34 Mr. Mac Cuutacn on the dynamical Theory of 
two radii of constant lengths, equal to OT and OT’ respectively, revolve within 
the surface, the cones B and B’ described by these radii will intersect each other 
at right angles, since they have the same focal lines. And supposing the axis of 
y to be the mean axis of the ellipsoid, so that the nodal diameters lie in the plane 
of xz, the axis of x will lie within one of the cones, as B, and the axis of z within 
the other cone B’. Now the angle contained by the two sides of either cone, 
which lie in the plane of wz, is given by the angles @ and 6’ which the direction 
of the right line OTT’ makes with the nodal diameters; because the angles 
which any side of a cone makes with its focal lines have a constant sum, or a 
constant difference, according to the way in which they are reckoned. But if the 
angles @ and 6’ be reckoned (as they may be) so that their sum shall be equal to 
the angle contained by the two sides of the cone B which are in the plane of az, 
their difference will be equal to the angle contained by the two sides of the cone 
B’ which are in the same plane; the contained angle, in each case, being that 
which is bisected by the axis of x. Therefore, the lengths OT and OT’, which 
we denote by 7 and 7’, are equal to two radii of the ellipse whose equation is 
these radii making with the axis of z the angles $(@+ 6’) and $(@ — @’) respec- 
tively. Hence 
1__sin?}(0-+-6') , cosh(0+ 0) 71 1 nie 1 : 
Fic mama 1s se iy anal ae aa omnes oC "de 
1 _sim*t(0— 0) _, cos't(0— 6’), lela ae 1 : 
Bo ge nt cs eae Ge 
These formule give the two velocities of propagation along a ray which makes 
the angles 0, 6’ with the nodal diameters. Subtracting them, we have 
1 1 ] 1 
—— ee ] 1 to 14 
i a (— =) sin @ sin @ (14) 
All the preceding equations, relative to the wave-surface, may be transferred 
to the index-surface, by changing the quantities a, b, ¢ into their reciprocals. 
For example, if the normal to a wave make the angles @,, 0, with the nodal dia- 
