PROBLEM II. In an oblique-angled spherical triangle given two sides to determine the variation produced in the third side by a small variation of the opposite angle. Let a, b be the two given sides, c the included angle, and c the opposite side. Then cos c = cos a cos b+ sin a sin b cos c cos (c+dc): = cos a cos b+ sin a sin b cos (c + dc); .. by subtraction, that is, cos (c + dc) — cos c = sin a sin b {cos (c + ¿c) — cos c}; 2 sin (cdc) sin dc = 2 sin a sin b sin (c+dc) sin do and dc is therefore the least possible when sin c is the least possible, that is, when C= 0. To find the expression for dc, in this case, restore what has been rejected, and we shall have In an oblique angled spherical triangle given two sides and the included angle, to find the variation in one of the opposite angles corresponding to a small variation in the included angle. Let a, b be the given side, c the included angle, to find what influence a small variation in the value of c will have on a opposite a. Substitute the expression for cos c above, in the corresponding expression for cos a, and 1 sin2 b for cos b, there results cot (a + da) sin (c+dc) — cot a sin c = cos b {cos c-cos The first member of this equation is equal to cot (A +da) { sin (c+dc) — sin c} + sin c{cot (c + ¿A) —cot ▲}; and the quantities within the brackets are respectively equal to (1) PART III. WHEN a ship sails from any known place, and a correct account is kept of her various directions, and rates of sailing, her situation at any time may be determined by the rules of Plane Trigonometry. The processes employed for this purpose constitute what is called Navigation. But, owing to the imperfection of the instruments with which a ship's course and the distance sailed are observed, it would be unsafe, after a long passage, to compute the place of the ship from the dead reckoning, as the observed direction and distance are called. In such cases recourse must be had to astronomical observations, from which the place of the ship or its latitude and longitude are computed by the rules of Spherical Trigonometry. The problem then becomes one of Nautical Astronomy. We shall treat successively of each of these important branches. NAVIGATION. DEFINITIONS. 96. 1. For the purposes of Navigation the earth may be considered as spherical. It revolves about one of its diameters, called its axis, in twenty-four hours. This rotation is from west to east, causing the heavenly bodies to have an apparent motion from east to west. 2. The great circle, whose poles are the extremities of the axis, is called the equator. The poles of the equator are called also the poles of the earth; the one being the north pole, and the other the south pole. 3. Great circles passing through the poles are called meridians. Through every place on the surface of the earth such a great circle may be drawn, and will be the meridian of the place. The meridian from which the meridians of other places are estimated is called a first meridian. The English have fixed upon the meridian of Greenwich Observatory for the first meridian, which has also been adopted in this country. |