VOL. XXXVII.] PHILOSOPHICAL TRANSACTIONS. 563 



about 60 tons burden, the smallness of which made it also more liable to be 

 lifted up and let down again by the waves : and if the difference of height oc- 

 casioned by that means was about 4 or 5 feet, as we judged it to be, it must 

 necessarily sink and raise the visible horizon by turns near one minute. The 

 computations of the sun's altitudes are all made for the latitude of 51° 28'; 

 whereas a good part of them were taken under sail, and on different tacks, the 

 vessel sometimes standing north-east or north, and at other times south-east, 

 for near a quarter of an hour at a time. 



Several of these circumstances inay probably have contributed to increase the 

 inconsistency of the observations; but as no particular notice was taken of them 

 at the time, they are here barely mentioned. 



Postscript. — The principle on which the contrivance of this instrument de- 

 pends, was laid down in the beforementioned Philos. Trans. N° 420, in one 

 proposition, and several corollaries, the 3th of which contains the grounds of 

 an approximation for correcting some small errors, which will arise if the plane 

 of the instrument be suffered to vary too much from the great circle passing 

 through the two objects, when the observation is taken. There appears reason 

 to think, that there will be very little occasion in practice for that correction ; 

 but it was necessary to mention it, in order to explain the nature of the instru- 

 ment ; and as the manner of deducing that corollary from the proposition may 

 not appear obvious to every reader, Mr. Hadley has here annexed the demon- 

 stration of it. 



Let DBG, fig. 1, pi. 15, represent an infinite sphere, at whose centre Rare 

 placed the two specula inclined to each other in any given angle ; and let their 

 common section coincide with the diameter org. Let ban be the circumfer- 

 ence of a great circle, to the plane of which the common section of the spe- 

 cula orc is perpendicular, and er its radius : let ban be the circumference of a 

 circle parallel to ban, and at the distance from it Bb : draw bo the sine, and br 

 the sine complement of the arch sb : bd is the versed sine of the same. Let a 

 be a point of an object placed in the circumference of the great circle ban, and 

 N the point in which its image is formed by the two successive reflections, as 

 before described; and let a be a point of another object placed any where in the 

 circumference of the parallel ban, and n its image; and let ahn be an arch of a 

 great circle passing through the points a and n. The point a is at the same 

 distance from the great circle ban, as the point b, i. e. at the distance nb. Draw 

 AR, an, rn, ar, an, rn, aR and ur. 



By corol. 4, the figures arn and arn are similar; consequently the line an is 

 to the line an, as ar or br, is to ar or br, i. e. as the radius is to the sine com- 

 4 c 2 



