METHOD OF VARIATIONS. 61 



Periodic Variations. 



A very large and important class of investigations are con- 

 cerned with Periodic Variations. We may define a periodic 

 phenomenon as one which, with the constant and uniform 

 change of the variable, returns time after time to the 

 same value. If we strike a pendulum it presently returns 

 to the point from which we disturbed it, and with the 

 uniform progress of time goes on making excursions and 

 returning, until stopped by the dissipation of its energy. 

 If one body in space approaches by gravity towards 

 another, they will revolve round each other in an elliptic 

 orbit, and return for an indefinite number of times to the 

 same relative positions. On the other hand a single body 

 projected into empty space, away from the action of any 

 extraneous force, would go on moving for ever in a 

 straight line, according to the first law of motion. In the 

 latter case the variation is called secular, because it pro- 

 ceeds during ages in a similar manner, and suffers no 

 TrepioSo? or going round. It may be doubted whether 

 there really is any motion in the universe which is not 

 periodical. Mr. Herbert Spencer long since adopted the 

 doctrine that all motion is ultimately rhythmical P, and 

 abundance of evidence may be adduced in favour of his 

 view. The so-called secular acceleration of the moon's 

 motion is certainly periodic, and as, so far as we can tell, 

 no body is beyond the attractive power of other bodies, 

 rectilinear motion becomes purely hypothetical, or at least 

 infinitely improbable. All the motions of all the stars 

 must tend to become periodic. Though certain disturb- 

 ances in the planetary system seem to be uniformly pro- 

 gressive, Laplace is considered to have proved that they 

 really have their limits, so that after an almost infinitely 

 great time, all the planetary bodies might return to the 



P 'First Principles,' 31-0! edit. chap. x. p. 253. 



