228 Mr. J. Cockle on Differential Covariants. 



then the above differential covariant may be written 



+ (-2b 4 + 4b*c-2b C2 b'-bf+bc , + cb'-c ,2 )y. 

 The binordinary differential critical function of this covariant 



1* 



* + \ 2K, ) dx\ 2K„ )' 



'2 



and is the same as that of 



"-C^r-V-** 



\ ^ 1V 2 



the accents denoting differentiation with respect to the indepen- 

 dent variable x. Each of the coefficients of this function is 

 critical. 



The parallelism between some of the foregoing results and 

 those of the algebraical covariant theory will perhaps be better 

 seen if we consider the complete form 



For this form we have, say, 



K^P-ac + ab'-ba', 



K 3 = 2b 3 - Sabc + aj- a (ab" - ba") + 2a\ab f -ba'), 



and the second and last coefficients of the differential covariant 

 obtained in this paper become, respectively, 



■p — bc— af-\- ac' — cd , 



q = ^-bf+bc , -cb , -2(K 2 -b')K 2 . 



The resemblance of these expressions to those which occur in 

 the ordinary Hessian, though not perfect, is marked. I first 

 deduced the differential Hessian (for the case a = l) by seeking 

 an expression for which the coefficients of transformation and of 

 substitution should be the same, and I afterwards found that an 

 operator, to wit A> reduced it to zero. I take this opportunity 

 of referring (in connexion with binordinary differential critical 

 functions) to my three other papers " On Linear Differential 

 Equations of the Second Order" in the first volume of the 

 Oxford, Cambridge, and Dublin Messenger of Mathematics. 



Brisbane, Queensland, Australia, 

 December 11, 1863. 



