264 SirW. H. llamilroii on some 



by interchanging / and </, and subtracting. Again, a type which 

 is in general still more subordinate, as including only }n(n— 1) 

 distinct equations, of 2(w— 2) terms each, maybe derived by the 

 same process from (120) ; namely the type, 



= l{fyh){hff+hffff); (126) 



or in a slightly more expanded form, 



{fg^Mf+cgg) = T{gfh){hff+hgg); . (137) 



which may also be easily derived, in the same way, from (121). 

 It will, however, be found, by pursuing a little further the ana- 

 lysis of [14.], that the equations of this last type, (126) or (127), 

 are always consequences of the equations of the intermediate 

 type, (124) or (125) ; the sum of the n—2 equations of the 

 form (125), which answer to the various values of e that remain 

 when / and g have been selected, being in fact equivalent to the 

 formula (126). It will also be found, by the same kind of ana- 

 lysis, that the intermediate equations of the type (124) or (125) 

 i\x^ generally deducible from those of the form (123). But on 

 the subject of these general reductions, connected with the elimi- 

 nation of the symbols {fg) or {gf)i it may be proper to add a 

 few words. 



[19.] Let us admit, at least as temporary abridgments, the 

 notations 



\.M='^{M-¥f)i Ifffel^^if^.ffeh); . (128) 

 where e, /, g are any three unequal indices, and h varies under S, 

 as before, from 1 to n. Then the formula (95) gives n — 1 distinct 

 equivalents for the symbol {fg), of which one is by (120) of the 

 form [fg'], and the n—2 others are each of the form [fge] ; in 

 such a manner that we may write, instead of (95), with these 

 last notations, the system of the two formulfx?/ 



(f9} = [M, {fy)=[M: ■ ■ ■ (129) 



whereof the latter is equivalent to a system of w— 2 equations : 

 and of course, instead of (97), we may in like manner write 



(.fff)=yf]. isn^yfii- ■ ■ ■ (^m 



The equations (99) may now be thus presented : 



(n-l){M = IM +2'[/<^«] =22(/Ae .geh) H 

 {n-l){fff) = [fff]+-S.'lff/e]=12{feh.ff/>e);S 



where e under the sign 2' is distinct from each of the two indices 

 y* and ^; but, under the double sign 2S, both e and /* may each 

 receive any one of the values from 1 to n. The two double sums 

 are equal, as in [14.], and therefore we must have, identically/, 



[M + ^'[M = [fffi +-^'l<)fe] : . ■ (132) 



the equation (100) being at the same time seen again to be a 



