330 TRANS. ST, LOUIS ACAD. SCIENCE. 



__ d" I i-sin^y- 



n V I — sin 2 i ' 

 as sin^ r =. ?i^ sin^ z and 



■ 2 ■_ ^' 



Eq. (ii) easily takes the form 



_l_ = I , . . • . (l2> 



fl/-2 -h (^/_^^^>)2 ^ >' 



_ I 



which is the equation of an ellipse whose semi-axes are 



i> -^ _ ^' + ^^^ 



The eccentricity of this ellipse is 



r I 1 r 



^ - r - -T I I d' , d' 



(i3> 



The value of e can only become unity if d' =z oo , d" z=z o, or 

 n =z I. 



In the first case, the transverse axis of the ellipse becomes 

 infinite, the conjugate axis remaining constant. In the second 

 case, the ellipse reduces to the limited straight line 6c, the semi- 



d' 

 axes becoming 6 = o and a = ~=^=^ . where /«2 _ j is the 



y « ' — I ^ 



cotangent of the critical angle. In the third case, where n = i ,. 

 since the axes become a = co and 6 = d" , the apparent and real 

 positions coincide, or D = d" . 



The minimum eccentricity physically possible is , in whichi 



case <a?' = o and a^= b , --, ; or for water, for the limiting 



case, a = 1-512 ^' 



This ellipse could therefore become a circle only for a sub- 

 stance having n ^ zo . 



The ellipsoid of revolution obtained by revolving the above- 

 ellipse about the vertical axis would therefore appear as a hori- 

 zontal disc exactly filling the cone whose angle is twice the criti- 

 cal angle, the diameter of this disc being equal to the transverse- 

 axis of the ellipse. 



