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About the centre C, with semi-diameter CG, let the circumference gRG be described, cutting the ray RC at R; and let RV be the perpendicular on CG. Then as the line N is to CG let CV be to CD, and let DI be drawn parallel to CM, cutting the Ellipse gMG at I; then joining CI, this will be the required refraction of the ray RC. Which is demonstrated thus.
But it appears that the said proportion of the refraction is less here than in the section through FEB; for it was there the same as the ratio of N to CG, that is to say, as 156,962 to 98,779, very nearly as 8 to 5; and here it is the ratio of N to CQ the major semi-diameter of the spheroid, that is to say, as 156,962 to 105,032, very nearly as 3 to 2, but just a little less.
And having regard to some other observations and phenomena of which I shall speak afterwards, I put N at 156,962 parts, of which the semi-diameter CG is found to contain 98,779, making this ratio 8 to 5-1/29.
"I am the same," replied the man with the rifle. "Anything I kin do for you?" "Yes; have you seen a piebald cayuse straying around loose-like, or anybody leading one CG being the brand?" "I did; it was straying." "An' which way did it go?" "Into the town pound." "What! Pond! What'n blazes is it doing with a pond? Couldn't it drink without getting in? Where's the pond?" "Right here.
As then, CK is to KO, so also is RC to CV. But KO is equal to N, and RC to CG: then as CK is to N so will CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then as CK is to CG so is CG to CD. And because DI is parallel to CM, the conjugate diameter to CG, it follows that KI touches the Ellipse at I; which remained to be shown.
Further it will be seen that the ray CI in emerging through the opposite surface of the crystal, ought to pass out quite straight, according to the following demonstration, which proves that the reciprocal relation of refraction obtains in this crystal the same as in other transparent bodies; that is to say, that if a ray RC in meeting the surface of the crystal CG is refracted as CI, the ray CI emerging through the opposite parallel surface of the crystal, which I suppose to be IB, will have its refraction IA parallel to the ray RC.
A procession slowly wended its way up the street, led by the marshal, astride a piebald horse bearing the crude brand of the CG. Three men followed him and numerous dogs of several colors, sizes, and ages roamed at will, in a listless, bored way, between the horse and the men.
Prosperity had indeed returned! "Where the CG ranch is I dunno, but I do know where one of their cayuses is," he mused, glancing between two of the corral posts at the sleepy animal. "If I has to auction it off to pay for its keep and the fine, the saddle will bring a good, round sum. I allus knowed that a dollar wasn't enough, nohow."
One sees then that as there is in the refraction of ordinary media a certain constant proportion between the sines of the angles which the incident ray and the refracted ray make with the perpendicular, so here there is such a proportion between CV and CD or IE; that is to say between the Sine of the angle which the incident ray makes with the perpendicular, and the horizontal intercept, in the Ellipse, between the refraction of this ray and the diameter CM. For the ratio of CV to CD is, as has been said, the same as that of N to the semi-diameter CG.
Nat Fisher, punching cows for the CG and tired of his job, leaned comfortably back in his chair in the Paradise and swapped lies with the all-wise bartender. After a while he realized that he was hopelessly outclassed at this diversion and he dug down into his pocket and brought to light some loose silver and regarded it thoughtfully.
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