728 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR 
zero upon it. (The sign at the top of V, would become — during this 
process. 
parca 7 will be unintelligible, unless it be remembered that if 
we were calculating for » = 1 instead of n = 4, Operation 6 would 
have completed the computation of B, itself; in which case the engine, 
instead of continuing its processes, would have to put B, on V.,; and 
then either to stop altogether, or to begin Operations 1, 2....7 all over 
again for value of 2 (= 2), in order to enter on the computation of B, ; 
(having hovever taken care, previous to this recommencement, to make 
the number on V, equal to éwo, by the addition of unity to the former 
2 = 1lonthat column). Now Operation 7 must either bring out a result 
equal to zero (if 2 = 1); or a result greater than zero, as in the pre- 
sent case; and the engine follows the one or the other of the two 
courses just explained, contingently on the one or the other result of 
Operation 7. In order fully to perceive the necessity of this experi- 
mental operation, it is important to keep in mind what was pointed 
out, that we are not treating a perfectly isolated and independent com- 
putation, but one out of a series of antecedent and prospective com- 
putations. 
1 @n—1 2n 
Cards 8, 9, 10 produce —3- 9,4 + B, aa 
see an example of an upper index which again becomes a value after 
having passed from preceding values to zero. V,, has sucessively been 
OV, ,5!Vi152Vis Vis 9V),3 and, from the nature of the office which V,, 
performs in the calculation, its index will continue to go through further 
changes of the same description, which, if examined, will be found to 
be regular and periodic. 
Card 12 has to perform the same office as Card 7 did in the pre- 
ceding section; since, if 2 had been = 2, the 11th operation would 
have completed the computation of B. 
Cards 13 to 20 make A,. Since A,»—, always consists of 2% — 1 
factors, A, has three factors; and it will be seen that Cards 13, 14, 
15, 16 make the second of these factors, and then multiply it with the 
first ; and that 17, 18, 19, 20 make the third factor, and then multiply 
this with the product of the two former factors. 
Card 23 has the office of Cards 11 and 7 to perform, since if 2 
were = 3, the 21st and 22nd operations would complete the computa- 
tion of B,. As our case is B,, the computation will continue one 
more stage; and we must now direct attention to the fact, that in 
order to compute A, it is merely necessary precisely to repeat the 
group of Operations 13 to 20 ; and then, in order to complete the com- 
putation of B,, to repeat Operations 21, 22. 
It will be perceived that every unit added to 7 in Bon —1, entails an ad- 
ditional repetition of operations (13...23) for the computation of Bon —1. 
Not only are all the operations precisely the same however for every 
such repetition, but they require to be respectively supplied with numbers 
from the very same pairs of columns; with only the one exception of 
Operation 21, which will of course need B, (from V.5) instead of B, 
(from V..). This identity in the colwmns which supply the requisite 
numbers, must not be confounded with identity in the values those 
columns haye upon them and give out to the mill. Most of those 
In Operation 9 we 
