720 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR 
applied to the engine) that a certain prism which is a part of the me- 
chanism (see Note C.), turns a new face, and thus presents a new card 
to act on the bundles of levers of the engine; the new card being per- 
forated with holes, which are arranged according to the peculiarities of 
the operation of addition, or of multiplication, &c. Again, the numbers 
in the preceding formula (8.), each of them really represents one of 
these very pieces of card that are hung over the prism. 
Now in the use made in the formule (7.), (8.) and (10.), of the 
notation of the integral calculus, we have glimpses of a similar new 
application of the language of the higher mathematics. 4, in reality, 
here indicates that when a certain number of cards have acted in suc- 
cession, the prism over which they revolve must rotate backwards, so 
as to bring those cards into their former position; and the limits 
1 to n, 1 to p, &¢., regulate how often this backward rotation is to be 
repeated. A. A. L. 
Norte F.—Page 688. 
There is in existence a beautiful woven portrait of Jacquard, in the 
fabrication of which 24,000 cards were required. 
The power of repeating the cards, alluded to by M. Menabrea in 
page 680, and more fully explained in Note C., reduces to an im- 
mense extent the number of cards required. It is obvious that this 
mechanical improvement is especially applicable wherever cycles occur 
in the mathematical operations, and that, in preparing data for cal- 
culations by the engine, it is desirable to arrange the. order and com- 
bination of the processes with a view to obtain them as much as pos- 
sible symmetrically and in cycles, in order that the mechanical ad- 
vantages of the backing system may be applied to the utmost. It is here 
interesting to observe the manner in which the value of an analytical 
resource is met and enhanced by an ingenious mechanical contrivance. 
We see in it an instance of one of those mutual adjustments between 
the purely mathematical and the mechanical departments, mentioned 
in Note A. as being a main and essential condition of success in the 
invention of a calculating engine. The nature of the resources afforded 
by such adjustments would be of two principal kinds. In some eases, 
a difficulty (perhaps in itself insurmountable) in the one department, 
would be overcome by facilities in the other ; and sometimes (as in the 
present case) astrong point in the one, would be rendered still stronger 
and more available, by combination with a corresponding strong point 
in the other. 
As a mere example of the degree to which the combined systems of 
cycles and of backing can diminish the nwmber of cards requisite, we 
shall choose a case which places it in strong evidence, and which has 
likewise the advantage of being a perfectly different kind of problem 
from those that are mentioned in any of the other Notes. Suppose it 
be required to eliminate nine variables from ten simple equations of the 
form— 
a%+ba,+ea,+da,+.....=p (l.) 
@x4blz,+elr,+da,+..+... =p! (2) 
&e. &e. &e. &e. 
We should explain, before proceeding, that it is not our object to con- 
