{ 
684 LL. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. | 
following* very simple example, in which we are to multiply 
(a + b2') by (A+ Bcos!z). We shall begin by writing 2°, 2, 
cos? 2, cos! 2, above the columns V,, V,, V,, V;; then, since from 
the form of the two functions to be combined, the terms which 
are to compose the products will be of the following nature, 
x#®.cos® x, x°.cos' xz, z'.cos° x, 21.cos! a; these will be inscribed 
above the columns V,, V;, V,, V,._ The coefficients of 7°, x', cos® 
x, cos! x# being given, they will, by means of the mill, be passed 
to the columns Vy, V,, V. and V;. Such are the primitive data 
of the problem. It is now the business of the machine to work 
out its solution, that is to find the coefficients which are to be 
inscribed on V,, V;, Vg, V;, To attain this object, the law of 
formation of these same coefficients being known, the machine 
will act through the intervention of the cards, in the manner in- 
dicated by the following table :— 
Coeffi- | Cards of the J 
cients. | operations. Cards of the variables. 
Columns 
above which a a S Columns 
are written Be bas | Columns on which Indication of 
the functions | < E 3s 5 = | onwhich | are to be | change of value 
of the 2} & | 2] | operations | inscribed | on any column Results of the operatio 
variable. 6] 2 5 O17 °] aretobe |theresults} submitted to 
5 2] 3 | performed. of the an operation. 
B Se a operations. 
° ° 
x? ececccces lV, a ” ” ” ” , ” ” 
zl seeeeesee ly, b ’ ” ” ” ” ” ” 
Cos? x . ANE A ” ” ” ” ” ” ” 
Cos} zx osaeu Vg B ” » ” ” ” 1 ” 1 ” 
V, = 1V, F 
2° cos? x ..°Vy|...,aA| 1 | xX [Vox!Vo=]!V, ...... {i a iy? f 1V,=aA coefficients of a 
pres IV) = °V 
2° cos! ..°V;|....@B} 2 | X |VoX!V3=]1V, ...... ene = iv} 1V;=aB 
ty, ="4V, 
x) cos? x ..°V,_|...,5A}] 3 | X PVyX!Vo=l!V¢e ...--- {ay 7, oy} IV = DA nes e 
pS a 2 
Br) 
zl cos! x ..°V,}..., BB} 4 | X }IV)X!V3=/!Vz ...... Wi oy. IVP Bence vice 
Sea 3 
It will now be perceived that a general application may be 
made of the principle developed in the preceding example, to every 
species of process which it may be proposed to effect on series 
submitted to calculation. It is sufficient that the law of forma- 
tion of the coefficients be known, and that this law be inscribed 
on the cards of the machine, which will then of itself execute all 
* See Note E. 
+ For an explanation of the upper left-hand indices attached to the V’s in 
this and in the preceding Table, we must refer the reader to Note D, amongst 
those appended to the memoir.— Nore sy TRANSLATOR. 
