VELOCITIES. &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 



1 d ( 



^ • I ' 



r cos d dt \ 



dt 



c. Let the axes of as, y, z be always parallel to the tangent, principal normal and normal 

 to the osculating plane of any curve. Then 



where de, dr denote respectively the angle between consecutive tangents, and that between 

 consecutive osculating planes. 



Hence 



tangential acceleration =f^=-~, 



... . . , , ^ ds de /ds\2 de 1 (dsY 



acceleration m principal normal = r, = -—.-— =-- .-^ = --^5 

 ^ ^ -^^ dt dt \dtl ds fj \dtj 



acceleration in normal to osculating plane =f^ = 0. 



SECTION II. 



Dynamical Applications. 



9. I propose here to consider the problem of the motion of any material system, so far as 

 it depends on external forces only, and to develop the solution in that case in which the entire 

 motion is determined by these forces, namely, in the case of an invariable system. 



1 0. This problem naturally resolves itself into two : for, since every system of forces is 

 reducible to a single force and a single couple, we have to investigate the effects of that force, 

 and the effects of that couple. Now we know that the resultant force determines the motion 

 of the centre of gravity of the system, be the constitution of the system what it may. In like 

 manner the resultant couple determines something relatively to the motion of the system about 

 its centre of gravity, which in the case of an invariable system defines its motion of rotation 

 about that point, but which in other cases is not usually recognised as a definite objective 

 magnitude, and has therefore no received name. This defect will be remedied by adopting 

 momentum as the intermediate term between force and velocity, and by regarding as distinct 

 steps the passage from force to momentum and that from momentum to velocity. In accordance 

 with this idea we proceed to shew that as in our first problem we shall be concerned with the 

 magnitudes, force, linear momentum or momentum of translation, and linear velocity or velocity 

 of translation, so in the other we shall be concerned with the corresponding magnitudes, couple, 

 angular momentum or momentum of rotation, and angular velocity or velocity of rotation ; and 

 that, as all these magnitudes possess the properties characteristic of the magnitude u in the 

 previous section, the Calculus there developed is applicable to them. 



