916 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR 
Variables and beconie mere Working-Variables ; Vos V4;, &c. being now 
the recipients of the ultimate results. 5 
We should observe, that if the variables cos 6, cos2 6, cos3 0, &c. — 
are furnished, they would be placed directly upon V,,, V4o, &c., like any 
other data. If not, a separate computation might be entered upon in 
a separate part of the engine, in order to calculate them, and place 
them on V,,, &c. 
We have now explained how the engine might compute (1.) in the 
most direct manner, supposing we knew nothing about the general 
term of the resulting series. But the engine would in reality set to 
work very differently, whenever (as in this case) we do know the law 
for the general term. 
The two first terms of (1.) ate 
(BA +2B,A,) + (BA, + B, A + 2B, A, .G0s0).4.... (4) 
and the general term for all after these is 
(B An +4B,-An—1 + Any2)cosnd...... odiasi0* el 
which is the coefficient of the (m + 1) term. The engine would 
calculate the two first terms by meatis of a separate set of suitable 
Operation-cards, atid would then need another set for the third term ; 
which last set of Operation-cards would calculate all the succeeding terms 
ad infinitum ; merely requiring certain new Variable-cards for each term 
to direct the operations to act on the proper columns. The following 
would be the successive sets of operations for computing the coefficients | 
of n + 2 terms := 
(Xs XH +), 0X5 Xs Xe Hy Hy +) (CK, +, XH, +). 
Or we might represent them as follows, according to the numerical 
order of the operations :— 
(1; 2 +664); (6,6 ..+ 10), (11, 12+..15). 
The brackets, it should be understood, point out the relation in which 
the operations may be grouped, while the comma marks succession. 
The symbol + might be used for this latter purpose, but this would be 
liable to produce confusion, as + is also necessarily used to represent 
one class of the actual operations which are the subject of that succes-— 
sion. In accordance with this meaning attached to the comma, care 
must be taken when any one group of operations recurs more than once, — 
as is represented above by x (11... 15), not to insert a comma after 
the number or letter prefixed to that group. #,(11...15) would stand 
for an operation n followed by the group of operations (11 .+.15);_ 
instead of denoting the number of groups which are to follow each 
other. ‘ 
Wherever a general term exists, there will be a recurring group of 
operations, as in the above example. Both for brevity and for dise 
tinctness, a recurring group is called a eycle. A cycle of operations, — 
then, must be understood to signify any se¢ of operations which is re- 
peated more than once. It is equally a cycle, whether it be repeated 
twice only, or an indefinite number of times; for it is the fact of a 
repetition occurring at all that constitutes it such, In many cases of 
