Page:Scientific Memoirs, Vol. 3 (1843).djvu/731

sider this problem with reference to the actual arrangement of the data on the Variables of the engine, but simply as an abstract question of the nature and number of the operations required to be performed during its complete solution.

The first step would be the elimination of the first unknown quantity $\scriptstyle {x_{0}}$ between the two first equations. This would be obtained by the form—

${\begin{aligned}&\scriptstyle {(a^{1}a-aa^{1})x_{0}+(a^{1}b-ab^{1})x_{1}+(a^{1}c-ac^{1})x_{2}+}\\\scriptstyle {+}&\scriptstyle {(a^{1}d-ad^{1})x_{3}+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots =a^{1}p-ap^{1}}\end{aligned}}$,

for which the operations $\scriptstyle {10(\times ,\times ,-)}$ would be needed. The second step would be the elimination of $\scriptstyle {x_{0}}$, between the second and third equations, for which the operations would be precisely the same. We should then have had altogether the following operations:—

$\scriptstyle {10(\times ,\times ,-),10(\times ,\times ,-),=20(\times ,\times ,-)}$.

Continuing in the same manner, the total number of operations for the complete elimination of $\scriptstyle {x_{0}}$ between all the successive pairs of equations, would be—

$\scriptstyle {9.10(\times ,\times ,-)=90(\times ,\times ,-)}$.

We should then be left with nine simple equations of nine variables from which to eliminate the next variable $\scriptstyle {x_{1}}$; for which the total of the processes would be—

$\scriptstyle {8.9(\times ,\times ,-)=72(\times ,\times ,-)}$.

We should then be left with eight simple equations of eight variables from which to eliminate $\scriptstyle {x_{2}}$, for which the processes would be—

$\scriptstyle {7.8(\times ,\times ,-)=56(\times ,\times ,-)}$,

and so on. The total operations for the elimination of all the variables would thus be—

$\scriptstyle {9.10+8.9+7.8+6.7.+5.6+4.5+3.4+2.3+1.2=330}$.

So that three Operation-cards would perform the office of 330 such cards.

If we take $\scriptstyle {n}$ simple equations containing $\scriptstyle {n-1}$ variables, $\scriptstyle {n}$ being a number unlimited in magnitude, the case becomes still more obvious, as the same three cards might then take the place of thousands or millions of cards.

We shall now draw further attention to the fact, already noticed, of its being by no means necessary that a formula proposed for solution should ever have been actually worked out, as a condition for enabling the engine to solve it. Provided we know the series of operations to be gone through, that is sufficient. In the foregoing instance this will be obvious enough on a slight consideration. And it is a circumstance which deserves particular notice, since herein may reside a latent value of such an engine almost incalculable in its possible ultimate results. We already know that there are functions whose numerical value it is of importance for the purposes both of abstract and of practical science to ascertain, but whose determination requires processes so lengthy and so complicated, that, although it is possible to arrive at them through great expenditure of time, labour and money, it is yet on these accounts practically almost unattainable; and we can conceive there being some results which it may be absolutely impossible in practice to attain with any accuracy, and whose precise determination it may prove

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