ON BABBAGE’S ANALYTICAL ENGINE. 719 
_ peculiar to itself, and not recurring for the remaining functions. 
_ We have given but a very slight sketch, of the principal general 
steps which would be requisite for obtaining an mth function of such 
a formula as (5.). The question is so exceedingly complicated, that 
_ perhaps few persons can be expected to follow, to their own satisfac- 
tion, so brief and general a statement as we are here restricted to on 
this subject. Still it is a very important case as regards the engine, 
and suggests ideas peculiar to itself, which we should regret to pass 
wholly without allusion. Nothing could be more interesting than to 
follow out, in every detail, the solution by the engine of such a case 
as the above; but the time, space and labour this would necessitate, 
could only suit a very extensive work. 
To return to the subject of cycles of operations : some of the notation 
of the integral calculus lends itself very aptly to express them: (2.) 
might be thus written :— 
(6.) (+), 2 (+1)? (x, —) or (1), 2 (+ 1? (2, 3), 
where p stands for the variable ; (+ 1)? for the function of the variable, 
that is, for ¢ p; and the limits are from 1 to p, or from 0 top — 1; 
each increment being equal to unity. Similarly, (4.) would be,— 
(7.) S41)" {(+), BC+" (x, —)} 
the limits of being from 1 to , or from 0 to x — 1, 
(8.) or 5(+1)" {(1), B(4+ 1) (2, 3) f- 
Perhaps it may be thought that this notation is merely a circuitous 
way of expressing what was more simply and as effectually expressed 
before ; and, in the above example, there may be some truth in this. 
But there is another description of cycles which caz only effectually be 
expressed, in a condensed form, by the preceding notation. We shall 
call them varying cycles. They are of frequent occurrence, and in- 
clude successive cycles of operations of the following nature :— 
(9) p (1, 2,-.m), p—1 (1,2...m),p— 2 (1,2..m)...p—n(1,2..m); 
where each cycle contains the same group of operations, but in which 
_the number of repetitions of the group varies according to a fixed rate, 
with every cycle. (9.) can be well expressed as follows :— 
pec might in many cases be developed through a set of processes 
(10.) Spi, 2 ..m), the limits of p being from p — x to p. 
_ Independent of the intrinsic advantages which we thus perceive to 
result in certain cases from this use of the notation of the integral cal- 
sulus, there are likewise considerations which make it interesting, from 
e connections and relations involved in this new application. It has 
een observed in some of the former Notes, that the processes used in 
alysis form a logical system of much higher generality than the ap- 
ications to number merely. Thus, when we read over any algebraical 
formula, considering it exclusively with reference to the processes of 
the engine, and putting aside for the moment its abstract signification 
as to the relations of quantity, the symbols +, x, &c., in reality re- 
present (as their immediate and proximate effect, when the formula is 
VOL, Ill, PART XII. 3B 
