691 
NOTES BY THE TRANSLATOR. 
4 ’ - 
Nore A.—Page 674. 
The particular function whose integral the Difference Kngine was 
constructed to tabulate, is 
= rN Uz = 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 
2 A7u,.=0 
_ being Uz=a+be+ecx?+ dx + ext + fas + gx", 
the constants a, b, c, &c. are represented on the seven columns of 
dises, of which the engine consists. It can therefore tabulate aceu- 
rately and to an unlimited extent, all series whose general term is com- 
_ prised 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 in- 
stance, 
J F (a, y, z, log x, siny, a, &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 accord- 
_ ingly in a certain corresponding order. One of the simplest cases 
_ would be, for example, to suppose that 
of F (a, y, 2, &e. &c.) 
is the particular function 
A*u, =0 
which the Difference Engine tabulates for values of x only up to 7. 
In this case the cards would order the mechanism to go through that 
succession of operations which would tabulate 
‘ Uz =at+betexr+....m2"-}, 
where n might be any number whatever. 
