ON BABBAGE’S ANALYTICAL ENGINE. 729 
values undergo alterations during a performance of the operations 
(13 ....23), and consequently the columns present a new set of values 
for the newt performance of (13....23) to work on. 
At the termination of the repetition of operations (13...23) in com- 
puting B,, the alterations in the values on the Variables are, that 
V;, =2n —4 instead of 2 n — 2. 
WHS Glas Ww VOUS Ae 
Wip Ove eee 1 
V,;= A, + A, B, + A, B, + A, B, instead of A,+ A, B, + A, B,; 
In this state the only remaining processes are first: to transfer the 
value which is on V,,, to V.,; and secondly to reduce V,, V,, V,; to 
zero, and to add* one to V,, in order that the engine may be ready 
to commence computing B,. Operations 24 and 25 accomplish these 
purposes. It may be thought anomalous that Operation 25 is repre- 
sented as leaving the upper index of V, still = unity. But it must be 
remembered that these indices always begin anew for a separate calcu- 
lation, and that Operation 25 places upon V, the first value for the new 
calculation. 
It should be remarked, that when the group (13...23) is repeated, 
changes occur in some of the wpper indices during the course of the 
repetition: for example, 9V, would become *V, and °V,. 
We thus see that when m = 1, nine Operation-cards are used ; that 
when 2 = 2, fourteen Operation-cards are used; and that when » > 2, 
twenty-five Operation-cards are used; but that no more are needed, 
however great » may be; and not only this, but that these same 
twenty-five cards suffice for the successive computation of all the Num- 
bers from B, to By» -, inclusive. With respect to the number of 
_ Variable-cards, it will be remembered, from the explanations in pre- 
_ vious Notes, that an average of three such cards to each operation (not 
__ however to each Operation-card) is the estimate. According to this 
_ the computation of B, will require twenty-seven Variable-cards; B, 
forty-two such cards; B, seventy-five; and for every succeeding B 
_ after B,, there would be thirty-three additional Variable-cards (since 
each repetition of the group (13...23) adds eleven to the number 
of operations required for computing the previous B). But we must 
now explain, that whenever there is a cycle of operations, and if these 
merely require to be supplied with numbers from the same pairs of 
columns and likewise each operation to place its veswlé on the same 
column for every repetition of the whole group, the process then 
_ admits of a cycle of Variable-cards for effecting its purposes. There is 
_ obviously much more symmetry and simplicity in the arrangements, when 
_ cases do admit of repeating the Variable as well as the Operation-cards. 
_ Our present example is of this nature. The only exception to a per- 
: Sect identity in all the processes and columns used, for every repetition 
of Operations (13.,.23) is, that Operation 21 always requires one of 
its factors from a new column, and Operation 24 always puts its result 
* It is interesting to observe, that so complicated a ease as this calculation 
of the Bernoullian Numbers, nevertheless, presents a remarkable simplicity in 
one respect ; viz., that during the processes for the computation of millions of 
_ these Numbers, no other arbitrary modification would be requisite in the ar- 
_ Yangements, excepting the above simple and uniform provision for causing one 
of the data periodically to receive the finite increment unity. 
