# Page:Scientific Memoirs, Vol. 3 (1843).djvu/701

691

NOTES BY THE TRANSLATOR.

Note A.—Page 674.

The particular function whose integral the Difference Engine was constructed to tabulate, is

${\displaystyle \scriptstyle {\Delta ^{7}u_{z}=0}}$.

The purpose which that engine has been specially intended and adapted to fulfil, is the computation of nautical and astronomical tables. The integral of

${\displaystyle \scriptstyle {\Delta ^{7}u_{z}=0}}$

being

${\displaystyle \scriptstyle {u_{z}=a+bx+cx^{2}+dx^{3}+ex^{4}+fx^{5}+gx^{6}}}$,

the constants ${\displaystyle \scriptstyle {a}}$, ${\displaystyle \scriptstyle {b}}$, ${\displaystyle \scriptstyle {c}}$, &c. are represented on the seven columns of discs, of which the engine consists. It can therefore tabulate accurately and to an unlimited extent, all series whose general term is comprised in the above formula; and it can also tabulate approximatively between intervals of greater or less extent, all other series which are capable of tabulation by the Method of Differences. The Analytical Engine, on the contrary, is not merely adapted for tabulating the results of one particular function and of no other, but for developing and tabulating any function whatever. In fact the engine may be described as being the material expression of any indefinite function of any degree of generality and complexity, such as for instance,

F(${\displaystyle \scriptstyle {x,y,z,\log x,\sin y,x^{p},}}$ &c.),

which is, it will be observed, a function of all other possible functions of any number of quantities. In this, which we may call the neutral or zero state of the engine, it is ready to receive at any moment, by means of cards constituting a portion of its mechanism (and applied on the principle of those used in the Jacquard-loom), the impress of whatever special function we may desire to develope or to tabulate. These cards contain within themselves (in a manner explained in the Memoir itself, pages 677 and 678) the law of development of the particular function that may be under consideration, and they compel the mechanism to act accordingly in a certain corresponding order. One of the simplest cases would be, for example, to suppose that

F(${\displaystyle \scriptstyle {x,y,z,}}$ &c. &c.)

is the particular function

${\displaystyle \scriptstyle {\Delta ^{n}u_{z}=0}}$

which the Difference Engine tabulates for values of ${\displaystyle \scriptstyle {n}}$ only up to 7. In this case the cards would order the mechanism to go through that succession of operations which would tabulate

${\displaystyle \scriptstyle {u_{z}=a+bx+cx^{2}+\ldots mx^{n-1}}}$,

where ${\displaystyle \scriptstyle {n}}$ might be any number whatever.