464 Prof. Sylvester on the Properties of the Test Operators 



used as new operators and applied in succession to the same 

 operand, the result is the same as if some single third algebraical 

 function of the extensors had operated alone on this operand. 

 The second great fact is, that the order in which the above- 

 described operations take place is indifferent, i. e. that the two 

 operators above described are commutable; in other words, we 

 have always 



^ iJ E 2 ,E4,.;.)*f(E 1 ;.:E 2) E 8) v..v)* ^ 



=n(E 1 ,E 2 ,E 3> ...)* y ..(C) 



##(E„ E„, E 3 , . . . )* 0(E 1; B 9 , Eg, . . . )*. J 



Thus, ex. gr., 



E''*Ei* = ¥(B. E, )*=E?*E<*, 



where, writing m = v-k^A, F represents the quasi hypergeorne- 

 trie series, 



E i+ J + • rj . m W + J- 2 . E 2 + ^'"^./p'-l) m 2 E Hi-4 g E 2 > 



Ij, ' J fj. fy 1.2 M 2 M 



and E* * E^'* will be expressible under a form ^w«m proxime ana- 

 logous. My immediate intention in this brief notice being 

 merely to call attention to the surprising properties of these 

 functions, I shall conclude with adding a slight extension of 

 theorem (B) above given, viz. 



e m x * -- (e T )sic. 



This may be regarded as a particular case of a more general 

 theorem which I have discovered, viz. 



E^fifci* =e tE i*E{*=(( d jYe T \ J 



a theorem which, with a simple change in the coefficients of T, 

 may be extended to the still more general form E{* e tE «>*, so as to 

 give a simple solution of the equation 



x*=(E a *)«i;*, 



where X is a form to be determined as an algebraical function 

 of E w , E 2w , E 3w , &c. . . . 



The cardinal problem to be solved in the theory of extensors 

 is the determination of 12 in formula (C), where yjr and 6 are 

 any given functional forms. 





