126 Sir W. R. Hamilton on some 



and let us inquire into the conditions under which this law 

 shall be fulfilled, for any 3 unequal or equal symbols of the 

 form t. 



If the conception of the polynomial expression 



P=2;wr=^oa?o+*ia?i+ • • +^«a?„, (1) 

 be no further restricted than it was in [1.], then each of the 

 three indices e, f, g, in the equation (52), may receive any one 

 of the n + 1 values from to w ; so that there are in this case 

 («-}- 1)^ associative conditions of this form (52), whereof eachj bv 

 comparison of the coefficients of the n-\-\ symbols C, breaks itself 

 up into n+ 1 separate equations, of the ordinary algebraical kind, 

 making in all no fewer than (71+ 1)"* algebraical relations, to be 

 satisfied, if possible, by the (w+1)^ constants of multiplication, 

 of the form [fgh) : respecting which constants, it will be remem- 

 bered that the general formula has been established, 



V*^=(/^0)^o+ • • + {f9hW+ . . + [fgnYn^ (7) 

 We may therefore substitute, in (52), the expressions, 



i'e('h=^k{ehk)ik, i'hi'g=^k(hgk)ii,',J 

 and then, by comparing coefficients of th} this associative formula 

 (52) breaks itself up, as was just now remarked, into (n-^iy 

 equations between the (»+ 1)^ constants, which are all included 

 in the following* : 



2^{fffh){ehk) = -S,^(efh){hffk}; .... (54) 



where the four indices efgk may each separately receive any one 

 of the 71+1 values from to ti, and the summations relatively 

 to k are performed between the same limits. 



[8.] Introducing next the simplification (10) of article [2.], 

 or supposing ^0=1^ which has been seen to reduce the number 

 of the constants of multiplication from (w + 1)^ to {n + l)n^j we 

 find that the number of the equations to be satisfied by them is 

 reduced in a still greater ratio, namely from (/» + l)'*to (71+ l)7i^. 

 For, if we suppose the index g to become 0, and observe that 

 each of the constants {fOh) and {Ofh) is equal, by (12) and (13), to 

 or to I, according as h is unequal or equal to /, we shall see that 



* This formula (54) may be deduced from the equation (214) in p. 239 

 of the writer's " Researches respecting Quaternions,'* published in the 

 Transactions of the Royal Irish Academy, vol. xxi. part 2, by changing 

 there the letters rstr'^tofhgeky and substituting the symbol {fgh) 

 for ng,/,h' Or the same formula (54) may be derived from one given in 

 page (30) of the Preface to the same author's Lectures on Quaternions, 

 (l^bhn, Hodges and Smith, 1853), by writing gfek instead offg g' h', and 

 changing each of the two symbols \gj, a, l^,/, *. to (Jgh). But the general 

 Ttductions of the present paper have not been hitherto pubhshed. 



