ON A CLASS OF DYNAMICAL PROBLEMS. 
[From the Proceedings of the Royal Society of London, vol. vm. (1857), pp. 506—511.] 
There are a class of dynamical problems which, so far as I am aware, have not 
been considered in a general manner. The problems referred to (which might be 
designated as continuous-impact problems) are those in which the system is continually 
taking into connexion with itself particles of infinitesimal mass (i.e. of a mass con 
taining the increment of time dt as a factor), so as not itself to undergo any abrupt 
change of velocity, but to subject to abrupt changes of velocity the particles so taken 
into connexion. For instance, a problem of the sort arises when a portion of a heavy 
chain hangs over the edge of a table, the remainder of the chain being coiled or 
heaped up close to the edge of the table; the part hanging over constitutes the 
moving system, and in each element of time dt, the system takes into connexion with 
itself, and sets in motion with a finite velocity, an infinitesimal length ds of the chain; 
in fact, if v be the velocity of the part which hangs over, then the length vdt is set 
in motion with the finite velocity v. The general equation of dynamics applied to the 
case in hand will be 
2 {(al- x ) - 7 ) % + - Z ) «*} dm + 2 (A«s?+A»s, + AwSf) J ( dp = 0, 
where the first term requires no explanation: in the second term £, r/, £ denote the 
coordinates at the time t of the particle d/x which then comes into connexion with 
the system; Au, Aw, Aw are the finite increments of velocity (or, if the particle is 
originally at rest, then the finite velocities) of the particle d/x the instant that it has 
come into connexion with the system ; 8%, 8rj, 8£ are the virtual velocities of the same 
particle d/x 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 
d/x which come into connexion with the system at the time t; of course, if there is 
only a single particle d/x, the summatory sign 2 is to be omitted. The values of