556 
ME. B. aOMPEETZ ON THE SCIENCE 
began to converge would render the development of perfect impracticability, though 
ultimately we could arrive as near as we chose to the insignificant value ‘000014 ; and I 
found that with such an annoyance it was no use to wrestle, no more than it would 
have been for David to wrestle with Goliath. I therefore was at the pains of finding a 
subterfuge by an interpolation which discards all the terms of a series alternately 
greatly positive and greatly negative, which so nearly destroyed each other’s effect when 
we arrived to very nearly the true but very insignificant value ‘000014. I have reason 
to think good mathematicians would think such a subterfuge was unattainable, but 
what follows will show the contrary. 
Art. 30. Calculating the value ^s®, ^s'®, ^s^®, we find the value to form the very 
swiftly converging series, ‘147424, ‘014424, ‘001441, -000188, ‘00008, and beyond this 
term perfectly insignificant ; so that, if we wished to express the value of where x is 
n whole number, by the common method of interpolation, we might easily do it thus : — 
^ 3 . X . 
x~\ x — 2 
~2 3“ 
A..^. 
x — 2 a?— 3 
“T"’ 4 ’ 
beyond which term the differences are of perfect insignificance, Aj, Aj, A 3 , A 4 being 
the first of the several orders of differences of the value ^s®, ^s'®, &c. ; and it is 
evident that A„ A 2 , A,, A^ are respectively k{z^—V), — ^(s® — 1)®, &c., which we 
may adopt for the sake of proof, instead of taking the actual difference. And as the 
form in which we obtain the value of requires for our purpose to be expressed by a 
series such as la-\-lyX^\■\x^-\-\x^^\-l^x‘^^ if the above value of be expanded, it ’will give 
Iq — Jc^ 
T 'a 1 a . 11 a 
I 2 — A 2— 2 Asfi- 2^A4 
where, as already observed, Ai=A:(a® — 1 ), A 2 =A:(a® — 1 )®, &c. ; and we find from the 
above values, I,,, Ij, I 2 , I 3 , 14 ; and as A'a®®is therefore in tii® 
room of 6x we write x, we shall have the equation 
J iX-j-J2X ^ + 14^^ 
if do — i-0? dj — 5^1’ ^2 — 52^2? d3 — 53^-3? d4 — 54^45 and WO havo 
I„=‘147422, I, = -‘253484, l 2 = + ‘158862, l3=-‘042453, l 4 = + ‘004068, 
d, =-‘050696, d2=‘0063545, d3=-@33962, d4=@65088, 
X-d4=2‘70497, Xd2=3‘80308, X-d3=5‘53100, Xd4=6‘81350. 
And with respect to the least value of a may be taken when «=60, as is so 
extremely insignificant if a be less than 60. I will consider the value of lO-v®®'*'® first, and 
consider afterwards if a exceeds 60 by J ; and I observe that |M/v®®, ( 0 ,;'®®+®, p®®+^®, 
p.('®®+^®, jO-!'®®'^®® form a regular series ; and here I stop, though this series is of a different 
character from the series A:s®, A:s®*, &c., because the terms of the latter series dwindle 
to such insignificance, but those of the other, on the contrary, continually increase. 
