1884.] identified with Sir J. Cockles Criticoids. 57 



unchanged after a change of one variable, and nltra- critical when it 

 so remains after a change of more than one, then seminvariants and 

 invariants are respectively critical and ultra-critical algebraical func- 

 tions. Criticoids may be called ordinary, or of the first class, when 

 the dependent variable is changed, and extraordinary, or of the second 

 class, when the independent variable is changed. A criticoid, whether 

 of the first or second class, is in the nature of a semin variant, not of 

 an invariant; it does not correspond with an invariant. It is critical, 

 not ultra-critical. In differential expressions we have critical functions 

 or criticoids. But we have not in general ultra-critical functions, or, 

 as it is proposed to call them, Invaroids. To obtain such functions it 

 is necessary to change two variables, the dependent and the indepen- 

 dent. In a recent letter to me Sir James Cockle suggests that in a 

 limited number of cases it may be possible by means of semicritical 

 relations* to form invaroids, that is, ultra-critical functions of the 

 calculus analogous to the invariants or ultra- critical functions of 

 algebra. Such functions may have a place in the general theory, but 

 it is not necessary to pursue the subject here. 



17. In tracing out the analogies between algebraical and differential 

 equations, Sir James Cockle has been led to consider another class of 

 criticoidal functions, namely, those connected with partial differential 

 equations. f How the criticoidal functions of the first and second 

 class may be combined, what are the uses of criticoids, and what is 

 the part they play in the theory of the solution of differential 

 equations,}; are points illustrated by numerous examples in the 

 memoirs from which I have drawn most of the materials of this 

 communication. 



The Society adjourned over the Christmas Recess to Thursday, 

 January 8th, 1885. 



* Let/(Q) be a function of the Q's and their differential coefficients with respect 

 to t, and lety(P) be the same function of the P's and their differential coefficients 



with respect to a?; also let <p(X) denote a function of X^ = {^-^j ^ with re- 

 spect to x. Then if 



f(Q)dt*-fP)dx» = <e(X.)dx* (A), 



he calls (A) a semicritical relation. Thus 



dt ) \ dx I 3 X* dx°- 



is a semicritical relation which, notation apart, includes (11) of Art. 53 of the con- 

 cluding paper " On Linear Differential Equations of the Third Order," cited in 

 footnote *, p. 51. In like manner (17) of Art. 55 of that paper is a semicritical 

 relation. 



f Cockle. " On Fractional Criticoids," ** Philosophical Magazine " for May, 

 1871, toI. xli, ser. 4, pp. 360-368. 



% In the concluding paper " On Linear Differential Equations of the Third 



