228 UNDULATOEY THEOEY OF LIGHT. 



indeterminate. "We have seen, however, that the direction is, in this case, fixed 

 by other considerations ; and it is furthermore demonstrable, that, as the point 

 of tangency approaches Q'", the line joining it with the foot of its correspond- 

 ing normal approaches perpendicularity to the principal section ; and that, in 

 'die limit, when the two points unite, the perpendicularity becomes absolute. 



In the discussion of the tangent plane AD, or DB, drawn parallel to one of 

 the circular sections of the surface of elasticity. Sir William Hamilton made 

 the remarkable discovery that the tangency is not confined to the points P and 

 Q in the principal section ; but that it extends throughout the circumference of 

 a minute closed curve, sensibly circular, of Avhich P and Q are only two points 

 of the circumference. The point W is, therefore, the vertex of a conoklal or 



umbilical depression ; and all the points of 

 the circumference of the circle of contact are 

 equally points in the wave front to which CQ"' 

 is normal, and which is parallel to the same 

 circular section of the surface of elasticity to 

 which the tangent plane is parallel. The 

 annexed figure represents this little circle, 

 p.^ g;; As, in this, CQ is the normal to the circular 



°' '^' section of the surface of elasticity, and CN* 



is the optic axis, we have — 



tanO CX=tana'= j- , / — ^,, and tanN0X=tan;3= ± - a / -^^>- 



c 



"Whence tana'=: -|- — tan^^*, [^^-^ 



In anhydrous sulphate of lime (anhydrite) in which the doubly refracting 

 power is uncommonly great, the ratio of c to a is .9725 to 1. The value of /3 

 is 14° 3.y, from which we deduce «'=13° 41' 11". And /?— a'=0° 22' 19". 



A general expression for the value of /? — a' may be found thus : 



tan/5— tana'=( 1 | tana'= tana' 



\c ; c . 



sin/j sina' a — c siua' 



Or 



cos/9 cosa' 



• n , r. • , <^ — c sma' , ^ a — c . , - 



smpcosa' — cos/Jsma'= ^,cosa'cos/3= — — sma'cosp 



c cosct' c 



(t — c 

 And sin (/?—«')= sina'cos/3, [^^-j 



In so far as the variation dependent on the trigonometrical function sina'cos/? 

 is concerned, we may easily determine the outside limit. For, since a' is less 

 than /?, sina'<sin/5'; and sina'cos/3<sin/5cos/3. But when s'mficosjS is at its 

 maximum, cos^;5;=sin^;9=sin^45^ = ^. Therefore, siuy'Jcos/^ = ^ also. And 

 sina'cos/5 is always less than ^. Hence, the sine of the angle between the 

 optic axes and the normals to the corresponding cu'cular sections is always less 

 than half the difference between the greatest and least axes of elasticity, divided 

 by the least axis. 



Inasmuch as all the points of the little circle QyP/^ are in the tangent 

 plane, it follows thatj if a ray should be incident upon a crystal in such a 

 manner as that CQ should be its direction for one nappe and CP for the other, 

 neither the ray CQ nor the ray CP would be confined to the point Q or P, but 

 both would spread themselves along the circumference Q^P/^. until, by blending 

 together, they should form a hollow cone. And as, at the umbilical point, the 



* The letter N should stand at the centre of the conoidal depression in the figui'e. 



