718 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR 
or in successive powers of 7 itself, the index of the function we are 
ultimately to obtain, in which case the general form would be . 
C+C,x+ C,n? + &e., 
and 2 would only enter in the coefficients. Again, other functions of 
z or of n instead of powers, might be selected. It might be in addition 
proposed, that the coefficients themselves should be arranged according 
to given functions of a certain quantity. Another mode would be to 
make equations arbitrarily amongst the coefficients only, in which case | 
the several functions, according to either of which it might be possible 
to develop the mth function of (5.), would have to be determined from 
the combined consideration of these equations and of (5.) itself. 
The algebraical nature of the engine (so strongly insisted on in a 
previous part of this Note) would enable it to follow out any of these 
various modes indifferently ; just as we recently showed that it can 
distribute and separate the numerical results of any one prescribed 
series of processes, in a perfectly arbitrary manner. Were it otherwise, 
the engine could merely compute the arithmetical nth function, a result 
which, like any other purely arithmetical results, would be simply a 
collective number, bearing no traces of the data or the processes which 
had led to it. 
Secondly, the daw of development for the mth function being selected, 
the next step would obviously be to develope (5.) itself, according to this 
law. This result would be the first function, and would be obtained 
by a determinate series of processes. These in most cases would in- 
clude amongst them one or more cycles of operations. 
The third step (which would consist of the various processes neces- 
sary for effecting the actual substitution of the series constituting the 
Jirst function, for the variable itself) might proceed in either of two 
ways. It might make the substitution either wherever z occurs in the 
original (5.}, or it might similarly make it wherever 2 occurs in the first 
function itself which is the equivalent of (5.). In some cases the former 
mode might be best, and in others the latter. 
Whichever is adopted, it must be understood that the result is to 
appear arranged in a series following the law originally prescribed for 
the development of the mth function. This result constitutes the se- 
cond function ; with which we are to proceed exactly as we did with 
the first function, in order to obtain the third function; and so on, 
nm — | times, to obtain the zth function. We easily perceive that since 
every successive function is arranged in a series following the same law, 
there would (after the first function is obtained) be a cyele, of a cycle, 
of a cycle, &c. of operations *, one, two, three, up to 2 —1 times, in 
order to get the zth function. We say, after the first function is ob- 
tained, because (for reasons on which we cannot here enter) the first 
* A cycle that includes x other cycles, successively contained one within an- 
other, is called a cycle of the n + 1th order. A cycle may simply include many 
other cycles, and yet only be of the second order. If a series follows a certain 
law for a certain number of terms, and then another law for anotlier number of 
terms, there will be a cycle of operations for every new law; but these cycles 
will not be contained one within another,—they merely follow each other. 
Therefore their number may be infinite without. influencing the order of a 
cycle that includes a repetition of such a series. 
