507 



are the coordinates at the time t of the particle dp which then comes 

 into connexion with the system ; AM, At>, Aw are the finite increments 

 of velocity (or, if the particle is originally at rest, then the finite 

 velocities) of the particle d\i the instant that it has come into con- 

 nexion with the system ; 8, Sij, are the virtual velocities of the 

 same particle dp considered as having come into connexion with and 

 forming part of the system. The summation extends to the several 

 particles or to the system of particles dp which come into connexion 

 with the system at the time t ; of course, if there is only a single 

 particle dft, the summatory sign S is to be omitted. The values of 



AM, At>, AM; are ^ -M, ^ -v, ** -w, if by ^, ^, ^ we under- 

 dt dt dt 3 dt dt dt 



stand the velocities of dp parallel to the axes, after it has come into 

 connexion with the system ; but it is to be observed, that considering 

 , i), as the coordinates of the particle dp, which is continually 

 coming into connexion with the system, then if the problem were 

 solved and , ?;, given as functions of t (and, when there is more 

 than one particle dp, of the constant parameters which determine 



the particular particle), , &c., in the sense just explained, cannot 



be obtained by simple differentiation from such values of , &c. : in 

 fact, , 77, so given as functions of t, belong at the time t to one 

 particle, and at the time t-\-dt to the next particle, but what is 

 wanted is the increment in the interval dt of the coordinates , 17, 

 of one and the same particle. 



Suppose as usual that x, y, z, and in like manner that , T;, are 

 functions of a certain number of independent variables 0, 0, &c., and 

 of the constant parameters which determine the particular particle 

 dm or dp, of which x, y, z, or , T?, are the coordinates, para- 

 meters, that is, which vary from one particle to another, but which 

 are constant during the motion for one and the same particle. The 

 summations are in fact of the nature of definite integrations in 

 regard to these constant parameters, which therefore disappear alto- 

 gether from the final results. The first line, 



may be reduced in the usual manner to the form 

 630 + *<ty + ____ 



VOL. VIII. 2 P 



