50 
of this ellipse upon the plane of incidence may, however, be 
easily found by the remark in p. 102 of the present volume ; 
the projecting cylinder is therefore known, and as the ellipse of 
vibration is a section of this cylinder, the question of deter- 
mining the ellipse is reduced to that of determining its plane. 
For this purpose Mr. Mac Cullagh gave the following rule. 
Having constructed the ellipsoid of indices (that whose axes 
are parallel to the axes of elasticity, and inversely proportional 
to the three principal velocities of propagation in the crystal) 
let its two planes of circular section intersect the aforesaid 
cylinder. The curves of intersection will be ellipses, which 
shall be supposed to have a common centre O in the axis of 
the cylinder. Let OP, OP’ be the greater semiaxes of these 
ellipses, and OQ, OQ’ the less semiaxes; the lengths of the 
two former being denoted by p, p’, and the lengths of the two 
latter by g, g’. Join the extremities P, P’ of the greater 
semiaxes, and the extremities Q, Q’ of the less semiaxes; and 
divide each of the right lines PP’, QQ’, in the ratio of 
Vp? —q? to Vp®—q”. Then a plane drawn through the 
centre O and the two points of division will be the plane of 
vibration. In the application of this rule some precautions 
are to be observed, but they need not here be insisted on. 
The foregoing rule was deduced (in the year 1843) from 
the general equations by a peculiar use of imaginary quanti- 
ties, after the author had several times tried in vain to obtain a 
geometrical interpretation of those equations by considerations 
of a more obvious and ordinary kind. This use of imaginaries 
is founded on a remarkable theorem relative to the ellipse, by 
which it appears, that the plane of an ellipse and its species 
(that is, the directions and the ratio of its axes) may be ex- 
pressed by two imaginary constants, just as the direction of a 
right line in space is expressed by two real constants. By 
means of this theorem—which it is unnecessary to repeat, as 
it has been published in the University Calendar (Examina- 
tion Papers of the year 1842, p. lxxxiv.)—we may find such 
