Let us consider a term of the form ; 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 ; that is to say, whatever be the number of multiplications required for elevating to the th power (we are supposing for the moment that is a whole number). Now, since the exponent indicates that is to be multiplied times by itself, and all these operations are of the same nature, it will be sufficient to employ one [Pg 21] single operation-card, viz. that which orders the multiplication. But when is given for the particular case to be calculated, it will be further requisite that the machine limit the number of its multiplications according to the given values. The process may be thus arranged. The three numbers , and will be written on as many distinct columns of the store; we shall designate them , , ; the result will place itself on the column . When the number has been introduced into the machine, a card will order a certain registering-apparatus to mark (), and will at the same time execute the multiplication of by . When this is completed, it will be found that the registering-apparatus has effaced a unit, and that it only marks (); while the machine will now again order the number written on the column to multiply itself with the product written on the column , which will give . Another unit is then effaced from the registering-apparatus, and the same processes are continually repeated until it only marks zero. Thus the number will be found inscribed on , when the machine, pursuing its course of operations, will order the product of by ; and the required calculation will have been completed without there being any necessity that the number of operation-cards used should vary with the value of . If were negative, the cards, instead of ordering the multiplication of by , would order its division; this we can easily conceive, since every number, being inscribed with its respective sign, is consequently capable of reacting on the nature of the operations to be executed. Finally, if were fractional, of the form , an additional column would be used for the inscription of , and the machine would bring into action two sets of processes, one for raising to the power , the other for extracting the th root of the number so obtained.