﻿On an Equation in Differences of the Second Order. 225 



of these currents is called into requisition for the works of nature 

 or of art, and the water is made to partake of the velocity of the 

 bed on which it rests, the store of force in our planet must be 

 wasted and the length of our day augmented. 



A more definite relation between the destruction of force in 

 this manner and the consequent change in planetary motion may 

 be shown by investigating the extent to which a satellite revol- 

 ving close to its primary has its orbit altered by tidal action 

 arising from the eccentricity of the ellipse which it describes 

 supposing the rotation adjusted for keeping the same side always 

 turned to the central body. To this problem other solutions 

 may be given besides that which I presented in the Philoso- 

 phical Magazine for December 1851. To seek for evidence of 

 the correlation of forces by physical inquiries of cases hitherto 

 untried or relating to phenomena presented in distant systems 

 is as legitimate as the course pursued by Newton and his 

 followers, who applied all the vast resources of mathematics 

 not for calculating the course of projectiles near the earth's sur- 

 face but for determining the orbits which solar attraction would 

 give bodies moving with immense velocities through the distant 

 realms of space. 



Cincinnati, January 21, 1869. 



[To be continued.] 



XXX. The Story of an Equation in Differences of the Second 

 Order. By J. J. Sylvester*. 



1Y/F Y recent researches into the order of the various systems 

 -LTX of equations which serve to determine the forms of redu- 

 cible cyclodes have led me to notice an equation in the second 

 order of differences which I imagine is new, and possesses a pe- 

 culiarly interesting complete integral. 

 If we call 



fx= {a*-tP) i (a*-b*f (&-&)* 



and (i,j, k,... &c. being given) determine a, b, c, . . . &c. so as 

 to make (/#)«+ (/'a?) 8 a complete square, and if we suppose the 

 indices i,j, k, . . . to consist of X integers of one value, //. inte- 

 gers of a second value, v of a third, and so on, the number of 

 solutions of the problem will in general depend not on i.j, k, but 

 on the derived integers \, p, v, . . . • and we may denote the 

 maximum value of this number by the type \X, /a, v, . . .]f. 



* Communicated by the Author, 

 f Ex. (jr. if 



/* = (* 2 - a 2 ) V - #/(*? _ e») V _ d ^ } 

 the type is [1, 1, 1, ].], of which the maximum value is 9; but if the sum 

 Phil. Mac/. S. 4. Vol. 37. No. 248. Mar. 1869. Q ^ 



