XXXV 



was followed up by the same writer in two papers on the " Theory of 

 Quinties," in the ' Quarterly Journal of Mathematics,' and also in an 

 exposition of Cockle's " method of symmetric products " in the 

 ' Phil. Trans.' for 1S60. The study of these papers led the late Pro- 

 fessor Cayley to investigate the subject, and his results were embodied 

 in a memoir entitled " On a New Auxiliary Equation in the Theory 

 of Equations of the Fifth Order," which appeared in the 'Phil. 

 Trans.' for 1861. Cockle had calculated the auxiliary equation for 

 one of the trinomial forms to which the quintic may be reduced 

 without any loss of generality, hence the simplicity of his result. 

 Cayley, employing an invariantive process, calculated the same equa- 

 1 ion for the complete quintic, that is, the quintic not deprived of any 

 of its terms, and not modified in any o£ its coefficients. The result 

 is, of course, less simple than that for the trinomial form, but it has 

 the advantage of being absolutely complete. Thus Cockle's labours 

 on the quintic invested the theory with a new interest, and the methods 

 he devised, and the results he obtained, largely directed the course of 

 subsequent speculation on the subject. 



His mode of dealing with the theory of differential equations was 

 equally marked by originality and independence of mind. Not con- 

 lining himself to the beaten track, he pushed his way into unexplored 

 regions, and succeeded in bringing to light important relations and 

 analogies between algebraic and differential equations. Two ex- 

 amples may be given. He found that from any rational and entire 

 algebraic equation of the degree n, whereof the coefficients are 

 functions of a single parameter, we can derive a linear differential 

 equation of the order n — 1, which is satisfied by any one of the roots 

 of the algebraic equation. Out of this germ has grown the theory of 

 Differential Resolvents. To Cockle also belongs the honour of bems: 

 the first to discover and develop the properties of those functions 

 called Criticoids or Differential Invariants, so called because they 

 remain unaltered when the differential equation is transformed by a 

 change of one of the variables, and are therefore analogous in this 

 respect to the critical functions or seminvariants of common algebra. 

 Criticoids seem destined to play an important part in the theory of 

 linear differential equations. 



But it would be impossible, within the limits at our disposal, to 

 discuss in detail Cockle's various discoveries in algebra and the 

 calculus. Enough to say here that his work was eminently initiatory. 

 He started theories, but left others to elaborate and perfect them. 

 Of his eighty or ninety papers given to the mathematical world, 

 many are no doubt slight and fragmentary, but there are few, even 

 among the shortest and least complete, which do not contain original 

 and valuable suggestions. He struck out ideas which have taken 

 root in other minds and borne fruit. 



