222 Mr. J. J. Sylvester on Extensions of 



other continental mathematicians, and because of the importance 

 of the geometrical and other applications of which it admits, and 

 of the inquiries to which it indirectly gives rise. We shall be 

 concerned in the following discussion with systems of homoge- 

 neous rational integral functions of a peculiar form, to which for 

 present purposes I propose to give the name of aggregative 

 functions, consisting of ordinary homogeneous functions of the 

 same variables but of different degrees, brought together into 

 one sum made homogeneous by means of powers of new variables 

 entering factorially. 



Thus if F, G, H . . . L be any number of functions of any 

 number of letters a:, y . . .t of the degrees tw, m — t, m—il.,,m — {c) 



respectively, F -\- G\* + VL/m^ + . . . L^^'^ will be an aggregative 

 function of the variables entering into F, G, &c., and of X, fjL,.J. 

 I shall further call such a function binary, ternary, quaternary, 

 and so forth, according to the number of variables contained in 

 the functions (F, G, H, &c.) thus brought into coalition. 



It will be convenient to recall the attention of the reader to 

 the meaning of some of the terms employed by me in the paper 

 above referred to. 



If F be any homogeneous function of x, y, s . . . t, the term 

 augmentative of F denotes any function obtained from F of the 

 form 



a?«./.a7^.../^xF. 



Agaiuj if we have any number of such functions F, G, H . . . K 

 of as many variables x, y, z . . . t, and we decompose F, G, H ... K 

 in any manner so as to obtain the equations 



F=a?«.P, +y*.P^+^.P3 + &c.... -hf'.i'P) 

 G=:x\Q, + yKQ^ + z'.Q^ + &c..., +t'.{Q) 

 H=.z^.Ili + y*.R2 + ^Mt3 + &c +^^.(1^) 



K=^.Si -f-y*.S2+^.S3 + &c. . . . +t^,{S), 

 and then form the determinant 



P, l\V,...(P) 

 Q, Q^Qs-.-CQ) 



R] Bg R3 . . . (R) 



S, S, S3 . . . (S) 



this determinant, expressed as a function oi x, y, 2 . . .t, is what, 

 in the paper referred to, I called a secondary derivec, but which 

 for the future I shall cite by the more concise and expressive 

 name of a connective of the system of functions F, G, H , . . K, 



