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sion for the acceleration of the resolved part of u along a given axis 

 as due to the motion of that axis. If u itself be variable, its variations 

 may be conceived to be due to an acceleration f in a definite direc- 

 tion, which in the time dt produces a quantity fdt in the direction 

 of f to be combined with u by the parallelogrammic law ; hence 

 result expressions for the changes in intensity and direction of u. 

 If, u being variable, the variations in its intensity estimated along a 

 given moveable direction be sought, it will consist of two parts : one, 

 that due to the resolved part off in the given direction ; the other, 

 that due to the motion of the axis, which is the same as if f had not 

 existed, or u had been constant : hence expressions for the total 

 acceleration of the resolved part of u along the given moveable axis. 

 If u be resolved along three rectangular axes, these expressions take 

 the forms of familiar kinematical and dynamical equations. 



These results furnish immediately expressions for the relative 

 velocities of a point with respect to moving axes when its absolute 

 velocities in their directions are given, and vice versd. They also 

 furnish very ready means of estimating accelerations in variable di- 

 rections ; as, for instance, the radial and transversal accelerations of 

 a point moving in a plane or in space, or the tangential and normal 

 accelerations in the same case. These are some kinematical appli- 

 cations of the calculus. 



The dynamical applications form the second part of the paper. 

 Here the general problem of the motion of a system, so far as it is 

 due to external forces, is divided into two steps : one from, force to 

 momentum, the other from momentum to velocity. If the momenta of 

 the particles of a system be reduced like a system of forces, they 

 produce a single linear momentum and a single angular momentum, 

 just as a system of forces produces a single force and a single couple. 

 The linear momentum is (in our received language) the momentum 

 of the mass of the system collected at its centre of gravity ; the an- 

 gular momentum is a magnitude the constancy of whose intensity 

 in a given axis is equivalent to the assertion of the principle of the 

 conservation of areas for that axis, and the constancy of whose direc- 

 tions determines the " invariable plane as a plane perpendicular to 

 it. The momentum, whether linear or angular, is a magnitude to 

 which the previous calculus applies, and the resultant force and 

 resultant couple are respectively the accelerators of the two kinds of 

 momentum : hence the equations obtained in the first part, inter- 

 preted with respect to these magnitudes, furnish equations in any 

 required form for the determination of the momenta at any instant. 

 The step from force to momentum is independent of the nature of 

 the system, that from momentum to velocity requires the system to be 

 particularized. In the paper the case of an invariable system only 

 is considered, and in particular its motion of rotation about its centre 

 of gravity. The axis of rotation or angular velocity is related in 

 direction to that of angular momentum, as the radius of the central 

 ellipsoid with which it coincides to the normal at its extremity. 

 Hence an angular momentum constant in intensity and direction, 

 in general gives rise to an angular velocity variable in both respects, 



