256 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



we took its variation with respect to the final coordinates, and so obtained results 

 which agreed with the known equations of motion, and which may be thus collected. 



8 1 r/8V\2 /8Vx2 /8Vx2-> _5U_ 



(M.) 



The same relation (F.), by being varied with respect to the quantity H, conducts 

 to the expression 



J_2 -LU—\' 



+ ©• 



+ 



cm - 



(N.) 



and this, when developed, agrees with the equation (K.), which is a new verification 

 of the consistence of our foregoing results. Nor would it have been much more dif- 

 ficult, by the help of the foregoing principles, to have integrated directly our integrals 

 of the first order, and so to have deduced in a different way our final integral system. 



6. It may be considered as still another verification of our own general integral 

 equations, to show that they include not only the known law of living force, or the 

 integral expressing that law, but also the six other known integrals of the first order, 

 which contain the law of motion of the centre of gravity, and the law of description 

 of areas. For this purpose, it is only necessary to observe that it evidently follows 

 from the conception of our characteristic function V, that this function depends on 

 the initial and final positions of the attracting or repelling points of a system, not 

 as referred to any foreign standard, but only as compared with one another ; and 

 therefore that this function will not vary, if without making any real change in either 

 initial or final configuration, or in the relation of these to each other, we alter at once 

 all the initial and all the final positions of the points of the system, by any common 

 motion, whether of translation or of rotation. Now by considering three coordinate 

 translations, we obtain the three following partial differential equations of the first 

 order, which the function V must satisfy. 



(O.) 



and by considering three coordinate rotations, we obtain these three other relations 

 between the partial differential coefficients of the same order of the same charac- 

 teristic function, 



