50 Kev. Robert Harley. Prof. Malet's Invariants [Dec. 18, 



■which will be found to possess distinct advantages over the other two ; 

 it is more direct, perfectly general, and easy of application ; moreover, 

 it determines the criticoids in their simplest, that is, basic forms. 



10. Besides the class of functions hitherto considered, there is 

 another class of a criticoidal nature, the discovery of which is also due 

 to Sir James Cockle. Treating the term Critical as generic, we have 

 at least two species, namely, the algebraical critical species, now com- 

 monly known as seminvariants, and the differential critical species, or 

 criticoids, of which there are at least two varieties, namely, the 

 ordinary, and that which Sir James Cockle calls the differential 

 criticoid. But in fact both varieties are differential criticoids, the 

 true distinction between them being that one set of functions remains 

 unaltered when the quantoid is transformed by factorial substitution, 

 and the other when the transformation is effected by change of the 

 independent variable. With the latter, Professor Malet's second class 

 of invariants may be readily identified. 



11. Consider the equation of the third order 



(l, P v P 2 , P 3 jA,l) 3 2/=0, 



and suppose that, by a change of the independent variable, this is 

 transformed into 



(i, Qi. Q 2 > Q3$J^)Ws 



then, denoting differentiations with respect to x by acute accents, and 



with respect to t by grave accents, and remembering that t' = \, we 



< * "' < "' ' ' ' x 



have, by the usual formula?, 



<fc=Pi»'-£, 



x 



q^v^y-F^+^y-l . x -, 



\x / ox 



and therefore, since P'=PV, 



Q3=P3'W 4 +3P 3 (^) 2 ^=§! . Q,^^3Q 3 (P^-Q 1 ) ; 

 or, what is the same thing, 



Similarly, it is shown that 



Q; + 2Q X 2 - 3Q 2 = (P/ + 2P/ - 3P 2 ) (xy. 



