508 



where, writing as usual 0', d.', &c. for , * &c., 



at at 



we have G _d_ dT _ dT dV 



dt dd' dd dd' 

 d dT dT , d\ T p 



(this supposes that Xdx+ Ydy + Zdz is an exact differential) ; only it 

 is to be observed that in the problems in hand, the mass of the system 

 is variable, or what is the same thing, the variables 6, <j>, &c., are intro- 

 duced into T and V through the limiting conditions of the summa- 

 tion or definite integration, besides entering directly into T and V in 

 the ordinary manner. And in forming the differential coefficients 



fL d F, ^Y, & c ., it is necessary to consider the variables e, 0, &c., 

 dt dd dd do 



in so far as they enter through the limiting conditions as exempt 

 from differentiation, so that the expressions just given for 9, *, &c., 

 are, in the case in hand, rather conventional representations than 

 actual analytical values ; this will be made clearer in the sequel by 

 the consideration of the before-mentioned particular problem. 

 Considering next the second line, or 



we have here 



where a, b, a', &c., are functions of the variables 6, $, &c., and of 

 the constant parameters which determine the particular particle d/.i. 

 The virtual velocities or increments 30, 20, &c., are absolutely arbi- 

 trary, and if we replace them by dd, df, &c., the actual increments of 

 0, ^, &c., in the interval dt during the motion, then , >?, $ will 



become ^ dt, ^L dt, ^f dt, in the sense before attributed to ^-, 

 at at at at 



dr, dt 

 dt' dt' 



The particle dp. will contain dt as a factor, and the other factor 

 will contain the differentials, or as the case may be, products of dif- 

 ferentials of the constant parameters which determine the particular 



