466 



while those of the Anthode may be denoted as follows : 



Xi -ZL — v~'^ x\ y, = — v~'^y', z, ■=! — v-'^ z' ; 



where w^ = o;'^ + ?/'^+ z''^. 



He had effected the passage from the theorem respecting 

 hodographic to that respecting anthodic isochronism, by the 

 help of his calculus of quaternions ; but had since been able 

 to prove both theorems by means of certain elementary proper- 

 ties of the circle. 



For a hyperbolic comet, the Anthode is a circular arc, 

 convex to the sun ; for a parabolic comet, the Anthode is a 

 straight line. And for comets of this latter class the theorem 

 of isochronism takes this curiously simple form : " Any two 

 diameters of any one circle (or sphere) in space, are anthodi- 

 cally described in equal times, with reference to any one point, 

 regarded as a common centre of force." By this last theorem, 

 the general problem of determining the time of orbital descrip- 

 tion of a finite arc oi a. parabola, is reduced to that of determin- 

 ing the time of anthodical description of a finite straight line 

 directed to the sun ; and thus it is found that " the interval 

 of time between any two positions of a parabolic comet, divi- 

 ded by the mass of the sun, is equal to the sixth part of the 

 difference of the cubes of the sum and difference of the diago- 

 nals of the parallelogram, constructed with the initial and final 

 vectors of slowness, as two adjacent sides." Another very 

 simple expression for the time of description of a parabolic 

 arc, to which Sir William Hamilton is conducted by his own 

 method, but which he sees to admit of easy proof from known 

 principles (though he does not remember meeting the expres- 

 sion itself), is given by the following formula : 



^ = ^ T tan (0 — tan -' ^ tan i B) ; 



where 61 is the true anomaly, and t is the time from perihelion, 

 while T is the time of describing the first quadrant of true 

 anomaly. 



