164 PROFESSOR TAIT ON THE APPLICATION OF HAMILTON'S 



(Trans. R.I.A., 1824-32), the path of a ray is assumed to be a straight line in 

 any one medium. Here the velocity depends only upon the direction of the ray, 

 as in homogeneous doubly refracting media, and the problem has no analogy with 

 the conservative case which is treated above. 



24. As an instance of an optical problem I take the following, due I believe to 

 Maxwell* If the refractive index of a medium be such a function of the distance 

 from a given point that the path of any one ray is a circle, the path of every other 

 ray is a circle ; and. all rays diverging from any one point converge accurately in 

 another. Or, in another form, find the relation between the velocity and the 

 distance from the centre of force that the brachistochrone may always be a circle. 



The symmetry shows that our investigations need only involve two dimensions. 

 Taking the centre of force as pole, the equation of a circle is 



r 2 - 2ar cos (6 - &) = f - a 2 , = b 2 suppose. 



Hence 



a = 6 — cos — ^ . 



2ar 



This is obviously the equation before written (3) in the form 



dr 



Hence 



a. 



da ** 



= a6-Jd 



da cos 



2av 



But, if v be the velocity (the reciprocal of the refractive index in the optical 

 problem), 



/dr\ 2 _l_/rfr\ 2 -l 

 \dr) + r 2 \dd) ~ V 2 ' 



Hence 



dT JT a 2 d f\ ^ b 2 -r 2 C b 2 + r 2 



dr~ W v 2 ~ r 2 _ drj da °° S 2ar ~ ~J da r J(4a 2 r 2 - (6 2 -r 2 ) 2 ) ' 



But v is not a function of a, so that we get by differentiation with respect to 

 that quantity 



a 



r 2 b 2 + r 2 ■ 



J\ _ a 2 ~ r V(4a 2 r 2 - (6 2 -r 2 ) 2 ) ' 



This is easily reduced to 



(b 2 + r 2 ) 2 _ (b* + r 2 ) 2 

 ~i(a 2 + b 2 ) - 4p 2 ' 



The condition, that v is a function of r and absolute constants only, thus leads 



* Cambridge and Dublin Math. Journal, IX., p. 9. 



