IV. On Hamilton's Numbers. Part II. 



By J. J. SYLVESTER, D.C.L., F.R.S., 



Savilian Professor of Geometry -in the University of Oxford, 



and JAMES HAMMOND, M.A., Cantab. 



Received March 9, -Read April 19, 1888. 



4. Continuation, to an infinite number of terms, of the Asymptotic Development 



for Hypotlienusal Numbers. 



"This was sometime a paradox, bat now the time gives it proof." 



(Hamlet, Act III., scene 1.) 



IN the third section of this paper ('Phil. Trans.,' A., vol. 178, p. 311) it was stated, 

 on what is now seen to be insufficient evidence, that the asymptotic development of 

 p q, the half of any Hypothenusal 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, f, 1, },!,*,... 



It was there shown that, when quantities of an order of magnitude inferior to that 

 of (q r) 1 are neglected, 



P ~ 1 = (q - r)* + I (q - r) + ft (q - r) + tf (q - r)' ; 



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

 next terra came out if 5 (q r)*, there was an infinite series of terms interposed 

 between this one and (q 7-)', viz., as proved in the present section, between (q r)' 

 and (q r) 1 there lies an infinite series of terms whose indices are 



B> T Sa 64> TS8 i 



and whose coefficients form a geometrical series of which the first term is -riiy and 

 the common ratio f . 



We shall assume the law of the indices (which, it may be remarked, is identical 

 with that given in the introduction to this paper as originally printed in the 

 ' Proceedings,' but subsequently altered in the ' Transactions ') and write 



MDCCCLXXXVm. A. K 1.6.88 



