184 The Rev. S. Haveuton on the Equilibrium and 
singular points of the surface ~ = 0. Applying this principle to the three forms 
of the surface of wave-slowness just given, it will appear that the intersections of 
r—1=0, (P—1)(e—1)—wH=0, 2z=0, 
will be singular points of the surface of wave-slowness in the plane (2, y) 3 also 
qg— 1=0, ‘(P= Dieg—1)—@=0, 7y=0, 
will give singular points in the plane (a, z); and 
p— l=0) (Ce) ge )) er 05 a = 105 
will give singular points in the plane (y, =). 
As the traces in the principal planes consist of a curve of the second and one 
of the fourth degree, we shall have in general eight points of intersection, real or 
imaginary, and therefore the surface of wave-slowness should have twenty-four 
singular points ; but as it is only the rea/ points of intersection that produce any 
effect in the physical problem, it becomes of importance to ascertain the number 
and position of the real singular points. 
I shall first prove that the curve of the fourth degree consists of two ovals, 
lying one inside the other, and not having any point in common. ‘The equation 
of the curve on the plane (2, 7) is 
(aa? + ny? — 1) (ny? + na? — 1) — 4n°*y’? = 0, 
and its polar equation 1s 
5 oie 1 A055 4 1 Sop : 
Acos-a ++ nsin°a — — }( Bsin’a + Nncos’a — — ) — 4n’sin*acos’a = 0, 
Pp p 
which is a quadratic equation with respect to —; the condition necessary for equal 
P 
roots 18 
[(aA + n) cos*a + (B + Nn) sin’a]? 
= 4(Acos’a + nsin*a) (Bsin?a + Neos’) — 16N°sin*acos’a ; 
and, if two radii vectores of the curve coincide, i. e. if the curve have a double 
point, this equation of condition must give a real value for a; arranging it with 
respect to tana, it becomes 
