ON BABBAGE’S ANALYTICAL ENGINE. 717 
analysis there is a reewrring group of one or more cycles; that is, a 
eycle of a cycle, or a cycle of cycles. For instance: suppose we wish to 
ide a series by a series, 
a+tbe+eu*+..... 
@)  @4+b'at+ea?+ ..... Z 
it being required that the result shall be developed, like the dividend 
and the divisor, in successive powers of z. A little consideration of 
(i.), and of the steps through which algebraical division is effected, 
will show that (if the denominator be supposed to consist of p terms) 
the first partial quotient will be completed by the following opera- 
tions: 
(2) {(+)p(% =)f or {1,9 @ 8)} 
that the second partial quotient will be completed by an exactly similar 
set of operations, which acts on the remainder obtained by the first set, 
instead of on the original dividend. The whole of the processes there- 
fore that have been gone through, by the time the second partial quo- 
tient has been obtained, will be,— 
(3) 214(+)p(x,—)} or2{(1),7(28)}, 
which is a cycle that includes a cycle, or a cycle of the second order. 
The operations for the complete division, supposing we propose to 
obtain » terms of the series constituting the quotient, will be,— 
x n{(+)p (x —)} orm {(1),p(28)}. 
It is of course to be remembered that the process of algebraical division 
in reality continues ad infinitum, except in the few exceptional cases 
which admit of an exact quotient being obtained. The number x 
inthe formula (4.), is always that of the number of terms we propose to 
ourselves to obtain; and the mth partial quotient is the coefficient of 
the (7 = 1)th power of «. ms 
_ There are some cases which entail cycles of cycles of cycles, to an 
indefinite extent. Such cases are usually very complicated, and they 
‘are of extreme interest when considered with reference to the engine. 
The algebraical development in a series, of the nth function of any 
given function, is of this nature. Let it be proposed to obtain the nth 
function of 
(.) ¢ (a,b,c ...... @), x being the variable. 
We should premise that we suppose the reader to understand what is 
meant by an vth function. We suppose him likewise to comprehend di- 
stinctly the difference between developing an nth function algebraically, 
and merely calculating an nth function arithmetically. If he does not, 
the following will be by no means very intelligible; but we have not 
_ Space to give any preliminary explanations. To proceed: the law, ac- 
cording to which the successive functions of (5.) are to be developed, 
must of course first be fixed on. ‘This law may be of very various 
cinds. We may propose to obtain our results in successive powers of x, 
n which case the general form would be 
C +C, 4+ C, a? + &é, 
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