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XIII. — On the Application of Hamilton's Characteristic Function to Special Cases 



of Constraint. By Professor Tait. 



(Read 20th March 1865.) 



1. One of the grandest steps which has ever been made in Dynamical Science 

 is contained in two papers, " On a General Method in Dynamics" contributed to 

 the Philosophical Transactions for 1834 and 1835 by Sir W. R. Hamilton. It is 

 there shown that the complete solution of any kinetical problem, involving the 

 action of a given conservative system of forces, and constraint depending upon 

 the reaction of smooth guiding curves or surfaces, also given, is reducible to the 

 determination of a single quantity called the Characteristic Function of the 

 motion. This quantity is to be found from a partial differential equation of the 

 first order, and second degree ; and it has been shown that, from any complete 

 integral of this equation, all the circumstances of the motion may be deduced by 

 differentiation. So far as I can discover, this method has not been applied to 

 inverse problems, of the nature of the Brachistochrone for instance, where the 

 object aimed at is essentially the determination of the constraint requisite to pro- 

 duce a given result. It is easy to see, however, that a large class of such 

 questions may be treated successfully by a process perfectly analogous to that of 

 Hamilton; though the characteristic function in such cases is not the same 

 function (of the quantities determining the motion) as that of the Method of 

 Varying Action. 



2. It is unnecessary to enter into any great detail with reference to the 

 present subject; because any one who is familiar with Hamilton's beautiful 

 investigations will have no difficulty in applying them, with the requisite slight 

 modifications, to the subject of this paper. I shall therefore content myself 

 with a brief explanation of the application of the method to the problem of the 

 Brachistochrone, and a mere indication of some other curious problems which 

 are easily solved in a similar manner. 



3. The problem of the Brachistochrone for a single particle is. in its simplest 

 form, as follows: — 



Find the form of the {smooth) constraining curve along which a particle will pass, 

 under the action of a given conservative system of forces, from one given point to 

 another in the least possible time, the initial velocity being given. 



The problem may easily be complicated by supposing, for instance, the 

 terminal points not to be definitely assigned, but to lie each on a given surface : 



VOL. XXIV. PART I. 2S 



