200 CELESTIAL MECHANICS. I. V. 19. 



§ 19. Of the principle of living force. It is only 

 true where the motions change hy imperceptible degrees. 

 Mode of estimating the alteration of the limng force in 

 the abrupt changes of the motions of a system. P. 51. 



[318. Definition. The product of the 

 mass of any body, into the square of its 

 velocity, is called its impetus or energy. 



319. Theorem. The joint impetus of any 

 system of bodies is equally increased or di- 

 minished by the action of any combination 

 of forces, provided that the initial and final 

 places of the system are the same, whatever 

 may have been the intermediate paths de- 

 scribed by the different bodies.] 



We may derive from the equation (P) of the last pro- 

 position several general principles of motion, which it will 

 be proper to -examine in detail. The variations ^x, ^y, 

 ^z, ^y, , . . , will obviously be subjected to all the condi- 

 tions of the connexion of the system, if they be supposed 

 proportional to the fluxions do;, dy, dz, da;', . . . , which 

 represent the actual motion ; we may, therefore, make 

 this substitution in the equation (P) and it will then 



becomeOrz 2 ^ mdx ( J^— P ) + mdy[--^— Q^ +mdz 



/ddz v7 d^2 + dv2 + dz2 

 l-TT^ R j ^ ; whence we have OziSm -^ 



- 2> {Pdx + Qdy + Rdz) and Xm ^^^±^1+J^ 



:=zC-h2lmf(Pdx'\'Qdy-\-Rdz), C being a constant 

 quantity. (Q) 



