L. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. 685 
the calculations requisite for arriving at the proposed result. If, 
for instance, a recurring series were proposed, the law of forma- 
tion of the coefficients being here uniform, the same operations 
which must be performed for one of them will be repeated for 
all the others; there will merely be a change in the locality of 
the operation, that is it will be performed with different columns. 
Generally, since every analytical expression is susceptible of 
being expressed in a series ordered according to certain func- 
tions of the variable, we perceive that the machine will include 
all analytical calculations which can be definitively reduced to the 
formation of coefficients according to certain laws, and to the 
distribution of these with respect to the variables. 
We may deduce the following important consequence from 
these explanations, viz. that since the cards only indicate the 
nature of the operations to be performed, and the columns of 
Variables with which they are to be executed, these cards will 
_ themselves possess all the generality of analysis, of which they 
_ are in fact merely a translation. We shall now further examine 
some of the difficulties which the machine must surmount, if its 
“assimilation to analysis is to be complete. There are certain 
~ functions which necessarily change in nature when they pass 
through zero or infinity, or whose values cannot be admitted 
4 _ when “2 pass these limits. When such cases present them- 
_ selves, the machine is able, by means of a bell, to give notice 
_ that the passage through zero or infinity is taking place, and 
it then stops until the attendant has again set it in action for 
» whatever process it may next be desired that it shall perform. 
— If this process has been foreseen, then the machine, instead of 
¢ _Tinging, will so dispose itself as to present the new cards which 
have relation to the operation that is to succeed the passage 
= zero and infinity. These new cards may follow the first, 
but may only come into play contingently upon one or other of 
the two circumstances just mentioned taking place. 
Let us consider a term of the form a6”; since the cards are 
but a translation of the analytical formula, their number in this 
particular case must be the same, whatever be the value of n; 
that is to say, whatever be the number of multiplications required 
for elevating 4 to the nth power (we are supposing for the mo- 
ment that n is a whole number). Now, since the exponent n 
indicates that b is to be multiplied » times by itself, and all these 
operations are of the same nature, it will be sufficient to employ 
