308 Royal Society : — 



where, writiusr as usual 0', (t>\ &c, for — , -^, &c., 

 ° at at 



we have g= — '^-— — + — 



dt d& dd dd' 



^ d dT dT , dY . 

 $== — — + — - , &c., 



dt d<p' d<p df 



(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 d, f, &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 



-, ^, —, —, &c., it is necessary to consider the variables Q, <{>, &c., 

 dt do' dd dd' ' ^ 



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

 from differentiation, so that the expressions just given for 0, *, &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 hue, or 



we have here 2^ = a Zd + bl<p + . . 



dt]=a'hd + b'd(t>+.. 



dC=a"^d + b'd(p + .., 

 where a, b, a', &c. are functions of the variables 6, (j), &c., and of 

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

 The virtual velocities or increments cd, ^<j), &c., are absohitely arbi- 

 trary, and if we replace them hy dO, d<p, &c., the actual increments of 

 0, f, &c., in the interval dt during the motion, then h^, Br], B^ will 



become — dt, — dt, ~ dt, in the sense before attributed to —t 

 dt dt dt dt 



dt) d^ 

 li'di' 



The particle c/ft 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 

 particle d^. "We have thus the means of expressing the second line 

 in the proper form ; arid if we write 



S {a" + a'' + a"^) dn=Adt 



^lb-+b'' + b"-)dix=Bdt 



Z{ab + a'b' + a"b") d^i=Ildt 

 S (au-^a'v + a"tv) rZ^= — Vdt 

 S (bu + b'v+b"iv) dft^ — Qdt, 



