294 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
Ol 
. o 1 
P<1 ~ 2 ’ 
V><f~ |-Vi a 2*. 
We may find a closer superior limit to p in terms of q as follows— 
t-> 1 ^Q 2 -Q R(R-l)(R-2) , S (S — 1) (S — 2) (S — 3) 
i L - or< ~2 6 + 24 ’ 
in which inequality it may be shown by inspection up to a certain point, and after 
that by demonstration, the tedium of writing out or reading which I spare my 
readers and myself, that P may be substituted for its flattened value P — 1. 
We have then 
T3 Q 2 — Q R 3 - 3R 3 , S* 
* < 2 6 + 24 
Let us suppose that S, P, are not lower in the scale of the E’s than 12, 48, respec¬ 
tively; so that P is not lower than E 6 , which is 409620. 
Then, as we have previously shown, 
Q 3 < P, P 2 < - 9 - Q, S 3 < - 9 - R. 
Moreover, we have 
P < ^ (Q 2 — Q), whence it follows that Q 2 > 2P -J- Q, 
and, a fortiori , 
Similarly 
and 
Now 
Q 3 >2P. 
R 3 > 2Q, 
S 2 > 2P. 
t> ^ Q 2 — Q Pd , P 2 , s* 
+ ¥ + 24 
Q 2 - 0 , 
< ~ i (2Q)* + i (¥ Q) + * (¥ W 
O 2 
<'t-^Q ? + HQ + *(¥) 3 Q j 
i.e,, 
P < i Q 3 — Q 5 4- Q. 
This result, expressed in terms of the reduced numbers p, q, takes the form 
