202 Sir James Cockle on Criticoids. 



show that they admit of. By a critical function of the roots or 

 coefficients of an algebraical equation I mean a function which 

 remains unaltered when each of the roots is increased or dimi- 

 nished by any (the same) quantity whatever, or which remains 

 unaltered when x-\- h is substituted for a? in a given equation 

 in x. The forms of these functions* are well known. The re- 

 searches of Spence in connexion with them are not, however, so 

 well known f- As seminvariants or as invariants, critical functions 

 play a conspicuous part in the grand theory of a later period, 

 the theory of covariants. 



2. 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. 

 I retain the word quantoid to signify the sinister of a linear dif- 

 ferential equation whereof the dexter is zero ; and that a general 

 quantoid has a quadricriticoid and a cubicriticoid I have already J 

 shown. Illustrations of the use of the quadricriticoid will be 

 found in places mentioned below §. The object of this paper is, 

 not to apply the theory, but to show the existence of criticoids 

 of higher degrees and actually to deduce expressions for quarti- 

 criticoids. 



3. In attempting to obtain quarticriticoids various courses 

 presented themselves. First, I deduced them, as in the lower 

 cases, by elimination. This course has its advantages. Its dis- 

 advantages are want of directness and of generality, and a failure 

 of proof of the existence of criticoids other than those actually 

 obtained or deducible therefrom by combination. Secondly, in 

 conformity with the analogies of the theory of algebraical equa- 

 tions, I assumed that the coefficients of a quantoid whereof the 

 second term is made to vanish are criticoids. This course, though 

 not devoid of advantage, labours under the same disadvantages 

 as the first, with the additional one that the results require test- 

 ing. Thirdly, I followed a new course which gives by a direct 

 process results perfectly general and makes manifest the exist- 



* Some information respecting them will be found in my papers respec- 

 tively entitled " On Critical and Spencian Functions, with Remarks upon 

 Spence's Theory" (Quarterly Journal of Mathematics, vol. iv. pp. 97-111), 

 and " On the General Forms of Critical Functions " (ibid. pp. 265-270). 



t Of Spence's (posthumous) i Outlines,' &c. it is said that there were 

 only eighty copies printed. 



% " Correlations of Analysis" (Phil. Mag. S. 4. vol. xxiv. pp. 531-534). 

 The same results are exhibited under a slightly different form in my paper 

 "On Quantoids" (Phil. Mag. for November 1865). 



§ See my " Notes on the Differential Calculus," published in the ' Mes- 

 senger of Mathematics ' (vol. hi. pp. 42-50, and pp. 247-256). In a paper 

 " On the Integration of Differential Equations" (Mathematical Reprint of 

 the Educational Times, vol. ix. pp. 105-112) I have to some extent em- 

 ployed the quadricriticoid. 



