8 
ON A CLASS OF DYNAMICAL PROBLEMS. 
[225 
Aw, Av, Aw are ~-—u, 
dt at 
^-w, if by 
df; drj d£ 
dt’ dt ’ dt 
we understand the velocities 
of dfjb parallel to the axes, after it has come into connexion with the system; but it 
is to be observed, that considering £, y, £ as the coordinates of the particle d/x which 
is continually coming into connexion with the system, then if the problem were solved 
and £, y, £ given as functions of t (and, when there is more than one particle dfx, of 
dP 
the constant parameters which determine the particular particle), -X , &c., in the sense 
just explained, cannot be obtained by simple differentiation from such values of £, &c.: 
in fact, £, y, £ 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 £, tj, £ of one and the same particle. 
Suppose as usual that x, y, z, and in like manner that £, y, £ are functions of a 
certain number of independent variables 0, </>, &c., and of the constant parameters which 
determine the particular particle dm or d/x, of which x, y, z, or £, y, £ are the coordi 
nates ; parameters, 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 altogether from the final results. The first term, 
2 {© - x ) ' + @ " Y )* + $ ~ z ) *} 
may be reduced in the usual manner to the form 
@80 + (P8(f) + ... 
where, writing as usual 0', <£', &c. for ^^, &c., we have 
(h) = ^ dT_dT dV 
~ dt d& d0 + d0’ 
(b _ddT_ dT dV 
dtdcf)' £ty + ety’ 
(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 0, <£, &c., are introduced into T and V through the limiting 
conditions of the summation or definite integration, besides entering directly into T and 
V in the ordinary manner. 
d dT 
And in forming the differential coefficients -j- , 
dt d0 
dT 
d0’ 
dV 
d0 * 
&c., it is necessary to consider the variables 0, </>, &c., in so far as they enter through 
the limiting conditions as exempt from differentiation, so that the expressions just given 
for ©, &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.