Mr. A. Cayley on a Class of Dynamical Problems. 307 



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 vdf is 

 set in motion with the finite velocity v. The general equation of 

 dynamics applied to the case in hand will be 



S 





dt 



where the first "line requires no explanation, in the second line ^ ??, ^ 

 are the coordinates at the time t of the particle rf/i which then comes 

 into connexion with the system; Au, Ac, Aio are the finite increments 

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

 velocities) of the particle du the instant that it has come into con- 

 nexion with the system ; h^, ht], c'C are the virtual velocities of the 

 same particle djx 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 dfx which come into connexion 

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

 particle d^i, the summatory sign S is to be omitted. The values of 



Am, At>, Aw are — — m, ——v, ~ —w, if by — , — , — we under- 



dt dt dt ^ dt dt dt 



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

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

 s, ri, 4 as the coordinates of the particle dj-i, which is continually 

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

 solved and ^, r), 4 given as functions of t (and, when there is more 

 than one particle d^, of the constant parameters which determine 



d^ 

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



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

 fact, £, r], C 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 ^, -q, ^ 

 of one and the same particle. 



Suppose as usual that x, y, z, and in like manner that ^, j/, ^ are 

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

 of the constant j)arameters which determine the particular particle 

 dm or dyi, of which x, y, z, or I, rj, '( 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 



eJ9 + 4>o0-t- 



X3 



