494 Sir W. R. Hamilton on some 



tion, ifffh) = ( fffh) . And thus the product of any two given poly- 

 nomes, P and V', of the form (1), combined in dt. given order as 

 factors, becomes equal to a third given polynome, P", of the same 

 general form, 



V" = 7?' = txfif.^x',ig='Zx\i„; ... (8) 



the summations extending still from to n, and the constituent 

 x"h of the product admitting of being thus expressed : 



a^', = l{fgh)a;fx', (9) 



As regards the subjection of the symbols t to the associative law 

 of multiplication, expressed by the formula, 



t.L'c" = ii'.i", 

 we shall make no supposition at present. 



[2.] As a first simplification of the foregoing very general* 

 conception, let it be now supposed that 



^0=1; (10) 



the n other symbols, t,, i^, . . t„, being thus the only ones which 

 are not subject to all the ordinary rules of algebra. Then 

 because 



tg=ig, ififj=if, (11) 



o''e 



it will follow that if either of the two indices /or ^ be =0, the 

 constant of multiplication {fgh) is either =1, or =0, according 

 as h is equal or unequal to the other of those two indices ; and 

 we may write, 



{Ofh) = {fOh)=0,iih^f; .... (12) 



(0//) = (yt)/) = l (13) 



With this simplification, the immber of the arbitrary or disposable 

 constants of the form (fgh), which arc not thus known already 

 to have the value or 1, is reduced from (n + 1)^ to {n-\-\)n^; 

 because we may now suppose that / and g are each > 0, or that 

 they vary only from 1 to n. For we may write, 



P=jo + ^, P'=y + fir', (14) 



where 



p =to^o =^o> ^ =tia?i + . . +LfXf-\- . . +i„a?„, 1 



p' = LqX'q = x'q, u/ = tj ^'j + . . -lf-l^x'g+ . . + LnOc'n ) ^ 



and then, by observing that p and p' are symbols of the usual 

 and algebraical kind, shall have this expression for the product 

 of two polynomes : 



P" = PP'=(;j + OT)(jo' + ^)=Joy+jOt!j'+Jo'w + wcr'; . (16) 



* Some account of a connected conception respecting Sets, considered 

 as including Quaternions, may be found in the Preface to the Lectures abready 

 cited. 



