ABSTRACT DYNAMICS 91 



formulae, the simple development of which provides 

 all the equations necessary for the solution of each 

 problem." "No diagrams will be found in this 

 work. The methods that I explain require neither 

 geometrical, nor mechanical, constructions or 

 reasoning, but only algebraical operations in 

 accordance with regular and uniform procedure. 

 Those who love Analysis will see with pleasure that 

 Mechanics has become a branch of it, and will be 

 grateful to me for having thus extended its domain." 

 If ever a work was conceived in the purely abstract 

 mathematical spirit it is surely this one. It is 

 very interesting to trace its effects. Lagrange's 

 idea of reducing the investigation of the motion 

 of a dynamical system to a form dependent upon 

 a single function of the generalized co-ordinates 

 and velocities of the system was further developed 

 by Hamilton and Jacobi into forms in which the 

 equations of motion of the system represent the 

 conditions for a stationary value of an integral 

 of a single function. The extension by Routh 

 and Helmholtz to the case in which "ignored co- 

 ordinates" are taken into account, was a step in 

 the desirable unification which would be attained 

 if the notion of potential energy were removed by 

 interpreting it as dependent upon the kinetic energy 

 of concealed motions included in the dynamical 

 system. The whole scheme of abstract Dynamics 

 thus developed upon the basis of Lagrange's work 

 has been of immense value in Physics, and parti- 



