686 L. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. ' 
one single operation-card, viz. that which orders the multiplica- 
tion. 
But when x 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 a, b and v will be 
written on as many distinct columns of the store; we shall de- 
signate them V,, V,, V.; the result a4” will place itself on the 
column V,;. When the number z has been introduced into the 
machine, a card will order a certain registering-apparatus to 
mark (x — 1), and will at the same time execute the multiplica- 
tion of b by 4. When this is completed, it will be found that 
the registering-apparatus has effaced a unit, and that it only 
marks (x — 2); while the machine will now again order the num- 
ber 4 written on the column V, to multiply itself with the product 
6? written on the column V,, which will give 4°. Another unit 
is then effaced from the registering-apparatus, and the same 
processes are continually repeated until it only marks zero. Thus 
the number 2” will be found inscribed on V;, when the machine, 
pursuing its course of operations, will order the product of 
b* by a; 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 ofn. If m were negative, 
the cards, instead of ordering the multiplication of a by 4", would 
order its division; this we can easily conceive, since every num- 
ber, being inscribed with its respective sign, is consequently ca- 
pable of reacting on the nature of the operations to be executed. 
Finally, if ~ were fractional, of the form 5 , an additional 
column would be used for the inscription of g, and the machine 
would bring into action two sets of processes, one for raising } 
to the power p, the other for extracting the gth root of the num- 
ber so obtained. 
Again, it may be required, for example, to multiply an expres- 
sion of the form aa” + 62” by another Az? + Ba’, and then 
to reduce the product to the least number of terms, if any of the 
indices are equal. The two factors being ordered with respect 
to x, the general result of the multiplication would be Aaa +p 
+Aba"t? + Baz"t?+ Boba +4, Upto this point the 
process presents no difficulties ; but suppose that we have m= p 
and n= q, and that we wish to reduce the two middle terms to 
