ON HAMILTON'S NUMBERS. 71 



Order f = $ - * tu* + (\ A - ^)l* 



A + A -v - A 8 - f fv + ( AV - A) > 3 - it- 



A 



Now the terms of the highest order in this equation must vanish when we write 

 t = u 9 , and therefore f i + fA -^5 = 0, which gives A = H- Substituting 

 this value for A, we find 



Order f = $ - \tu* + 



A + A<-V- 



A + 



which is a mere repetition of equation (8), with all the letters moved forward one 

 place. Hence it is evident that, if we treat this equation as we treated (8), we shall 

 find B = ^J-, arriving, at the same time, at another equation which will be merely a 

 repetition of (8), with all its letters moved forward two places ; and this process can 

 be continued as long as we please. 

 Thus we arrive at the result 



A = B = C = D = E=... =H, 



and the asymptotic development for Hypothenusal Numbers 



is established. 



Comparing this with the corresponding formula for Hamiltonian Numbers, 



given at the beginning of the third section (at the top of p. 302, where the last term 

 is incorrectly printed H), it will be noticed that each of the two developments begins 

 with an irregular portion consisting respectively of four and one terms, followed by a 

 regular series. In the one case the regular portion is ^ (q r)*, multiplied by a 

 series whose general term is f I (q r) (4> " ; in the other it consists of a series of terms 

 of the form <? {1> " multiplied by 



