Page:On Mr. Babbage's new machine for calculating and printing mathematical and astronomical tables.pdf/7

 and cause them to transfer their results to a common axis, with which the printing apparatus might be connected. Such a machine would, amongst other tables, compute one from the expression


 * A sin θ + B sin 2 θ + C sin 3 θ

the utility of which, in astronomy, is well known. In fact Mr. Babbage is of opinion that it would not be impossible to form a machine which should tabulate almost any individual equation of differences.

Amongst the singular and curious powers produced by small additions to the machinery, may be reckoned the possibility of tabulating series expressed by the following equation:

$$\bigtriangleup ^2u_3 = $$ the units figure of $$u^3$$

$$\bigtriangleup ^3u_3 = 2x^3$$ the figures found in the tens place of $$u_{3+x}$$

$$\bigtriangleup ^3u_3 = 4x$$ the figures found in the units and the tens place of $$u_{3+x}$$

and many others similar thereto.

Again, let the machine be in the act of tabulating any series, a part may be attached by means of which, whenever any particular figure (a 6 for example) occurs in the units place, any other number (23 for instance) shall be added to that and all the succeeding terms: and when, in consequence of this, another figure 6 occurs in the units place, then 23 more will be added to that and all the succeeding terms. Or, if it be preferred, the number added shall be added to the term ending in 6 only, and not to all succeeding ones.

These views may appear to some persons more curious than useful. They lead however to speculations of a very singular and difficult nature in determining the laws which such series follow: and they are not altogether so remote from utility as may be imagined. I avoid alluding to many other curious properties which this machine is capable of exhibiting, as they will scarcely be intelligible till the machine itself is more known in the world. Indeed I fear I have already tired your patience with this long letter.

Francis Baily