306 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



It may also be easily shown, that 



with other analogous equations : the perturbation of the coordinate |j may therefore 

 be thus expressed. 



(Q^.) 



■+"8^'.yo 8/3, '^'" 8/3.yo 8^', "^^ 





. . (R^) 



and the perturbations of the two other coordinates may be expressed in an analogous 

 manner. 



It results from the same principles, that in taking the first differentials of these 

 perturbations (R''.), the integrals may be treated as constant ; and therefore that we 

 may either represent the change of place of the disturbed point in^, in its relative orbit 

 about m^, by altering a little the initial components of velocity without altering the 

 initial position, and then employing the rules of binary systems ; or calculate at once 

 the perturbations of place and of velocity, by employing the same rules, and altering 

 at once the initial position and initial velocity. If we adopt the former of these two 

 methods, we are to employ the expressions (O^.), which may be thus summed up. 





(S'.) 



and if we adopt th^ latter method, we are to make, 



Af3', = 2, .7n,y^ -jj-dt,A^, = - 2,, . m^f^ -J^dt, 



A y, = 2„ . m,J^ ^dt,Ay, = -^„. m,f^ 15_ ^ t. 



The latter was the method of Lagrange : the former is suggested more immediately 

 by the principles of the present essay. 



(T'.) 



