210 Sir James Cockle on Criticoids. 



r]n-\ f d n ~ l v 



-H»-l)«?£S + &«=•}. («*) 



»¥ ^ £^rf*y) = e -/^{»^ « 2 £^-&c.}.(54) 



Hence, collecting results and making the proper substitutions 

 and changes in (51), that equation becomes 



„ , d n y , w — 1 / 2 da x \d n - 2 y Q , KK . 



which may be replaced by 



/(y) = (l,0,B 2 ,B 3 ,..B„J^, 1)"*, (56) 



But if we transform the quantoid 



f(e/W*i,)=e/^(l,A i ,A 2 ,..A n £~,iy'!/ . . . (57) 



in such manner as to deprive it of its second term, we shall be 

 conducted to (56). Thus, as the transformation of the quantic 

 by the linear substitution left the critical functions unaltered, so 

 the transformation of the quantoid by the factorial substitution 

 leaves the criticoids unaltered. In either case the primary trans- 

 formation is such as to deprive the quantic or quantoid of its 

 second term ; and the critical functions and criticoids so obtained 

 are what I term primary. I do not continue the development 

 commenced in (55), because the criticoids of the second, third, 

 and fourth degrees are hereinbefore exhibited. 



11. Independently, then, of the resolution of a quantic into 

 linear factors, or of a quantoid into linear symbolical factors, 

 and whether such resolution be theoretically possible or not, 

 there are certain functions of the coefficients possessing characte- 

 ristic properties. Such functions are obtained by means of the 

 primary transformation, and are invariable under the appropriate 

 (linear or factorial as the case may be) substitution. They admit 

 of a third definition, which may be expressed as follows. 



12. If, after expanding the sinister of (13), we replace the 

 indices of a by suffixes (i. e. write a c .instead of a c ) and leave the 

 indices of u standing (i. e. u c remains u c ), and if we call the 

 sinister, so modified, the quantic corresponding to A m , then a 

 critical function is a function wherein the substitution for each 

 coefficient a r of its corresponding quantic A,, causes no change in 

 the function, from which, indeed, u disappears. Again, if we 

 divide either side of (8) by u and call the quotient on the sinister 



