348 Chief Justice Cockle on Quantoids. 



3. "Now, designating the differential co variants of which I 

 have already spoken (in the March and September Numbers of 

 this Journal for 1864) as covaroids, the functions y n> y mi and 



dPy 



-r~ are covaroids for all values of m, n, p and q, and conse- 

 quently any functions of D, A, and the higher criticoids, and 

 also of any number of values of y m and of -~ are covaroids, and 



vice versa. Accordingly the covaroid which I have called the 

 differential Hessian may, after a slight modification, be put under 

 the form 



o*-( A *■■£>• 



and is only one of an infinite number of quadricovaroids. For 

 this particular case (the differential Hessian), if we so determine 

 u as to cause the middle term of the quadricovaroid to disappear, 

 we shall have transformed the quantoid y 3 into a quantoid Y 3 , 

 whereof the coefficients will satisfy the relation 



dk, 

 dec 



4. This transformation, the possibility of which is thus mani- 

 fested a priori, leads, when discussed by change of the dependent 

 variable, after expulsion of superfluous terms, to the result 



A,A 8 -A 8 +^ = (11) 



(«^+a?)«+»n*=ft • • oi 



which is linear in u. A variety of other possible transformations 

 is thus indicated a priori. 



5. The symbolical decomposition 



(A. IV— -iV — 



\dx x/\dx x) da? ' 



is remarkable, and leads to the conclusion that if a^a^o^,.., a n _ 2 

 be constant, and 



j-.^^ w) 



the equation y n — is depressible by one order. By reversing 

 the symbolical factors on the sinister of (13) we obtain a result 

 which, though slightly more complicated than (14), is still com- 

 paratively general. 



Brisbane, Queensland, Australia, 

 August 16, 1865. 



