204 CELESTIAL MECHANICS. I. V. 19. 



action of the bodies concerned, whenever, in the course of 



1 • /» , . • dx .d?/ _ .^ 



the motion, any of the variations A-rr*^T7, • . • , are tinite : 



and the preceding equation affords a very easy method of 

 determining this diminution. 



At every abrupt variation of the motion of the system, 

 we may conceive the velocity of m to be divided into two 

 portions, the one v, which it retains, theother F, destroyed 

 by the actions of the other bodies [, for, even if the velocity 

 be increased, we have only to suppose that a negative 

 portion of it has been destroyed, in order to justify this 

 expression of Dalembert, which is so often used by 



Laplace] : now the velocity of m being V -r^ , 



before this decomposition, and afterwards 



(dx + Adxy + (dy + Ad2/y + (dz + Adzf . ., 

 ^ _ ^ ,t IS easy to see 



that F2- (^ 57)'+ (a ■^y+ [a ^)^ [since the diagonal 



of a parallelepiped, of which the square is equal to the sum 

 of the squares of its sides, may be divided into two portions 

 of which the squares must be respectively equal to the 

 sums of the squares of the parts of those sides : in fact 

 ± V must be simply equal to the square root of this 

 quantity; since the sum of the squares of the finite dif- 

 ferences of the velocities, in the three orthogonal direc- 

 tions, must necessarily give tlie square of the difference 

 of the actual velocity :] and the preceding equation may be 

 expressed in this form, Imv^zz C—X^XmV^ + 2lfm{Pdx 

 '^Qdy-\-Rdz). 



[Scholium 2. It is very doubtful whether an abrupt 

 change of velocity ever takes place in nature, though the 



