1884.] identified with. Sir J. Cockle s Criticoids. 



49 



wards given in a paper on Criticoids,* where other results obtained 

 by Professor Malet are anticipated. For example, it is shown that 

 when, by the usual process, the differential equation 



(i,p 1 ,p 2) ...p^,i)W 



is deprived of its second term, it takes the form 



(l l5 0, H„ a, I + 3H2, . . . J^i)y=o ; 



and this is identical with Professor Malet's equation (2). 



8. In the same paper the author defines what he means by criticoids, 

 and indicates the different methods by which these functions may be 

 calculated. He says, " A criticoid stands in the same relation to a 

 factorially transformed linear differential equation that a critical 

 function fulfils with respect to a linearly transformed algebraical 

 equation." The word quautoid is used to signify "the sinister of a 

 linear differential equation whereof the dexter is zero." H is called a 

 quadricriticoid, Gr a cubicriticoid, and I a quarticriticoid of the 

 general quantoid, the degrees of the criticoids being the greatest 

 suffices which occur in them respectively. These criticoids are called 

 basic; they are so called 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." There are no quadricriticoids 

 which are not functions or multiples of H, and no cubicriticoids 

 which are not functions or multiples of G ; but all functions of 



I+XH2, 



where \ is arbitrary, are quarticriticoid s. 



9. In concluding an inquiry into the analogies between algebraical 

 and differential critical functions, Sir James Cockle anticipates Pro- 

 fessor Malet's proof of the general theorem on which the doctrine of 

 criticoids rests, showing that as the transformation of a quantic by a 

 linear substitution leaves the critical functions unaltered, so the trans- 

 formation of a quantoid by a factorial substitution leaves the criticoids 

 unaltered. (Compare Cockle on " Criticoids," Article 10, with Malet 

 on a " Class of Invariants," p. 752.) There are three ways of calcu- 

 lating ordinary criticoids. First, by elimination; secondly, by de- 

 priving the quantic of its second term ; thirdly, by means of hyper- 

 distributives,! a process which it is not necessary to explain here, but 



* Cockle. " On Criticoids," " Philosophical Magazine" for March, 1870, vol. 

 39, ser. iv, pp. 201-211. 



f Cockle. " On Hvperdistributives," " Philosophical Magazine" for 1872, vol. 

 43, ser. iv, pp. 300-305. 



VOL. XXXVIII. E 



