PROCEEDINGS OF THE POLYTECHNIC ASSOCIATION. 577 



any series of imnibcrs, the ?!th oi'der of differences may be regarded as 

 equal to zero, where n may be any number whatever. To produce such a 

 series by mechanical means, the only condition necessary is that there be 

 as many variable wheels indicating the numbers as we have differences. If 

 n was a very large number, the mechanism would become cumbersome and 

 unwieldy. It is found in practice that for the great majority of useful 

 computations, four orders of differences are sufficient. 



This machine is constructed for that number, and will consequently com- 

 pute any series, in which the fourth (or any inferior) order of differences 

 are equal. 



This will be more easily understood by a simple illustration. Suppose 

 it is desired to tabulate the series of square numbers beginning with unity. 

 Let us first see how these numbers can be produced by means of successive 

 differences. We arrange them for convenience in the following table: 



Number. Square. 1st diff. 2d diff. 3d diff. 



11 



3 

 2* 4 2 



5 



3 9 2 



1 



4 16 



Having given the number (1) square of 1, and the first difference 3 ~: 

 (2'— r) and the second diff, 2 = (3^— 2=)— (2'— P), the squares of all suc- 

 cessive numbers may be found by continued additions. 



The difference between (2^ — 1") r= 3 is the first diff.; the second diff. 

 (2) is constant. Then, 



(3)' = 9 = (2)' -f 1st diff. (3) + 2d diff. (2), 



or 9 = 4 + (3+2), 



or 9 = 4 +5, (5) being the second number in column of first differences 



And the same process may be repeated to any extent. 



What now is required iu a machine, is, first to be able to produce the 

 first order of differences, having given the first difference 3, and the second 

 difference (2) constant. 



Suppose we have a wheel, on the circumference of which is inscribed, at 

 equal distances apart, the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. If this wheel 

 should be set so that the figure 3 should coincide with a fixed mark, and 

 by means of any simple mechanism, it should be made to advance two di- 

 visions at every motion of a lever, we should successively cause the num- 

 bers 3, 5, 1, 9, to coincide with the mark; and if when the wheel has made 

 a complete revolution, it should cause another wheel placed by its side to 

 advance one division. The process maj' be continued so as to form the 

 series of first differences, in which every successive number would be 

 greater by 2 than the number precedi)ig. 



This principle of successive additions is exemplified in an ordinary clock; 

 for at every oscillation of the pendulum the second-hand advances one di- 

 vision, and after this has made a complete revolution, the minute-hand has 

 also advanced one division. By a slight change in the escapement wheel, 

 the second wheel could be made to advance 2, 3, or 4 divisions at every 

 [Am. Ixst.] K* 



