602 REPORT — 1887. 



By tliis process any polynomial, whether umbral or penumbral, may be developed. 

 Analogy requires that (.1-1- y)^ or (.i+y^o ^^s interpreted by .f (,?/(„ {.v + y + z)^, or 



(.r + 1/ + =5 hy .royo=o> and so on. 



The author shows how readily this system of notation lends itself to the 

 determination of both critical and criticoidal lorm.s. By critical forms are meant 

 those algebraical functions which remain unchanged when one of the variables is 

 augmented or diminished by any assignable quantity. As the leading coefficients 

 of covariants they are sometimes called seminvariants, being reduced to zero by 

 one only of the operators which reduce to zero an invariant. By criticoidal forms 

 are meant those differential expressions which remain unchanged when either the 

 dependent or the independent variable is changed. .Such expressions might per- 

 haps be called se^tiiniaroids. Criticoids which are unafiected by a change of the 

 dependent variable the author proposes to call decriticoids, and those which are 

 unafiected by a change of the independent variable he proposes to call incriticoidt^. 

 Sir James Cockle, to whom we owe the discovery of these forms, has termed the 

 first class ' ordinary criticoids,' and the second ' difierential criticoids ' ; but in fact 

 both are differential criticoids. 



To determine the general form of critical functions, the author considers the 

 eifect of the substitution of .r + uy for .r in the potence (.f + «?/)„ , a being an umbra, 

 and u, X, y radices. Writing A in place of « + u, the result obtained is 



F„(A) = (A-^;)^ = («-y=F.(«), 



a formula by means of w^hich critical functions may be calculated with great ease 

 and rapidity. When r = l, both sides vanish identically. When r = 2, 3, &c., 

 critical functions of the second, third, and higher degrees are readily found as 

 follows : — 



"o 



F4(a) = ;^(V«4 ~ ^tty^ffjffj + ewyfli'-rtj - .3«i'), &c. 

 Let TT be an operator such that 



TTrt,. = rflr-i, ""'■«,■ = ?•(;•— 1) «r_2, &C. ; 



then 



A^ = (a + u)^ = a^ + rjtf^_ ,?< + r.^a^._., m^ + &c. 



= flr +unar + rr~Tr"(ir + r-.j-o"'''^!- + &c. = e"''rt,. . 



And if we extend the meaning of rr so as to make it operate on powers and 

 products, thus 



i"(«p) = mpa'^'^Op.j, n{aj,a^) = agirfi,, + ft^.Trff^ =i^«p-i«!? + 1"i'^<i-v ^'^■> 



it is easy to see that when u is infinitesimal 



^ (A)^^(ff) + un<^ (a), 



where (p is integral with respect to «j, a.,, &c., and tt does not operate on a,, (or, 

 what is the same thing, jra^, = «„). Then, by a process similar to that commonly 

 -employed in the proof of Taylor's theorem, it is shown generally that 



^(A) = (f){a) + UTr(f){a) + -^ ..,7r-^(«) + ■ro73 7r'^(«) + &C. 



= e'"(^((j). 



