724 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR 
tion. We remarked in Note B., that any set of columns on which num- 
bers are inscribed, represents merely a general function of the several 
quantities, until the special function have been impressed by means of 
the Operation and Variable-cards. Consequently, if instead of re- 
quiring the value of the function, we require that of its integral, or of 
its differential coefficient, we have merely to order whatever particular 
combination of the ingredient quantities may constitute that integral 
or that coefficient. In aa”, for instance, instead of the quantities 
Vo Vv, V5 V3 
A Eo 
— _~Hs LS 
ax” 
being ordered to appear on V, in the combination aa”, they would 
be ordered to appear in that of 
ang-}, 
They would then stand thus :— 
V, V, V, Vie 
[«| | n | | iz | anan—| 
Shae Ean «ar el 
anxzr—1l 
Similarly, we might have < ia+), the integral of a2, 
An interesting example for following out the processes of the engine 
would be such a form as 
a daz 
Ve—-x 
or any other cases of integration by successive reductions, where an 
integral which contains an operation repeated times can be made to 
depend upon another which contains the same n—1 or n —2 times, 
and so on until by continued reduction we arrive at a certain ultimate 
form, whose value has then to be determined. 
The methods in Arbogast’s Caleul des Dérivations are peculiarly 
fitted for the notation and the processes of the engine. Likewise the 
whole of the Combinatorial Analysis, which consists first in a purely 
numerical calculation of indices, and secondly in the distribution and 
combination of the quantities according to laws prescribed by these 
indices. 
We will terminate these Notes by following up in detail the steps 
through which the engine could compute the Numbers of Bernoulli, 
this being (in the form in which we shall deduce it) a rather compli- 
cated example of its powers. The simplest manner of computing these 
numbers would be from the direct expansion of 
wv 1 
Gilet ah site Ars at ote il (1.) 
—1 \2 x? xs =” 
et = a &e, 
egg eat OF 
