20 Mr. A. Cay ley on the Cones which pass through 



that is the perpendicular position, the changes in the inclination 

 of the difiracted ray will be much less rapid, or the indications 

 more crowded together, than near the parallel position (90^), or 

 the plane of diffraction. 



But we can observe the changes only in the plane of polariza- 

 tion. If, then, these changes are more slow near the perpen- 

 dicular (0°), the planes of polarization are parallel to those of 

 vibration; if near the parallel (90^), then they are perpendicular 

 to those of vibration. 



A series of experiments of the most elaborate and accurate 

 kind unequivocally show the latter to be the fact. 



63. Thus experiment obliges us to adopt FresneVs hypothesis of 

 vibrations perpendicular to the plane of polarization^ and by con- 

 sequence (as here shown) either increased density or some other 

 property eocpressed by the same law. 



In thus being compelled to relinquish the hypothesis of equal 

 density, or at least the formulas expressing it, we do not, in fact, 

 sacrifice anything in point of simplicity, the same amount of 

 analysis being requisite to deduce the expressions on either sup- 

 position ; and in giving up the beautiful geometry of Maccullagh, 

 we do ample justice to his more substantial discoveries, — the 

 general laws of equivalence of vibrations (whatever difference 

 may arise on a subordinate point), and their connexion with the 

 principle of conservation of vis viva, on which the whole theory 

 reposes. There are, however, some other points hinted at in 

 what precedes, which may demand further inquiry at a future 

 opportunity. 



II. On the Cones which pass through a given Curve of the Third 

 Order in Space. By A. Cayley, Esq.^ 



THE following investigation is connected with the theory of 

 the cubic (a, b, c, d^jc, y)^, and in particular with a theorem 

 that the determinant formed with the second differential coeffi- 

 cients of the discriminant gives the square of the discriminant. 



Consider the coefficients a, 6, c, d as linear functions of co- 

 ordinates, the equations 



flc-Z>«=0, bc'-ad=0, bd—c^=0 



(equivalent, of course, to two equations) belong to a curve of the 

 third order in space, the edge of regression of the developable 

 surface obtained by putting the discriminant equal to zero, or 

 which has for its equation 



-a«<^ + Gabcd- 4a<^ - 4b^d+ Sb^c^ = 0. 

 And, moreover, the above forms are the general representations 

 * Communicated by the Author. 



