406 D'ALEMBERT. 



are several bodies acting on each other, as by being con- 

 nected through inflexible rods, or by mutual attraction, 

 or in any other way that may be conceived; suppose an 

 external force is impressed upon these bodies, they will 

 move not in the direction of that force as they would were 

 they all unconnected and free, but in another direction ; 

 then the force acting on the bodies may be decomposed 

 into two, one acting in the direction which they actually 

 take, or moving the bodies without at all interfering with 

 their mutual action, the other in such direction as that 

 the forces destroy each other, and are wholly extin- 

 guished; being such, that if none other had been im- 

 pressed upon the system, it would have remained at 

 rest.* This principle reduces all the problems of dy- 

 namics to statical problems, and is of great fertility, as 

 well as of admirable service in both assisting our investi- 

 gations and simplifying them. It is, indeed, deducible 

 from the simplest principles, and especially from the 

 equality of action and re-action; but though any one 

 might naturally enough have thus hit upon it, how vast a 

 distance lies between the mere principle and its applica- 

 tion to such problems, for example, as to find the locus or 



* Lagrange's statement of the principle is the most concise, but 

 I question if it is the clearest, of all that have been given. " If there 

 be impressed upon several bodies, motions which they are compelled 

 to change by their mutual actions, we may regard these motions as 

 composed of the motions which the bodies will actually have, and of 

 other motions which are destroyed; from whence it follows, that the 

 bodies, if animated by those motions only, must be in equilibrio." 

 (<Mec. An.' vol. i. p. 239, Ed. 1811.) It is not easy to give a 

 general statement of the principle, and I am by no means wedded 

 to the one given in the text. A learned friend has communicated 

 one which the reader will find in Appendix II., together with a 

 statement, by another excellent geometrician, of the real benefit 

 derived from the Principle. 



