466 
while those of the Anthode may be denoted as follows : 
Ca UO r* gg = "es eS Oe ee 
where v? = a’? + y/?+ 2", 
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 are, 
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 are of 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: 
é= 37 tan (06 —tan— dtan1 @); 
where @ is the trué anomaly, and ¢ is the time from perihelion, 
while r is the time of describing the first quadrant of true 
anomaly. 
