292 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
If, then, the law to be proved is true for all the consecutive terms of the upper series 
it will obviously be true for the second series, abstraction being made of its first terra, 
provided that no antecedent is less than its consequent in the series 
0,-2 
S - 4 
S 
which is true a fortiori if 
Q E 
3 ’ 4 ’ 5 ' 
continually decrease, as is obviously the case, inasmuch as 
Q, R, S,. . . 
form a descending series. 
In order, then, to establish the necessary chain of induction, it only remains to show 
that 
3P(P — i)-Q(Q — i)(Q-2) 
is positive. 
Now 
(P - Q) - 
and a fortiori P — 
(Q - R) 2 
2 
is positive for a reason previously given. 
And, if in the series 3, 4, G, 12, 48, 924, ... we make exclusion of the first three 
terms, we have always 
and consequently 
T> ^ Q 
It = or < — > 
4 
p>w , 
32 
And, since under the same condition (P — 1)/(Q — 1) > 4, 3P (P — l) — Q-(Q — 1), 
and a fortiori 3P (P — 1) — Q (Q — 1) (Q —2), is positive if 12P — Q 2 is .positive, 
which is the case, since P > 9Q 3 /32. 
Hence, since the theorem to be proved is true for the several series 
0 ) 
4.3 
3.2.1 
1.2 
“ 172 ^ ! 
i 
( 2 ) 
6.5 
4.3.2 
1.2 
_ 1.2.3 
> 
( 3 ) 
12.11 
6.5.4 
+ 
4.3.2.1 
1.2 
1.2.3 
1 .2.3.4 
( 4 ) 
48.47 
12.11.10 
+ 
6 .5.4.3 
1.2 
1.2.3 
1 .2.3.4 
* The proof that the ratio of each term of the series 4, 6,12,48, 924, ... to its antecedent continually 
increases is too easy and too tedious to be worth setting forth in the text. 
