﻿366 Sir James Cockle on Fractional Criticoids. 



since, in the critical case, X vanishes. Therefore we may now 

 write 



x (z) = aV + 2abs + bH + $z, 

 where 



$ = a*p + 2abo- + b 2 T + e{a% + b v )+aff. . . (7) 

 But 



( db 7 db\ de . de ' 



Hence, by substitution in (7), 

 . / da -. da\ > /, c$ , db\ de T de , 



And this is the criticoid, very general in its nature ; for inas- 

 much as the two equations of the group (1) are now equivalent, 

 one to the other, ? is to be determined from the linear partial 

 differential equation of the first order, 



4 + *| + '=°' < 8 > 



and the condition (6) will then be satisfied identically. 



9. If the sum of the first two terms of S, viz. 



(da b da\ «- / « db db\ 

 dx a dy) \ b dx dy) 



be proportional, term for term, to ag+bj), then 



da b da __db a db . 



dx a dy~~ dy b dx' ' * ' 



and when (9) holds, then, in virtue of \=0, or of (8), the criti- 

 coid becomes, after reduction, 



(da b da \ (de b de \ , << 



* = \d X + l7y- e ) e -\<kc+alty-&> ' ' < 10 > 



and we have the paradox of a definite criticoid obtained by an 

 arbitrary transformation. The explanation of this anomalous 

 result is as follows. 



10. Let D and A be symbolical operators defined thus, 



D— —4-6— A- 2^!±2 J~+i 2 — • 

 ~~ dx dy* "" dx 2 dxdy ay 2 ' 



then, an operator operating upon all that follows, 



D 2 — DD — A4- ( ^—4-b—*\—4-( —4-h^\-' 

 \ dx dy J dx \ dx dy) dy 



