734 PHILOSOPHICAL TRANSACTIONS. fANNO 1780. 



particles of the plane similarly situated in respect to ab, also qb, pa, perpendicular 

 to LCM. Now the centrifugal force of p, or its force in the direction ap, is 6 

 X ap, and that of q in the direction bq, is q X bq; to determine now how these 

 forces will affect the motion of the plane, we may observe in the first place, that 

 the force p X ap, acting at « in the plane, must tend to give it a motion about 

 an axis perpendicular to the plane; but as an equal force q X qb acts at q to give 

 it a motion in a contrary direction, it is evident that the two forces will destroy 

 each other, so far as they tend to generate any motion in the plane about an axis 

 perpendicular to it; and hence it is manifest, that if the parts of the plane ayb, 

 AZB, be similar, and similarly situated in respect to ab, the plane, after the 

 commencement of the motion, will have no tendency to revolve about an axis 

 perpendicular to it. Also, as the centrifugal force of each particle acts in a 

 direction parallel to ab, it can give the plane no tendency to revolve about that 

 line as an axis, and consequently the plane of rotation will be preserved as in 

 ipso motus initio. Conceiving therefore the plane on each side the line ab to 

 be similar, and similarly situated, suppose another plane to be fixed on this, 

 whose parts on each side ab are similar, and similarly situated, and the force to 

 act as before; then it is manifest, that as each plane endeavours to preserve the 

 same plane of rotation, the two planes connected will also continue to move in 

 the same plane of rotation ; for the action of one plane on another, on each side 

 the plane of rotation, being equal, cannot tend to disturb the motion in that 

 plane; and as this must be true for any number of planes thus similar and simi- 

 larly situated, it is evident, that if a force should act on a body, and each sec- 

 tion, perpendicular to the direction of the force, should be similar on each side 

 the plane passing through the direction of the force, and the centre of gravity 

 of the body, that that plane would be the plane of rotation in which the body 

 would both begin and continue its motion. It appears also from what has been 

 proved, that if every section on each side of that plane had not been similar, 

 the plane of rotation would not necessarily have continued the same after the 

 commencement of the motion. Hence all bodies, formed by the revolution of 

 any plane figure, will have the axis about which they were generated, a fixed axis 

 of rotation. To determine however, every other axis of a body about which it 

 would continue to revolve, would be foreign to the subject of this paper. Sup- 

 posing therefore the plane of rotation to continue the same (for in this paper 

 Mr. V. means to confine his inquiries to such cases) imagine all the particles of 

 the body to be referred to that plane orthographically, which supposition not af- 

 fecting the angular motion of the body, the centrifugal force of all the particles, 

 to cause the body to revolve about an axis perpendicular to that plane, will re- 

 main unaltered. Let lmno (fig. 7) be that plane, and suppose a force to act at 

 a in the direction pa lying in the same plane, which produce till it meets ln. 



