which occur in the Calculus of Invariants. 463 



l. e. 



-*(.£+»£+*£..> 



Then if I be any function of the coefficients a,h,c, . . . ; a', V, c' } . . . 

 in the algebraic forms (a, b, c, . . .) (x, l) p ; {a',b', c 1 , . . .) (x,1)p . . . , 

 and Ij is what I becomes when we substitute for a, b, c,. . .; 

 a 1 , b', c\ . . . , the values which these coefficients assume when 

 x + h is written in place of h, it is, or ought to be, well known 

 that 



I 1 =I + E 1 *IA+(B l *) 2 I T ^+(E 1 *) 8 I T -| : g+ i .. 



Here 



■^■.•■.-».n{«£+«|+-..). 



it will therefore become convenient slightly to depart from the 

 notation applied to the general form </>, and to write 



■.-*(«*+»£ + ■•■■) 



E " =2 V a S- +ni S£ +,, -F- &„ + '--> 



where « w , £„, c w , . . . are used to express the elements n, steps 

 more advanced than a, b } c, . . . respectively ; we have then by 

 the general theorem 



e'Ei*=( e T)*, (B) 



where T now takes the form 



I propose to give to the E series of operators the general name 

 of Extensor Operators, or simply Extensors. 



The first remarkable, I may say marvellous, property of these 

 extensors is, that they form a sort of closed group ; i. e. any two 

 algebraical functions whatever of the extensors regarded as alge- 

 braic functions of the quantities a,b } c, . . . ; -=, — , . . . being 



