BB Dr Anderson on Curves generated by 



and may be, therefore, said to fall under the form of an equa- 

 tion to a straight line. Thus, in the equation, cos I = cos a 

 sin X sin j8 + cos X cos /3, we have, when X and /3 are constants, 

 and I and a the variables, cos I = M cos a + N, in which 

 sin X sin /3 = M, and cos X cos /S = N. 



In like manner, in the case of a double reflection, when X 

 and jS are constants, we may assume (1 + 2 cos 2 /3), sin X sin 

 /3 = M, and (1 -2 cos 2/3) cos X cos jS—N. We thus have 

 cos ^ = M', cos a + N. Hence these expressions may be re- 

 garded as equations to a straight line of a transcendental 

 order. 



Having prosecuted, to an extent that may be perhaps deemed 

 an abuse of analytical research, the investigation of formulae 

 for determining the angle of incidence in a variety of cases, 

 we shall now proceed to point out, with all brevity, the method 

 of constructing the curves to which the fugitive reflections 

 give birth. To do this, however, at great length, both in the 

 case of double and single reflections, would lead into a wider 

 field of illustration than we should be justified in entering 

 upon, even if the subject possessed a higher degree of import- 

 ance than can be claimed for an inquiry so purely speculative. 

 If we begin with the simplest case, that of a single reflection, 

 and suppose the plane on which the reflected images are pro- 

 jected to be parallel to the plane of rotation, it will be conve- 

 nient still to retain that circle as the primitive, and to repre- 

 sent the difi'erent quantities involved in the problem according 

 to the principles of the stereographic projection of the sphere. 



The two leading circumstances which may be regarded as 

 constituting the elements for the construction of the resulting 

 curves are, in the first place, the magnitude of the angle 

 which the direction of the luminous ray makes with the axis 

 of rotation ; and, secondly^ the inclination of the mirror itself 

 to the plane of rotation. When these are equal, the magni- 

 tude of the principal axis of the generated curve is inversely 

 as the sine of that angle, and that of the greatest ordinate di- 

 rectly as the tangent of half the same angle, or, as it is some- 

 times termed, the semitangent of that angle. 



Let ABCD, the primitive, fig. 3, (Plate II.) represent 



