178 [June 19, 



cate a remarkable symbolical resemblance between det (P, Q) and 

 the product PQ. For instance, it may be proved that if P, P', Q be 

 any three straight lines, then the resultant of det (P, Q) and det (P', Q) 

 is det (P+P', Q), where P + P' denotes the resultant or complete 

 sum of P and P'. 



Again, it may be shown that the complete differential coefficient of 

 det (P, Q) is exactly similar in form to the differential coefficient of 

 the product P, Q. 



9. Furnished with these propositions, it is easy to extend the 

 formulae of the first chapter to lines moving in space of three dimen- 

 sions. 



Let A denote the instantaneous axis about which, and the angular 

 velocity with which, a line R, whose length at time t is r, revolves, 



then the complete differential coefficient of R is the resultant of - 



ut 



in the direction of R, and the determinant of A to R, or 



This is the fundamental proposition concerning the differential coeffi- 

 cient of a line. Applying it to the component parts of the last for- 

 mula, we obtain a somewhat remarkable expression for the second 

 differential coefficient of a line, and therefore also for a particle's 

 acceleration. That expression easily leads to the formulae for the 

 acceleration of a particle relatively to any moving axes, and also to a 

 very simple proof of Corioli's beautiful theorem concerning relative 

 motion. The use of this method is illustrated by showing how it 

 enables us at once to write down the equations for the motion of the 

 simple pendulum, taking the earth's rotation into account. 



10. I now pass to the dynamics of a rigid body. 



Compounding the momenta of the different particles of a body as 

 if they were forces, they may be reduced to a single momentum at O 

 and a couple of momenta. The former I call the body's single mo- 

 mentum, and denote it by U ; the latter I call the body's momentum 

 couple, and denote it, or rather denote its axis, by H. If now the 

 external forces acting on the body be similarly reduced to a force P 

 at O, and a couple whose axis is G, it may be shown that D' Alembert's 

 principle is contained in the proposition " that P and G are respect- 

 ively the complete differential coefficients of U and H." 



