PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 141 



in which the distance r is determined as a function of the time t and of the elements 

 T, », ^, by the 5th equation (Q^.)^ J^^d in which 



dr X . 

 -irdr 



y/\2{^ + 2 M/(r) - -^^ .^} 



q being still the minimum of r, when the orbit is treated as constant, and being still 

 connected with the elements z, ^m,, by the first equation of condition (186.). In astro- 

 nomical language, M is the sun, m a planet, | ?? ^ are the heliocentric rectangular co- 

 ordinates, r is the radius vector, & the longitude in the orbit, u the longitude of the 

 perihelion, v of the node, & ^ u is the true anomaly, ^ — v the argument of latitude, 

 yij the constant part of the half square of undisturbed heliocentric velocity, diminished 

 in the ratio of the sun's mass (M) to the sum (M -|- m) of masses of sun and planet, 



X is the double of the areal velocity diminished in the same ratio, — is the versed sine 



of the inclination of the orbit, q the perihelion distance, and r the time of perihelion 

 passage. The law of attraction or repulsion is here left undetermined ; for Newton's 

 law, /!/» is the sun's mass divided by the axis major of the orbit taken negatively, and 

 X, is the square root of the semiparameter, multiplied by the sun's mass, and divided 

 by the square root of the sum of the masses of sun and planet. But the varying 

 ellipse or other orbit, which the foregoing formulae require, differs essentially (though 

 little) from that hitherto employed by astronomers : because it gives correctly the 

 heliocentric coordinates, but not the heliocentric components of velocity, without dif- 

 ferentiating the elements in the calculation ; and therefore does not touch, but cuts, 

 (though under a very small angle,) the actual heliocentric orbit, described under the 

 influence of all the disturbing forces. 



38. For it results from the foregoing theory, that if we differentiate the expressions 

 (V^.) for the heliocentric coordinates, without differentiating the elements, and then 

 assign to those new varying elements their values as functions of the time, obtained 

 from the equations (S^.), and deduce the centrobaric components of velocity by the 

 formulae (P.), or by the following: 



then these centrobaric components will be the same functions of the time and of the 

 new varying elements which might be otherwise deduced by elimination from the in- 

 tegrals (Q^.), and will represent rigorously (by the extension given in the theory to 

 those last-mentioned integrals) the components of velocity of the disturbed planet w, 

 relatively to the centre of gravity of the whole solar system. We chose, as more 

 suitable to the general course of our method, that these centrobaric components of 

 velocity should be the auxiliary variables to be combined with the heliocentric co- 

 ordinates, and to have their disturbed values rigorously expressed by the formulae 



