one *division*. Therefore, if desired, we need only use three operation cards; to manage which, it is sufficient to introduce into the machine an apparatus which shall, after the first multiplication, 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 repeated. In the preceding example we have seen, that to find the value of we must begin by writing the coefficients , , , , , , upon eight columns, thus repeating and twice. According to the same method, if it were required to calculate 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 discussed, we will take the two coefficients and , which are each of them to enter into *two* different products, that is into and , into and . These coefficients will be inscribed on the columns and . If we commence the series of operations by the product of into , these numbers will be effaced from the columns and , that they may be transferred to the mill, which will multiply them into each other, and will then command the machine to represent the result, say on the column . 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 product, 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 and , whence in short they would not finally disappear, to be reproduced 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 requisite for resolving the two^{[1]} above equations of the first degree, may be definitively represented in the following table:—

- ↑ See Note D.