184 Sir W. R. Hamilton on some 



and making ^=/, in either (95) or (97), we obtain the equation, 



(f) = -S.(feh)(fhe),\ie%f. (98) 



For each of the n symbols (/), there are n—\ distinct expres- 

 sions of this last form, obtained by assigning different values to e ; 

 and when these expressions are equated ,to each other, there 

 result 7i(7i— 2) equations between the symbols of the form (fgh). 

 For each of the ^n(n — 1) symbols of the form {fg), where /and 

 g are unequal, there are w — 1 expressions (95), and n — \ other 

 expressions of the form (97), because, by (33) and (36), (^/) = [fg) ; 

 and thus it might seem that there should arise, by equating 

 these 2n—2 expressions for each symbol {fg), as many as 2n—Z 

 equations from each, or ^n[n — 1) (2/i— 3) equations in all between 

 the symbols {fgh) . i3ut if we observe that the sums of the n — \ 

 expressions (95) for {fg)j and of the n — \ expressions (97) for 

 {gf)> ai'e, respectively, 



[n-\){fg) = l.,Mgeh)(flie), "1 



(n-\)(gf)=^,-%,{feh)(ghe);i 



where the summations may all be extended from 1 to w, because- 

 {ffh) and (ggh) are each =0, by (35), since /i >0; and that 

 these two double sums (99) are equal; we shall see that the 

 formula 



{9f) = {f9), (100) 



though true, gives no information respecting the symbols (fgh) : 

 or is not to be counted as a new and distinct equation," in com- 

 bination with the n — \ equations (95), and the ti— 1 equations 

 (97). In other words, the comparison of the sums (99) shows 

 that we may confine ourselves to equating separately to each 

 other, for each pair of unequal indices /and g, the n — \ expres- 

 sions (95) for the symbol {fg), and the n—V other expressions 

 (97) for the symbol {gf), without proceeding afterwards to 

 equate an expression of the one set to an expression of the other 

 set. We may therefore suppress, as unnecessary, an equation of 

 the form (100), for each of the ^n{n — \) symbols of the form 

 {fg), or for each pair of unequal indices / and g, as was stated 

 by anticipation towards the close of paragraph [9.]. There 

 remain, however, 2(n— 2) equations of condition, between the 

 symbols {fgh), derived from each of those ^n{n—\) pairs; or as 

 many as n{n — \){n^2) equations in all, obtained in this manner 

 from (95) and (97), regarded as separate formulae. Thus, with- 

 out yet having used the formula (96), we obtain, with the help 

 of (98), by elimination of the symbols (/), {fg), {gf), through 

 the comparison of ti— 1 expressions for each of those n^ symbols, 

 rP{n — 2) equations of condition, homogeneous and of the second 



