PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 253 



other principle which has been introduced in the present paper, under the name of 

 the law of varying action, in which we pass from an actual motion to another motion 

 dynamically possible, by varying the extreme positions of the system, and (in general) 

 the quantity H, and which serves to express, by means of a single function, not the 

 mere differential equations of motion, but their intermediate and their final integrals. 



V^erifications of the foregoing Integrals. 



4. A verification, which ought not to be neglected, and at the same time an illus- 

 tration of this new principle, may be obtained by deducing the known differential 

 equations of motion from our system of intermediate integrals, and by showing the 

 consistence of these again with our final integral system. As preliminary to such veri- 

 fication, it is useful to observe that the final equation (6.) of living force, when com- 

 bined with the system (C), takes this new form, 



^^•^{(i^y+(iir+(m=^+«^ ^^-^ 



and that the initial equation (7.) of living force becomes by (D.) 



2 



^4{(iiy+(f^y+(m=u«+H <«•) 



Tliese two partial differential equations, initial and final, of the first order and the 

 second degree, must both be identically satisfied by the characteristic function V: they 

 furnish (as we shall find) the principal means of discovering the form of that function, 

 and are of essential importance in its theory. If the form of this function were known, 

 we might eliminate 3 /i — 1 of the 3 n initial coordinates between the 3 n equations 

 (C.) ; and although we cannot yet perform the actual process of this elimination, we 

 are entitled to assert that it would remove along with the others the remaining initial 

 coordinate, and would conduct to the equation (6.) of final living force, which might 

 then be transformed into the equation (F.). In like manner we may conclude that 

 all the 3 n final coordinates could be eliminated together from the 3 n equations (D.), 

 and that the result would be the initial equation (7-) of living force, or the transformed 

 equation (G.). We may therefore consider the law of living force, which assisted 

 us in discovering the properties of our characteristic function V, as included recipro- 

 cally in those properties, and as resulting by elimination, in every particular case, 

 from the systems (C.) and (D.) ; and in treating of either of these systems, or in con- 

 ducting any other dynamical investigation by the method of this characteristic func- 

 tion, we are at liberty to employ the partial differential equations (F.) and (G.), which 

 that function must necessarily satisfy. 



It will now be easy to deduce, as we proposed, the known equations of motion (3.) 

 of the second order, by differentiation and elimination of constants, from our interme- 



MDCCCXXXIV. 2 L 



