On Total Reflexion. 251 



divide each of the right lines PP', QQ', in the ratio of 



- q z to vV 2 - q' 2 . Then a plane drawn through the 

 centre 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* we may find 

 such properties of elliptical vibrations as are analogous to those 

 of rectilinear vibrations ; and it was in this way that the above 

 rule was discovered. It is analogous (though it scarcely appears 

 so at first sight) to the rule by which, in the theory of Fresnel, 

 the direction of rectilinear vibrations is determined, when the 

 plane of the wave is given. 



* "Examination Papers" of the year 1842, p. Ixxxiv. [The theorem is as 

 follows : 



Given an ellipse in space, the origin of co-ordinates being taken at the centre. 

 Let the points #, y, z, x", y', a 7 , be the extremities of conjugate diameter. Then 

 the imaginary quantities 



y + y'v/^i 2 + z'y/^ 



x + x'^/^l x + z'v/- 

 are constant for every system of conjugate diameters.] 



