204 Sir James Cockle on Criticoids. 



theory can be completed in a satisfactory manner, the operator 

 would, it seems to me, affect separately the parts involving and 

 the parts free from differentiation, and so render a further test 

 requisite. 



4. It will be convenient to use the following notation. Put 



1 d r u , 



ud^= U » W 



then we have, identically, 



^-£=0, (2) 



^-3^ + 2^-^=0, (3) 



5. If the quantoid 



dx* 



d 3 u 

 ^-4^3-3^+ 12 w 2 M2 _6wJ— -7-3=0. (4) 



a (l, a v a 2 , . . «nj^' 1 ) n y=tfn ... (5) 

 be transformed by the factorial substitution 



. y=uY (6) 



into the quantoid 



A (l,A 1) A 2> ..A„X^,l)»Y = Y„, . . (7) 

 Y n being equal to y n , then generally 



(1, a„ « 2 , . . am£j-> I) m u=A m u. ... (8) 



efficients of the original quantoid. In this case the criticoids are not iden- 

 tical after transformation, but one is a factor of the other. 

 Secondly. If we study the divided quantoid (y n )-i-y, wherein 



(y«)= (i> «i> a 2 , • . a «X^ l ) n y> 



we obtain results very closely conforming to some of the analogies of alge- 

 braical notation ; for, adopting the notation signified by equation (1) of 

 the text, we have, applying that notation on the dexter only, 



y 



provided that after expanding the dexter we change exponents into suffixes. 

 And the transformation (6) of the text will be indicated as to its results by 



™=(Y+A)», 



on the dexter of which the like change is to be made, and for which equa- 

 tion (13) of the text is supposed to hold. The use of the divided quantoid 

 facilitates the definition of a semicriticoid. 



