Prof. Encke on Hadley's Sextant. 85' 



of a correct measurement ; viz. the line of collimation of the 

 telescope must be parallel 

 to the plane of the sextant, 

 and both mirrors must be 

 perpendicular to it. In this 

 supposition, let O A be 

 parallel to the line of col- 

 limation, and p that pole 

 of the plane parallel to the 

 small mirror which answers 

 to the back of it; in the same 

 manner let P be the pole of 

 the plane of the great mir- 

 ror, but the one answering 

 to its reflecting surface. By 

 our assumption. A, p, P 

 are in the great circle of 



the plane of the sextant. In order to find the position of the 

 objects whose angle is measured by this position of the mirrors, 

 we will not follow the path of the ray of light in its real direc- 

 tion from the object to the eye, but in the contrary direction 

 from the eye to the object. It is well known that the path of 

 the ray is the same if we exchange the luminous and the il- 

 luminated points. The direct ray has the direction OA. The 

 doubly reflected one coinciding with it has at first the same 

 direction. In this direction it strikes the small mirror, and 

 is reflected by it to the large mirror. If we make p B on the 

 other side o{ p =■ p K in the great circle of the plane of the 

 sextant, BO will be the direction of the ray after the first re- 

 flexion. In the same manner OC will be its direction after 

 the second reflexion, if we make PC = PB on the other side 

 of P. The objects whose angle is now measured are A and C. 

 This construction immediately shows the law on which this 

 measurement depends ; for as p bisects the arc AB, and P the 



arc B C, we have/J P = ^ A C or the angle of the poles 



of the two mirrors, which is equal to that of the planes of the 

 mirrors themselves, is always one half of the real angle. On 

 the sextant the double angles are accordingly always marked. 



In this manner it is possible to follow the path of the ray 

 after three, four, or any number of reflexions. For greater 

 inmibers, however, the analytical form would be more conve- 

 nient, and the symmetry of the formulaj would likewise be more 

 ))crfect, if throughout the same poles, corresponding eillier to 

 the rcMecting surfaces or to the opposite ones, were made use 

 of, as also eilhcr the points where the ray issues from, or those 

 wlmro it enlers inlo (he surface of the Sj)hcrc. 



Tiie errors to which this measurement is liable, independent 



of 



