TRANSACTIONS OF SECTION A. 603 



When the coefficients a^, «„ a^, &c., are replaced by a, b, c, &c., respectively, 

 the operator rr becomes equivalent to 



«§(, + 2bb^ + Scd^ + &c., 

 and we recognise in the last result a well-known theorem. When 7r<f>(a) = 



we have ^(A) = (^(«), and the umbral notation enables us to exhibit one form 

 of <f>, viz., 



<^(«)=F,(«) = (a-^^)^, 



where r is any positive integi'al number, not less than 2 and not greater than n, the 

 highest suffix of a. 



The notation is next applied to the determination of decriticoids. Any linear 

 differential expression of the wth order, 



d"ii d"~h/ dv 



where ff^, «,, . , . a,, are functions of x, may be changed into the non-linear 



form 



1 d'^u 1 d"-^i/ 1 (/// 



y d.v» y dx-^-^ y dx "' 



either by dividing by a^ y and replacing ^ by a,., or by making ff^ = 1 and dividing 



ty y- Write y,. for - -^ ; then the above non-linear form will be expressed, in 



the umbral notation, by 



{y + n)n- 

 Consider the effect of substituting uy for y in the differential expression, u being 



any function of .r. This substitution being made in -^ ^, we obtain - *X?f?0 



y dx'' uy dx'' ' 



which is readily shown to be equal to (K + y),., u and y being both umbral. It 

 hence appears that the substitution of n + y for y in the umbral form is equivalent 

 to the substitution of uy for y in the ordinary differential form. Effecting the 

 substitution and expanding, we have 



(y + at + a)„ = y„(it + a)g + Mjy„_j(2< + a)i+ • • • +"iyi(M + «)„-, + (« + a)„, 

 so that, writing A for m + a, the changed coefficients are 

 Ao = (m + «)o=«o--Jo = 1. 



Ai = (m + «)i = Z«j + CTj, 



Aj = (m + a)^ = M2 + Swjflj + «,, 



A^=(u + a)^ = u^. + 7\Ur_.ya^ + r.,>/,._..a.,+ . . . +r2K„ar_2 + riMja,._j + n,.. 

 And since Aj - aj = m,, therefore 



(?'Ai_rf''aj_d'w, 

 «tf^ dx'' dx'' ' 

 an equation whose dexter may be developed in terms of Mj, u„, . . . u,.+^. Repre- 

 aenting this development by 0^+^ {ti) the author shows that 



a criticoidal relation. In determining the form of 0, two theorems are used, viz. 



dUr -ad, 



-^ =U,.+l-U,Un and ^,(" -«l)r=(«- «l)r+l -'•(«- «'l)2(«-«l),-l- 



These were given, without demonstration, by Sir James Cockle in his paper on 

 Hyperdistributives. 



The former is readily proved ; for 



dur^d_/l d''u\ _ 1 dr+hi_l du A d'-u\ 

 dx dx\u ' dx") u ' dir''+' u' dx' \u' dx^J 



