I have formed other series of the same elass, and have succeeded in express- ins any term independent of all the rest by two distinct processes. Thus have incidentally been able to integrate the equations I have mentioned: I will just state one other of a simple form ; it is the equation A m units fig. « z ~ whose integral is ^ = ^ + go . where a is that one of L numbers 1, 2, 3, 4, which taken from . leaves the remainder divisible by 4, and b is the quotient of that dms.on. The table is as follows : ^ 4 8 4-16 22 24 28 8-36 42 One of the general questions to which these researches give rise is, suppo- sing the law of any series to be known, to find what figure will occur m the ftth place of the nth term. That the mere consideration of a mechanical engine should have suggested these inquiries, is of itself sufficiently remarkable; but it is still more singular, that amongst researches of so very abstract a nature i should have met with and overcome a difficulty which had presented itselt in toe form of an equation of differences, and which had impeded my progress several years since, in attempting the solution of a problem connected with the game of chess.
