FRESNEL ON DOUBLE REFRACTION. 315 



pression which we shall put equal to —, so that the variable {t) 



may represent unity divided by the square of the radius vector. 

 We obtain thus the polar equation of the ellipsoid 



of which Petit has made so elegant an application to the general 

 discussion of surfaces of the second degree. 



To express that the particular radius vector we are considering 

 is contained in the plane ?/=jo^* + 9'^, we must write l=j9« H-g/S; 

 an equation which being differentiated with respect to («) and 

 (/3) gives 



^ = - ^. 

 da. q' 



If we differentiate in the same way the polar equation of the 



elUpsoid, considering (/3) and [t) as functions of («), we have 



d & p 



or, putting for -r^ the above value — — , 



2qfci- 22ih^ - 2tqoL + 2tp^ = (l+a^ + ^S) __ . 



whence we get 



dt _ 2 qfx — 2ph(B - 2iqot + 2tp fi 



d^ ~ l+a2 + /32 



When the radius vector attains its maximum or its minimum, 

 (/) is at its maximum or minimum, and consequently -r- = ; 



therefore 



2qfa.- 2ph(i-2tqu + 2tp^ = 0, 

 or 



ocq[t-f)-^p{t-h)=0. 



If we join to this relation the equation of condition, 



pci. + q^ = l, 

 which expresses that the radius vector is contained in the plane 

 of the elliptical section, we obtain the following values of («) 

 and (/3), corresponding to the maximum and minimum values 

 of the radius vector, 



« - y (^ - ^) a _ Q{t-f) 



p^{t- h) + q-' [t -/)' ^ ~ p^ {t-h)+ q^ {t -/)• 



