PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 285 



they subtend also equal angles at the other. Reciprocally, if any plane curve possess 

 this property, when referred to a fixed point in its own plane, whicii may be taken as 

 the origin of polar coordinates r, 0, the curve must satisfy the following equation in 

 mixed differences: 



cotan(^y a4- = (A + 2)^-|:, (115.) 



which may be brought to the following form, 



{TQ + d¥)T = ^^ (116.) 



and therefore gives, by integration, 



^ ~ 1 + £? cos (5 - -Br) ' (117.) 



the curve is, consequently, a conic section, and the fixed point is one of its foci. 



The properties of parabolic are included as limiting cases in those of elliptic mo- 

 tion, and may be deduced from them by making 



H^ = 0, or a= 00; . .(118.) 



and therefore the characteristic function w and the time t, in parabolic as well as in 

 elliptic motion, are functions of the chord and of the sum of the radii. By thus 

 making a infinite in the foregoing expressions, we find, for parabolic motion, the par- 

 tial differential equations 



and in fact the parabolic form of the simplified characteristic function w may easily 

 be shown to be 



w = 2^^{^^T^^^J^^), (U3.) 



r being, as before, the chord, and a the sum of the radii ; while the analogous limit 

 of the expression (S^.), for the time, is 



' = 5-7;{(^+^)*+(^-^)'}= (v'-) 



which latter is a known expression. 



The formulae (K^.) and (L^.), to the comparison of which we have reduced the 

 study of elliptic motion, extend to hyperbolic motion also ; and in any binary system, 

 with Newton's law of attraction, the simplified characteristic function w may be 

 expressed by the definite integral 



i^=y \/^4^^^3^.^r, (W3.) 



•/ _T V (T -I- T 4a ' 



this function w being still connected with the relative action V, by the equation 

 (H3.) ; while the time t, which may always be deduced from this function, by the 

 law of varying action, is represented by this other connected integral, 



MDCCCXXXIV. 2 P 



