604 KEPORT — 1887. 



And a proof of the latter is briefly indicated below : we have 



(ll — Ml),. = (U + )-U^J r = 2?'„,2<i"'2<r_m» 



r ! 



where ?•„ = (—)'"- '— — . and the summation extends from m-0 to m=r, 



{)• - m) \in\ 



[r^ = 1]. By the first theorem we have 



-5- (Mi ""«,._,„) = 2<i"'(m,._„,+i - MlM,_„) + m(M-Ml)2Mi"'-' !/,._„, 



and therefore 



— (ll—U^,. = 2{r,„Wi"'(?<r-m+l — MiMr_„,) + mr,„{u — Ui).^Ui"'-^Ur-m} • 



On effecting the summation between the assigned limits m = and m = r, and 

 reducing by means of the relations 



'm - '■»,-! = ('■ + l)m> ''"'^d ""■», = - '■ ('— l)r-P 

 the truth of the second theorem becomes apparent. 



By the aid of these theorems it is easy to calculate the non-differential portions 

 of decriticoids. Write U for the penumbral form u — u^, then 



and by successive differentiations we obtain 



d.v^ dx 



«.r' dx 



^^ (") = S' = i. (U4 - ^u,,o = U5 - 10 u,U3, 



dx"^ dx 

 d^ 

 'dx 



e^ (u) = "5^1 = Ug - 15 U.,U, - 10 U3- + 30 U,', &c. 



Hence 



^3 (''') = (« - «i).> = «2 — O'l'*, 



e^ (a) = {a- a,)3 = «3 - 3 a,a.^ + 2 a j', 



5( («) = (« — a,) , - 3 (a — ai)^- = a^ - 4 ajff, - 3 «./ + 12 ffj^ffj - 6 «!*, 



^0 ("') = ('^ — «i)5 — 10 (« — «i)2(« ~ ^''Js = ''^s — 5 OyU^ — 10 rtjffj + 20 a^-a^ + SOa^a^* 

 — 60 Ui^a^ + 24 tti'', 



e^ (a) = (« - aJs - 15 (« - o!i), (« - a,)^ - 10 (« - a,)l + 30 (« - a,)l = &c. 



and the law of derivation is obvious. 



The umbral notation is equally effective in deaUng with incriticoidal forms. 

 Various examples are given in the paper, and the author carries his investigation 

 as far as the determination of the quadrincriticoid, that is to say, the incriticoid 

 of the fourth degree, the degrees of criticoids being the greatest suffices which 

 occur in them respectively. It is proposed to call a decriticoid of the m-th degree 

 an ?H-ide, and the incriticoid of the ?n-th degree an wt-ine. 



4. On Criticoids. By Robert Rawson, F.B.A.S. 



The method proposed in this paper was suggested by a study of the Rev. 

 Robert Harley's paper entitled Professor Malet's Classes of Invariants identified 

 with Sir James Cockle's Criticoids, printed in the ' Proceedings of the Royal 



