OF THE MOTiON OF A SYSTEM. 



205 



loss of force by friction, and by the change of the form 

 of aggregation may sometimes produce almost the same 

 phenomena : but the investigation of such cases scarcely 

 requires to be conducted in a very general manner, or in 

 great detail. It may be of more utility to insert here a 

 geometrical demonstration, subservient to the illustration 

 of the principle of the preservation of impetus or living 

 force, though it might, without impropriety, have been 

 introduced somewhat earlier, since it relates to a single 

 moving point only. 



321. Corollary. " 245.'' Two bodies 

 being attracted towards a given centre, with 

 equal forces at equal distances, if their velo- 

 cities be once equal at equal distances, they 

 will always remain equal at equal distances, 

 w^iatever their direction may be. 



Let one of the bodies descend in the A 

 right line AB towards C, and let the other 

 describe the curve AD, and let the velocities ^ 

 at B and D be equal; let DE, in the tangent ^ 

 of AD, be the space which would be de- 

 scribed in an evanescent portion of time by 

 the velocity at D, FG the arc of a circle of 

 which the centre is C, and GE its tangent ; 

 and while BF would be described by the 

 velocity at B, let FH be added to it by the 

 attractive force; draw the arc HI and its ^ 

 tangent IK, and EL parallel to DC, and KL perpendi- 

 cular to DK, then DG : DE::GI : EK::EK : EL, by 

 similar triangles ; therefore GI is to EL m the duplicate 

 ratio of DG to DE, or as the square of DG to the 



