Page:Memoirs of Royal Astronomical Society Volume 01.djvu/335

312 other numbers, in order to produce the next portion of the table. This opera- tion must be repeated more or less frequently according to the nature of the table. The more numerous the order of differences, the less frequent will this operation become requisite. The chance of error in such computations arises from incorrect numbers being placed in the engine : it therefore becomes de- sirable to limit this chance as much as possible. In examining the analytical theory of the various differences of the sine of an arc, I noticed the property which, it possesses of having any of its even orders of differences equal to the sine of the same arc increased by some multiple of its increment multiplied by a constant quantity. With the aid of this principle an engine might be formed which would require but little attendance, and I believe that it might in some cases compute a table of the form A sin 0 from the 1st value of 0=0 up to 0=90° with only one set of figures being placed in it. It is scarcely necessary to observe what an immense number of astronomical tables are comprised under this form, nor the great accuracy which must result from having reduced to so few a number the preliminary computations which are requisite. In pursuing into its detail the principle to which I have alluded, which lends itself so happily to numerical application, I have traced its application to other species of tables, and am enabled to point out a course of analytical investiga- tion which will in all probability afford ready methods for constructing tables, even of the most complicated transcendent, in a manner equally easy. I will now advert to another circumstance, which, although not immediately connected with astronomical tables, resulted from an examination of the engine by which they can be formed. On considering the arrangement of its parts, I observed that a different mode of connecting them would produce tables of a new species altogether different from any with which I was acquainted. I therefore computed with my pen a small table such as would have been formed by the engine had it existed in this new shape, and I was much surprised at discovering that no analytical method was yet known for determining its n th term. The following is the first series I wrote down : Digitized by