465 



APPENDIX. 



I. 



NOTE ON D'ALEMBEET'S PRINCIPLE, 



Professor Playfair (' Ed. Eev.' xi., 253) has by no means 

 been nappy in his enunciation of the Principle. " If the 

 motions which the particles of a moving or a system of 

 moving bodies have at any instant be resolved into each two, 

 one of which is the motion which the particle had in the pre- 

 ceding instant, then the sum of all these third motions must 

 be such that they are in equilibrium with one another." 



The following are the observations referred to in p. 396, note. 



The great utility of this principle proceeds from the uni- 

 versality of its operation, and from its supplying the place 

 of the detached artifices and ingenious assumptions by which 

 dynamical problems had hitherto been treated, by a rule 

 directly applicable to the circumstances of the motion of 

 one or more bodies whose motions were any other than those 

 immediately proceeding from the direct and unfettered action 

 of the motive force. 



The principle applies equally to the most elementary and 

 the most difficult problems to the motion of a body down 

 an inclined plane the vibrations of a simple pendulum or 

 to the theory of the radiation of heat the vibrations of a 

 chord : two subjects previously of insuperable difficulty, to 

 which the illustrious author applied his new method, and 

 which became remarkable in his hands, not only for the 

 solutions which he obtained, but also for the manner of 

 them for it was his singular good fortune, by a further 

 invention, to overcome the analytical difficulties into which 

 the fecundity of his dynamical principle had led him. 



The great utility of this principle wiU not appear from 

 the comparison of the solutions of any one problem obtained 

 by its means, with the detached artifices previously employed; 

 2 H 



