306 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
term in the formula by its integer portion, and in the series on the right stop at the 
term immediately preceding the first term for which 
E q® = 1. 
Thus, when 
we have 
and 
So also, when 
p — 462 and q = 24, 
= 4, 
E{*(Eg*+Eg*)}=E{»(4 + 2)}=4. 
p = 41881603128, q= 204810, 
E (Yzr) = 319> 
and 
E (f(Eg* + Eg* + Eg* + Eg*) } = E (f(452 + 21 + 4 + 2) } = E (-ip-) = 319. 
But, if we had included the term Eg 1 ' 6 , the result would have been 
Again, when 
and 
E {f (452 + 21 +4-}-2+l)} = 320. 
p = 3076736843548289379224261404637538760216584, 
q = 1754062953145429399086, 
E = 27921159919, 
E || (Eg" + Eg* + Eg- + Eg* + Eg-'- + Eg*)} 
= E (1(41881534751 + 204649 + 452 + 21 + 4 + 2) } = 27921159919.* 
We will now proceed to consider afresh the asymptotic development of any 
Hypothenusal Number p — g in terms of its antecedent g — r, and to reduce to 
apodictic certainty results which in the first section were partly obtained by observa¬ 
tion. It has already been shown in that section that 
p > g 3 
12 8 
8 1 
T 
£ 
2 
when p is not lower than 204810 in the scale 2, 3, 6, 24, 462, 204810, . . . , i.e., when 
g is not less than 462. 
Hence 
p > g 3 - 2g ; + g + (ff g* - f g), 
* The authors must be understood merely to affirm the possibility of the theorem being true, and to 
offer no opinion on the strength of the presumption raised that it is so. 
