Sir James Cockle on Criticoids. 203 



ence of criticoids of all degrees. Lastly, I did not attempt to 

 employ an operative symbol; for I am not satisfied with the 

 theory of the symbol as it at present stands*; and even if such 



* See papers " On Differential Covariants " (Phil. Mag. for March 

 1864) and "On the Operating Symbol of Differential Covariants" (ibid, 

 for September 1864), in which I used or implied the use of the term differ- 

 ential covariants in reference to certain functions which in the paper " On 

 Quantoids " (ibid, for November 1865) I called covaroids. But such func- 

 tions are analogous, not to covariants, but to a more restricted form of 

 algebraical function. From the two forms 



(1, a lt a 2) . .a n $x, ^) n , (l, A 1} A 2 , . . Ajjjr, l) n 



we may, by making n = \, 2, 3, &c. successively, derive two sets of qualities. 

 If the linear substitution of x+h for x converts the last quantic of one set 

 into the last quantic of the other, it will convert every quantic of one set 

 into the corresponding quantic of the other. Let us then call all the quan- 

 tics of a set uniform. Then a function of uniform functions is uniform, 

 for the linear transformation will transform it into a corresponding func- 

 tion. If a uniform function, using the term in its general sense, be poris- 

 matic with respect to x (that is, if x disappears from it), the uniform function 

 is critical. If x disappears partly, but not wholly, the terms in which it oc- 

 curred being replaced by critical functions, the uniform function so modified 

 may be termed a semicritical function. The differential functions which I 

 have called covaroids, but which ought rather to be called semicriticoids, 

 are the analogues of these algebraical semicritical functions, which latter 

 may be denned as functions of critical and of uniform functions. If we di- 

 vide each side of equation (8) of the text by u, either side of the equation 

 so divided may be regarded as the form of a set of uniform differential 

 functions, and any function of such functions will be uniform under facto- 

 rial substitution. If u disappears from a uniform function, such function 

 is a criticoid. If u disappears partly, but not wholly, the disappearing 

 terms being replaced by criticoids, we have a semicriticoid, which may be 

 defined as being a function of criticoids and of uniform differential func- 



dPy q 

 tions. In my paper " On Quantoids," the expression — — should be re- 

 placed by B p yq ; and, in conformity with the terminology of this footnote, I 

 should now paraphrase art. 3 of that paper to some extent by saying that 

 y m , y n , and 8 p y q are uniform, and that any function of any number of such 

 expressions and of criticoids is a semicriticoid. The paper " On Differen- 

 tial Covariants " suggests two remarks. 



First. If we study the quantoid y n under the homogeneous form 



y n =(a , a v . . a»X^' l T y> 



and call its basic criticoids k 2 , k 3 , &c, and those of the transformed quan- 

 toid K 2 , K 3 , &c, we shall in general have 



U m k m = Km, 



wherein, for instance, 



k 2 = a Q a 2 — a\— («o-p — «i -^J> h= Szc, 



and Km is the same function of A , Ai, &c, the coefficients of the trans- 

 formed quantoid that k m is of a , a } , &c. the corresponding respective co- 



