226 Mr. J. J. Sylvester on Extensions of 



of terms to be eliminated will be n-fjo— *, arising from Fm, 

 Gh, H;,; and n-\-p—2t, from Fm-t, Gn-t, Hjo-t; together 

 making up the proper number 2wH-2j9 — 3*. 



Each Connective contains ternary combinations of the coeffi- 

 cients, viz. one of the coefficients belonging to that part of 

 U, V, W which contains r*, and two coefficients from the other 

 part ; the dimensions of the resultant in respect of the coefficients 

 of the former will hence be readily seen to be equal to the num- 

 ber of connectives -|- the number of terms in the augmentatives 

 into which z^ enters, i. e. will equal m-\-n-^p—2c; the total 

 dimensions of the resultant in respect to all the coefficients of 

 U, V, Wwill be Sm-'r{2n-\-2p — m—SL),i.e.2m + 2n-{-2pSL; 

 and consequently, in respect to the coefficients of Fm; G»; H^, 

 will be of 



{2m-\-2n + 2p—SL) — {m + n-\-p — 2t), 



«. e. o{ m + n+p — t dimensions. This result, which is of con- 

 siderable importance, may be generalized as follows. 



Returning to the general system (A.), (for which we have 

 proved that the total dimensions of the resultant are 



(r~l)(m + w+ ...p)-r{i + i'+ . . . + (t)), 

 let the coefficients of the column of partial functions 



Fn, 

 G„ 



be called the first set ; the coefficients of the column 



F»i— I 

 Gn— I 



the second set, and so forth ; then the dimensions in respect of 

 the 1st, 2nd ... (r — l)th sets respectively are s, s— -t, s— t' . . . 

 s— (t), where 



s = m-\-n-{-Scc. -j-j9 — (tH-t'H-&c. +(0)- 



The important observation remains to be made, that all the 

 above results remain good although any one or more of the 

 indices of dimension of the partial functions in the system 

 (A.), as m—i, m^i!, n—i, &c., should become negative, pro- 

 vided that the terms in which such negative indices occur be 

 taken zero, as will be apparent on reviewing the processes already 



