I. 



ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 



[American Journal of Mathematics, vol. n. pp. 49-64, 1879.] 



Formation of a new Indeterminate Formula of Motion by tlie Sub- 

 stitution of the Variations of the Components of Acceleration for 

 the Variations of the Coordinates in the usual Formula. 



The laws of motion are frequently expressed by an equation of the 

 form x + ( F _ m ^ Sy + ( Z - m ^$ z ] = 



in which 



?7i denotes the mass of a particle of the system considered, 

 x, y, z its rectangular coordinates, 

 x, y, z the second differential coefficients of the coordinates with 



respect to the time, 



X, F, Z the components of the forces acting on the particle, 

 Sx, Sy, Sz any arbitrary variations of the coordinates which are 



simultaneously possible, and 

 Z a summation with respect to all the particles of the system. 



It is evident that we may substitute for Sx, Sy, Sz any other 

 expressions which are capable of the same and only of the same sets 

 of simultaneous values. 



Now if the nature of the system is such that certain functions 

 A, B, etc. of the coordinates must be constant, or given functions of 



the time, we have 



_ (dA . , dA s , d A \ A ' 

 2 -y ooj+^7 oy-\ T6Z ) = 0, 

 \dx dy dz J 



f ^.Sx+^^^-^ I (2) 



,dx dy 



etc. 



These are the equations of condition, to which the variations in 

 the general equation of motion (1) are subject. But if A is constant 

 or a determined function of the time, the same must be true of 

 A and A. Now 



f dA , . dA , . dA .\ 



dx" r dy y r dz 



*A A ^ . , dA . , dA .. 



G. II. 



