20 Prof. Poweirs Abstract o/M, Cauchy's 



lation and the time. is established, at least in a general and 

 theoretical point of view. 



Lastly, in the Memoires of the Academy of Sciences, torn. x. 

 1831, p. 293, there is a paper by the same author, entitled 

 " M^moire sur la theorie de la lumiere," read to the Aca- 

 demy 31st May 1830. This contains a more full exposition 

 of the physical application of the theory of waves to the va- 

 rious phaenomena, especially those of polarized light; and 

 exhibits its accordance with the experimental laws of Brewster 

 and Arago, and with the formulae of Fresnel. 



Preliminary Property of Surfaces of the second degree. 



It may be convenient here to premise a brief statement of 

 some points in the general investigation of surfaces of the se- 

 cond degree, which is given by M. Cauchy in the third volume 

 of his Exercices de Mathematiques^ liv. 25, and of which im- 

 portant use is made in the theory of waves. 



Assuming, in the first instance, the equation of the second 

 degree with three variables in its most general form, the 

 author deduces the equations of diametral planes, and shows 

 what conditions give them perpendicular to each other, or prin- 

 cipal planes ; and whether they all three intersect in one point, 

 or whether their intersections lie in a given line, such line be- 

 ing called a central line, and such a point the centre. In this 

 last case the lines of their intersections are the principal dia- 

 meters or geometrical axes of the surface. 



The centre being taken as the origin, the equation is 



lux'^-VMy^-\-^z^-\-2?yz + 2Qzx + 2Rxy = 1 ; 



and if a straight line passing through the origin be inclined 

 to the three axes at angles a, /3, y, its equations being 



cos a cos /3 cos y * 

 whence likewise we have 



cos^ a + cos^ /3 + cos^ y =. \, 

 If also we have the three following values equal, 

 L cos a + R cos ^ + Q cos y 

 cos a ' 



^ R cos a + M cos p + P cos y 



"* cos /3 



_ Q cos a + P cos /3 4- N cos y 

 "" cos y 



