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



47 



which is Professor Malet's function H. 



4. In the third, paper of the same series it is noticed that the 

 "characteristic" H remains unchanged after the substitution of fY 

 for y ; and that if the given linear differential equation be soluble, 

 then 



f| + 2Q 1 4^+f ^1 + Q 1 2 -H)r=0, 



dx 2 dx \dx J 



and therefore also 



^ + 2(P 1 + a)^+(P 3 + 2Pa + a 2 )Y = o 

 dx z ax 



are soluble, the last two equations being connected by the relation 



Q 1 = P 1 + a. 



5. But these and other results are included in the more general 

 forms subsequently deduced from the linear differential equation of 

 the nth order. It is shown that the form of the function H is 

 independent of the order of the differential equations. 



6. " There are critical functions of the coefficients of differential 

 equations analogous to those £ critical functions ' of the theory of 

 algebraic equations which, in the theory of quantics, are termed 

 leading coefficients of covariants, penin variants, or semin variants."* 

 By a critical function of the roots or coefficients of an algebraic eqna- 

 tion is meant a function which remains unaltered when each of the 

 roots is increased or diminished by any quantity whatever, or which 

 remains unaltered when x + h is substituted for x in a given equation 

 in x ; it is in fact what Professor Cayley calls a seminvariant, that is, 

 a function which is reduced to zero by one only of the operators 

 which reduce to zero an invariant. The forms of these functions are 

 now well known. About forty years ago Sir James Cockle called 

 attention to them in the pages of the " Mathematician. "f Since then 

 he has discussed them more fully in other journals, % pointing out the 



# Cockle. " Correlations of Analysis," " Philosophical Magazine" for 1862, vol. 

 24, ser. iv, p. 532, § 2. 



f Cockle. " On the Transformation of Algebraic Equations," " Mathematician," 

 vol. i, No. 2, March, 1844, pp. 82-84 (see also vol. i, p. 299) ; vol. iii, No. 4, 

 November, 1848, pp. 176-178 ; and Supplement, No. 7, September, 1850, pp. 27-34. 



X Cockle. " On the Existence of Finite Algebraic Solutions of the General 

 Equations of the Fifth, Sixth, and Higher Degrees," " Philosophical Magazine," 

 vol. 28, ser. iii, March, 1846, pp. 190, 191. " On Critical and Spencian Functions, 

 with Remarks upon Spence's Theory," " Quarterly Journal of Mathematics," vol. 

 iv, pp. 97-111. " On the General Forms of Critical Functions," ibid., pp. 

 265-270. 



