100 



On Hamilton s Numbers, 



[Apr. 19, 



number, could be expressed as a series of powers of q— r, the half of 

 its antecedent, in which the indices followed the sequence 2, -§-, 1, -f, 



5 1 



85 2' • * • • 



It was there shown that, when quantities of an order of magni- 

 tude inferior to that of (q — r) 1 are neglected, 



P-9. = (2-0 2 +f(2-r) f + H(2-r)+^(g-r) ; 



but, on attempting to carry this development further, it was found 

 that, though the next term came out yj-f-g (q — r) f , there was an in- 

 finite series of terms interposed between this one and (q— r)K 



In the present section it will be proved that between (q— r) 1 and 

 (q—ry there lies an infinite series of terms whose indices are — 



and whose coefficients form a geometrical series of which the first 

 term is yff-g- and the common ratio § . 



We shall assume the law of the indices (which, it may be re- 

 marked, is identical with that given in the introduction to this paper 

 as originally printed in the 4 Proceedings ') and write — 



p-z = + i(q-r)- + + Mv-ry 



2 3 . 2 4 2 5 



+ 33 A( ff -r)l + ^(2-0* + ^C(q-r)U 



2 6 2 7 

 + -36 — + gyE(gr— r)iV* + & c ., ad inf. 



+ 0* r(l.) 



The law of the coefficients will then be established by proving 

 that — 



A = B = C = D = E = .... = |i. 



If there were any terms of an order superior to that of (q— r) 4 , 

 whose indices did not obey the assumed law, any such term would 

 make its presence felt in the course of the work ; for, in the process 

 we shall employ, the coefficient of each term has to be determined 

 before that of any subsequent term can be found. It was in this 

 way that the existence of terms between (q— and (q— r)* was 

 made manifest in the unsuccessful attempt to calculate the coefficient 

 of (q—ry. 



It thus appears that the assumed law of the indices is the true one. 

 It will be remembered that p, q, r, . . . . are the halves of the 



# In the text above G represents some unknown function, the asymptotic value 

 of whose ratio to (q — r)* is finite. 



