280 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



stants hy H^, with the extreme lengths of the radius vector r, and with the angle ^ 

 described by this radius in revolving from its initial to its final direction, is the equa- 

 tion of the plane relative orbit ; and the other equation of condition (T^.), connecting 

 the same two constants with the same extreme distances and with the time, gives 

 the law of the velocity of mutual approach or recess. 



We may remark that the part V^ of the whole characteristic function V, which 

 represents the relative action and determines the relative motion in the system, 

 namely, 



may be put, by (I^.), under the form 



^•=^:\j:A^-^'S^r, (w^.) 



or finally, by (79.), 



y^^^^ »,,^,/(r) + H, ^^, (X2.) 



the condition (I^.) may also itself be transformed, by (79.), as follows : 



^ = *X^^- • • • ■ (Y^-) 



results which all admit of easy verifications. The partial differential equations con- 

 nected with the law of relative living force, which the characteristic function V, of 

 relative motion must satisfy, may be put under the following forms : 



I 



AVA2 1 AVA2_ 2;;.,^ 1 



\lr) "*"r2\8d/ — m^ + m^y^ ^ '^ih 



/^Z/V . L PJL\' - Ab^ m -1- H V ! 



r 



{ZK) 



and if the first of the equations of this pair have its variation taken with respect to r 

 and S^, attention being paid to the dynamical meanings of the coefficients of the cha- 

 racteristic function, it will conduct (as in former instances) to the known differential 

 equations of motion of the second order. 



On the undisturbed Motion of a Planet or Comet about the Sun : Dependence of the 

 Characteristic Function of such Motion, on the chord and the sum of the Radii. 



15. To particularize still further, let 



/W = T. , • • • (101.) 



that is, let us consider a binary system, such as a planet or comet and the sun, with 

 the Newtonian law of attraction ; and let us put, for abridgement, 



;7Ji-i-m2 = ^, -=;?, -2-^7 = a (102.) 



