290 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
• Or in other words, writing /3 0 E, = 1, /3 1 E i _ 1 = E z -_ 1? and replacing i — 1 by 
Ei = 1 + /3 2 E z _i — /3 3 E 2 _2 + . . . + (—)' +1 A +1 E 0 
for all values of i greater than zero. 
This is eminently a practical formula, as all the numerical calculations made use of 
to obtain any E are available for finding the E which follows. Dispensing with the 
symbol /3 , we may deduce all the values of E successively from those that go before 
by means of the equivalence 
S = (1 — t'f° + t{l — tf 1 + t* (1 — tf‘ + . . . = 1 — 2 1, 
which, by equating the powers of t on the two sides of the equivalence, gives 
E 
o 
= 3, 
E x = 
E,= 
Eo = 
, 3.2 
1 + H2 
1 + 
1 + 
4.3 
1.2 
6.5 
L2 
4, 
3.2.1 
rji = 0 - 
4.3.2 3.2.1.0 
1.2.3 1.2.3.4 
= 12, 
and so on. 
I use the term equivalence and its symbol in order to convey the necessary caution 
that the relation indicated is not one of quantitative equality ; tor, although the series 
on the left-hand side of the symbol converges for all positive values of t less than 2, 
it is never equal to the expression on the right-hand side except when t = 0. Thus, 
e.g., when t is unity the two terms of the equivalence are 0 and — 1, and when t = 4 
they are 
2“ E o _]_ 2~ El_1 + 2 -E2-2 -f- . . . and 0, respectively ; 
and for all values of t within the limits of convergence the value of the left-hand 
side is in excess of the value of the right-hand side of the equivalence by a finite 
quantity which decreases continuously as t decreases from 2 to 0, and which vanishes 
when t = 0. # 
In a word, the generating equation is not an equation in the usual sense of the 
term. Conceiving each term of the series S to be expanded in ascending pow r ers of t, 
and like powers of t to be placed in columns under and above each other, the double 
* Of the truth of the statement that the excess never changes sign, and continually decreases, I ha\e 
scarcely a doubt, but it requires proof. Mr. Hammond remarks that 
( i _ t f o +1 (1 - 0 El + d (1 - 0 E * + • • • + t n (1 - tf" = (1 - 2 f) +t\ 1 - tf"~ 2 F„ (0 - t n+1 (1 - tf'~ \ 
where F„ (t) is positive for all positive values of t. Probably a proof of the point in question might be 
deduced from this expression, but I have not thought it necessary to investigate the matter. 
