ON BABBAGE’S ANALYTICAL ENGINE, 721 
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 
2) between the two first equations. This would be obtained by the 
form— 
(aa —aa')x, + (ab —ab')x, + (e—ac') x + 
+ (ai'd—ad')xa,4+..-...+-.--- Jy Pe, =a'p—ap', 
for which the operations 10 (x, X, —) would be needed. The second 
step would be the elimination of x, between the second and third equa- 
tions, for which the operations would be precisely the same. We should 
then have had altogether the following operations :— 
10(x, X,—),10(x, xX, —), =20(x, X,—) 
Continuing in the same manner, the total number of operations for the 
complete elimination of x, between all the successive pairs of equations, 
would be— 
9.10(x, X,—) =90(x, x, —). 
We should then be left with nine simple equations of nine variables 
from which to eliminate the next variable x, ; for which the total of the 
processes would be— 
8.9(X, X,—) = 72( x, xX; —)- 
We should then be left with eight simple equations of eight variables 
from which to eliminate 2, for which the processes would be— 
1 B(x, eyes Hb 0K) «, —); 
_ and so on. The total operations for the elimination of all the variables 
_ would thus be— 
9.10+8.9+7.84+6.7.+5.64+4.54+3 .442.34+1.2 = 330. 
So that three Operation-cards would perform the office of 330 such cards. 
If we take » simple equations containing x — 1 variables, x 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 
obyious enough on a slight consideration. And it is a circumstance 
which deserves particular notice, since herein may reside a latent 
_yalue of such an engine almost incalculable in its possible ultimate 
results. We already know that there are functions whose numerical 
| yalue it is of importance for the purposes both of abstract and of prac- 
tical 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 
: 3B2 
sy 
4 
* 
