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XLI. On an Extension of the Dynamical Principle of Least 

 Action. By James H. Cotterill, St. John's College, Cam- 

 bridge *. 



WHEN a material body, or system of bodies, is exposed to 

 the action of force, the points of application of that 

 force move to a greater or less degree, and the motion of these 

 points of application, by reason of the physical connexion of the 

 points of the system, causes a general relative displacement of 

 every one of them, thereby calling into play forces which in- 

 crease with the extent of that displacement, and at length be- 

 come sufficiently great to balance the applied forces ; the dis- 

 placement then ceases, and a state of equilibrium is attained. 



Thus in every case of the balance of forces by the resistance 

 of matter, a certain amount of energy is expended by the ap- 

 plied forces, and a certain amount of work done in partially 

 overcoming the resisting forces ; and it is a well-known case of 

 the general law of the conservation of energy, that the energy so 

 expended is equal to the work so done. 



But further, if any portion of the system be considered apart 

 from the rest, the forces generated at the points of junction 

 must form a system of forces in equilibrium, and the energy 

 expended by them considered as applied to the detached portion 

 must be equal to the work done in that detached portion ; and 

 these are general conditions to which the forces generated at 

 every point of the system are subject. Now Mr. Moseley has 

 shown that if any number of pressures are in equilibrium, some 

 of which are resistances, then each of these resistances is a 

 minimum, subject to the conditions imposed by the equilibrium 

 of the whole — a principle which he has called the principle of 

 Least Resistance ; let us assume this principle, and let us further 

 suppose, for the present, that it is generalized so as to include 

 the case of the resisting forces generated as above described ; 

 then each of those resisting forces is a minimum, subject to the 

 general conditions stated above : and further, the relative dis- 

 placements which are the cause of those forces must also be 

 the least possible, and the work done the least possible. Thus 

 in the assumption mentioned, to which I shall return in the 

 sequel, it appears that the work done is a minimum, subject to 

 the law of conservation of energy and the statical conditions of 

 equilibrium; and this principle, analogous to the dynamical 

 principle of Least Action, it is the object of this article to con- 

 sider and apply. 



If the work done be expressed in terms of the resisting force 

 at all the points of the system, or some of them, then, the law of 



* Communicated by the Author. 

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