Page:Somerville Mechanism of the heavens.djvu/106

Fermat extended the same principle to the refraction of light supposing the velocity of a ray of light to be less in the denser medium, he found that the ratio of the sine of the angle of incidence to that of the angle of refraction, is constant and greater than unity. Newton however proved by the attraction of the denser medium on the ray of light, that in the corpuscular hypothesis its velocity is greater in that medium than in the rarer, which induced Maupertuis to apply the theory of maxima and minima to this problem. If IS, a ray of light moving in a rare medium, fall obliquely on CD the surface of a medium that is more dense, it moves uniformly from I to S; but at the point S both its direction and velocity are changed, so that at the instant of its passage from one to the other, it describes an indefinitely small curve, which may be omitted without sensible error: hence the whole trajectory of the light is ISR; but IS and SR are described with different velocities; and if these velocities be v and v', then the variation of ${\text{IS}}\times v+{\text{SR}}\times v'$ must be zero, in order that the trajectory may be a minimum: hence the general expression $\delta \int vds=0$ becomes in this case $\delta ({\text{IS}}\times v+{\text{SR}}\times v')=0$, when applied to the refraction of light; from whence it is easily found, by the ordinary analysis of maxima and minima, that $v\sin {\text{ISA}}=v'\sin {\text{RBS}}$. As the ratio of these sines depends on the ratio of the velocities, it is constant for the transition out of any one medium into another, but varies with the media, on account of the velocity of light being different in different media. If the denser medium be a crystallized diaphanous substance, the velocity of light in it will depend on the direction of the luminous ray; it is constant for any one ray, but variable from one ray to another. Double refraction, as in Iceland spar and in crystallized bodies, arises from the different velocities of the rays; in these substances two images are seen instead of one. Huygens first gave a distinct account of this phenomenon, which has since, been investigated by others.

 

Motion of a Particle on a curved Surface.

81. The motion of a particle, when constrained to move on a curve or surface, is easily determined from equation (7); for if the