250 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



the first order of the function U. But notwithstanding the elegance and simplicity 

 of this known manner of stating the principal problem of dynamics, the difficulty of 

 solving that problem, or even of expressing its solution, has hitherto appeared insu- 

 perable ; so that only seven intermediate integrals, or integrals of the first order, with 

 as many arbitrary constants, have hitherto been found for these general equations of 

 motion of a system of n points, instead of 3 w intermediate and 3 n final integrals, in- 

 volving ultimately 6 n constants ; nor has any integral been found which does not 

 need to be integrated again. No general solution has been obtained assigning (as a 

 complete solution ought to do) 3 n relations between the n masses w^, mg, . . . m , the 



3 n varying coordinates ^i, 3/1, ;2;i, . . . a? , «/ , « , the varying time t, and the 6 n initial 



data of the problem, namely, the initial coordinates a^, b^, Ci, . . .a , b , c , and their 



initial rates of increase, a\, b\, c\, . . .a! ,b' ,c ; the quantities called here initial 



being those which correspond to the arbitrary origin of time. It is, however, possible 

 (as we shall see) to express these long-sought relations by the partial differential co- 

 efficients of a new central or radical function, to the search and employment of which 

 the difficulty of mathematical dynamics becomes henceforth reduced. 

 2. If we put for abridgement 



T = J2.w(.r'2-|-y2+«'2), (4.) 



so that 2 T denotes, as in the Mecanique Analytique, the whole living force of the 

 system ; (a?', y', z', being here, according to the analogy of our foregoing notation, 

 the rectangular components of velocity of the point m, or the first differential coeffi- 

 cients of its coordinates taken with respect to the time ;) an easy and well known 

 combination of the differential equations of motion, obtained by changing in the for- 

 mula (1.) the variations to the differentials of the coordinates, may be expressed in 

 the following manner, 



dT-dV, (5.) 



and gives, by integration, the celebrated law of living force, under the form 



T = U -f- H (6.) 



In this expression, which is one of the seven known integrals already mentioned, 

 the quantity H is independent of the time, and does not alter in the passage of the 

 points of the system from one set of positions to another. We have, for example, an 

 initial equation of the same form, corresponding to the origin of time, which may 

 be written thus. 



To = Uo + H (7.) 



The quantity H may, however, receive any arbitrary increment whatever, when we 

 pass in thought from a system moving in one way, to the same system moving in 

 another, with the same dynamical relations between the accelerations and positions 

 of its points, but with different initial data; but the increment of H, thus obtained. 



