680 LL. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. 
one division. Therefore, if desired, we need only use three ope- 
ration cards; to manage which, it is sufficient to introduce into 
the machine an apparatus which shall, after the first multiplica- 
tion, for instance, retain the card which relates to this operation, 
and not allow it to advance so as to be replaced by another one, 
until after this same operation shall have been four times re- 
peated. In the preceding example we have seen, that to find 
the value of 2 we must begin by writing the coefficients m, n, 
d, m', n', d', upon eight columns, thus repeating ” and n! twice. 
According to the same method, if it were required to calculate 
y likewise, these coefficients must be written on twelve different 
columns. But it is possible to simplify this process, and thus 
to diminish the chances of errors, which chances are greater, 
the larger the number of the quantities that have to be inscribed 
previous to setting the machine in action. To understand this 
simplification, we must remember that every number written on 
a column must, in order to be arithmetically combined with 
another number, be effaced from the column on which it is, 
and transferred to the mill. Thus, in the example we have dis- 
cussed, we will take the two coefficients m and n', which are each 
of them to enter into ¢wo different products, that is m into ma! 
and md', n’ into mn’ and n'd. These coefficients will be in- 
scribed on the columns Vy, and V,. If we commence the series 
of operations by the product of m into n!, these numbers will be 
effaced from the columns V, and V,, that they may be trans- 
ferred to the mill, which will multiply them into each other, 
and will then command the machine to represent the result, 
say on the column Vg. But as these numbers are each to be 
used again in another operation, they must again be inscribed 
somewhere; therefore, while the mill is working out their pro- 
duct, the machine will inscribe them anew on any two columns 
that may be indicated to it through the cards; and, as in the 
actual case, there is no reason why they should not resume their 
former places, we will suppose them again inscribed on V, and 
V,,, whence in short they would not finally disappear, to be re- 
produced no more, until they should have gone through all the 
combinations in which they might have to be used. 
We see, then, that the whole assemblage of operations requi- 
site for resolving the two* above equations of the first degree, may 
be definitively represented in the following table :— 
* See Note De 
