respectively become 



S&-«J + «.-B» (32) 



^_2^ + 3« 2 « 1 + ff8 =B 3 (33) 



^ + 6«J-12MX + 3w| + 6« 2 M 2 + 4a 3 M,+a 4 = B 4 . . (34) 



Now replace, as we may do, u x by (—#,), and (32) and (33) re- 



