r .. y (cos a + √=I sin a) = 2 √ =I √/ { − {1⁄2 + √ (Art. 159.) Similarly, y (cos a r - √=I sin a) = − 2 √ = √ {− 3 − Therefore, by subtracting and dividing by 2 = 1 2 31 321. COR. To find the numerical value of this quantity, firstly, suppose q positive. Thus is found the numerical value of one root of this cubic p2 equation. In the second case, must be greater than 4 93 27' for, otherwise the sine would be greater than the radius. Therefore, the remaining two roots are imaginary. (Wood, 331.) Thus the nth root by a binomial, consisting of a real and an imaginary quantity, may be found by trigonometrical tables. 323. PROB. If A and B be two positions, of which the distance can be measured; and C and D two objects visible from A and B, but inaccessible. To show how to find the distance of C and D. C D B At A take the angles, BAC= A, BAD = A'. When the distance of the objects from the stations A and B is very great compared with AB and CD, a, and a, will be nearly of the same magnitude, and B" will be very small. In this case, the subsidiary angle S' must be used. Whenever a, and a, are nearly of the same magnitude, and B" nearly 180°, which will never happen except when B is near the middle point of CD, the angle S must be used. When the four objects are in the same plane, = 324. PROP. In a plane triangle, of which the sides are a, b, c, and a, ß, y, the angles converted into arcs to radius unity, if a and ẞ are very small. For a c. B, a = c. sin a sin y sin a |