730 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR. 
on a new column. But as these variations follow the same law at 
each repetition, (Operation 21 always requiring its factor from a co- 
lumn one in advance of that which it used the previous time, and Ope- 
ration 24 always putting its result on the column one in advance of 
that which received the previous result), they are easily provided for 
in arranging the recurring group (or cycle) of Variable-cards. 
We may here remark that the average estimate of three Variable- 
cards coming into use to each operation, is not to be taken as an ab- 
solutely and literally correct amount for all cases and circumstances. 
Many special circumstances, either in the nature of a problem, or in 
the arrangements of the engine under certain contingencies, influence 
and modify this average to a greater or less extent. But it is a very 
safe and correct general rule to go upon. In the preceding case it 
will give us seventy-five Variable-cards as the total number which will 
be necessary for computing any B after B,. This is very nearly the 
precise amount really used, but we cannot here enter into the minutize 
of the few particular circumstances which occur in this example (as 
indeed at some one stage or other of probably most computations) to 
modify slightly this number. 
It will be obvious that the very same seventy-five Variable-cards may 
be repeated for the computation of every succeeding Number, just on 
the same principle as admits of the repetition of the thirty-three Varia- 
ble-cards of Operations (13...23) in the computation of any one Num- 
ber. Thus there will be a cycle of a cycle of Variable-cards. 
If we now apply the notation for cycles, as explained in Note E, we 
may express the operations for computing the Numbers of Bernoulli in 
the following manner:— 
1.7), (24,25) -eeceeececeeceeees gives B, = Ist number; (7 being 
OR CRONC CD RO B, Sand WR iC eee 
1...7), (8.12), (1323), (24, 25) +0 eeeeee B, =Srd..... : (werree 
(17); (812), 2 (13.93), (24, 25) -. +++. B, =4th..... SR 
(1.7); (S12), SC41)"=2(18..98), (24,25) eBan—1=nth vee {8 
Again, 
(1x7); (245 25), 5(+1)" J (doe ),(Sue12); B(2 +2) (13.023),(24y 25 yt 
limits 1 to 2 limits 0 to (n + 2) 
represents the total operations for computing every number in succes- 
sion, from B, to Bo» —1 inclusive. 
In this formula we see a varying cycle of the first order, and an or- 
dinary cycle of the second order. ‘The latter cycle in this case includes 
in it the varying cycle. 
On inspecting the ten Working-Variables of the diagram, it will be 
perceived, that although the value on any one of them (excepting V, 
. and V,) goes through a series of changes, the office which each per- 
forms isin this calculation fixed and invariable. Thus V; always pre- 
pares the nwmerators of the factors of any A; V, the denominators. 
V, always receives the (2n — 3)th factor of Agn-1, and Vy the 
(2m —1)th. Vj,. always decides which of two courses the suc- 
ceeding processes are to follow, by feeling for the value of » through 
