726 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR 
Multiplying every term by (2.3...2%), we have 
1 2n—1 2n 2Qn.2n—1.2n—2 
Soe ache eB tee Be = 
‘ roast (B)+ ; 2.3.4 y+ @) 
QN .2N—1 wveseeeee 2n—4 i 
+B; 2.3.4.5.6 Veet Ben 
which it may be convenient to write under the general form :— 
0=A,+ A,B, + A; B, + A,B, +... + Bon-1--.-+-. (9) 
A,, Az, &e. being those functions of » which respectively belong to 
B,; B,, &e. 
We might have derived a form nearly similar to (8.), from D.,, _ 1 the 
coefficient of any odd power of x in (6.); but the general form is a 
little different for the coefficients of the odd powers, and not quite so 
convenient. 
On examining (7.} and (8.), we perceive that, when these formule 
are isolated from (6.) whence they are derived, and considered in them- 
selves separately and independently, x may be any whole number what- 
ever; although when (7.) occurs as one of the D's in (6.), it is obvious 
that n is then not arbitrary, but is always a certain function of the di- 
stance of that D from the beginning. If that distance be = d, then 
2n+1=d,andn2= a 5 : (for any even power of z.) 
2n=d, anda= 5 (for any odd power of 2.) 
It is with the independent formula (8.) that we have to do. Therefore 
it must be remembered that the conditions for the value of n are now 
modified, and that ” is a perfectly arbitrary whole number. This cir- 
cumstance, combined with the fact (which we may easily perceive) 
that whatever n is, every term of (8.) after the (x + 1)th is = 0, and 
that the (n + 1)th term itself is always = Ben 1-4 = Boni, enables 
us to find the value (either numerical or algebraical) of any nth Number 
of Bernoulli Be, —1, 2” terms of all the preceding ones, if we but know 
the values of B,, B,....Bo,z—3. We append to this Note a Diagram 
and Table, containing the details of the computation for B,, (B,, B,, 
B, being supposed given). 
On attentively considering (8.), we shall likewise perceive that we 
may derive from it the numerical value of every Number of Bernoulli 
in succession, from the very beginning, ad infinitum, by the following 
series of computations :— 
lst Series —Let n = 1, and calculate (8.) for this value of n. The 
result is B,. 
2nd Series. —Let n = 2. Calculate (8.) for this value of n, substitu- 
ting the value of B, just obtained. The result is B,. 
3rd Series—Let » = 3. Calculate (8.) for this value of , substitu- 
ting the values of B,, B, before obtained. The result is B,. And so 
on, to any extent. 
The diagram* represents the columns of the engine when just prepared 
* See the diagram at the end of these Notes. 
