304 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
and thus 
p = cf — | q (p + + 2 * + g* + • • • + q (iy ) + 3, 
as was to be shown.* 
# This theorem may be rigorously demonstrated, and reduced to a more precise analytical form, as 
follows:— 
For the sate of brevity, we may call — p\q + q the relative deficiency of p, and denote it by A. 
First it may be noticed that, if in the equation 
F( 2 )=XoV“ S -2-0 
we write log q = h, 
k A 
1.2.3 .7 
+ 
k' a 
1 . 2 . 3.4.5 . 31 
+ 
W 
1 . 2.3.4.5 . 6 . 7 . 127 
which is always convergent. 
Moreover, the value of F (q) may be calculated for any given value of q within close limits. For, if 
we call IT the right-hand branch of the series in q, beginning with z 
,-l 
, the terms of U will easily be 
seen to lie between those of two geometrical series of which z — z -1 is the first term, and of one of which 
-§, and of the other (z* + a -1 ) -1 , is the common ratio. 
Hence U is intermediate between 2 (z 2 — l)/z and (z 2 — 1) (z + l)/z (z — z i + 1). 
[The difference between these limits, it may be parenthetically observed, is 
0 
- 1 ) 
(z i 
,-l)2 
-1+z- 
which, when z is nearly unity (the limit to which g®' converges), is nearly equal to Ar( z — 2 -1 ) 3 ; i-e., if 
z = 1 + t, the difference between the limits (for t small) is very near to r 3 /2.j 
Now on p. 309 (post) it is shown that */ q — r + s — |- v' r = e, and that, when the rank of q is taken 
indefinitely great, e converges to j. Hence e always lies between finite limits. 
[For, iu general, * being any one of a series of increasing numbers, and ^ (x) a function of x which is 
always finite for finite values of x, but ultimately converges to c, by taking for x a value of L sufficiently 
great, we make the series of terms for x > L intermediate between c + 3 and c — 3, where 3 is any 
assigned positive quantity; and consequently, if p, v, are the greatest and least values of yjr ( x ) when x 
does not exceed L, the greater of the two values, c + 3, p, and the lesser of the two, c — 3, v, -will be 
superior and inferior limits to the value of (x) for all values of *.] 
Hence, writing 
\/p — q + r — f Vq - e l5 
V q — r + s —• -§• vV = 6 2 , 
Vr — s + t — % s = e 3 , 
• • ..J 
,/6 — 3 + 2 — f v/3 = 6 *_ 15 
we obtain, by summation, 
V p — g + 3 ( ^ g + b s + ... + 6) = S« — 2 + ^ 3 , 
and, consequently, 
bp — g + i.(\/ g + b r+ b s + .. . + b 6)=/j.r, 
where p is always a finite quantity lying between determinable limits. But again (p. 307)— 
p—(q-~eV q) 2 , 
where 0 (whose ultimate value is •§■) is always a proper fraction. Hence 
2-b — 2 (q — bp) — o~. 
