242 Royal Astronomical Society. 



since the heliocentric velocity of the earth =v, we have 



n=*/r(Vr+l) and r=\(\ + 1 n — */l +4n). 



Using this formula, Mr. Chevallier deduces from the Durham ob- 

 servations, May 6 and 10, a mean radius vector which agrees very 

 nearly with that computed by Mr. Graham. 



On a formula for reducing Observations in Azimuth of Circum- 

 polar Stars near Elongation, to the Azimuth at the greatest Elon- 

 gation. By Captain Shortrede. 



In trigonometrical surveys the direction of a chain of triangles is 

 often deduced from the observed azimuth of a circumpolar star at its 

 greatest elongation. The method is convenient, as an accurate 

 knowledge of the time is not requisite, and the latitude is generally 

 sufficiently well-known. It is indeed necessary to have the polar 

 distance of the star with the utmost precision. In the northern 

 hemisphere Polaris and 8 Ursse Minoris are usually employed. 



When the time is known with tolerable certainty, it is much more 

 satisfactory to observe the star frequently, both before and after its 

 greatest elongation ; and Captain Shortrede in this memoir gives a 

 demonstration of the formula which he has found most convenient 

 for reducing such a series of observations of azimuth to the azimuth 

 at the greatest elongation. 



He deduces an exact expression for the tangent of the difference 

 of each azimuth from the greatest azimuth, in which the only vari- 

 able quantities are the time from the greatest elongation, and the arc 

 joining the position of the star at its greatest elongation with its 

 position at the time of observation. The formula is easy of compu- 

 tation ; and when many observations have been made, would greatly 

 facilitate their reduction. 



Captain Shortrede shows, that in most cases an approximate com- 

 putation of the principal variable, viz. of the secant of the arc above- 

 mentioned, would be sufficient, and that this may be very readily 



vpt" si n 

 effected by a formula involving log — ^ — v . A table of this function 



for all arcs up to 1 h is added to the memoir. 



On a Regulated Time-ball. By Professor Chevallier. 



" The usual method of indicating the time by a ball is by permitting 

 the ball to fall freely, the motion being a little accelerated at first by a 

 spring. It is evident that this method is subject to some uncertainty 

 as to the particular instant of time which is to be observed. There 

 is also some inconvenience arising from the derangement to which 

 the apparatus is liable by the sudden stoppage of the motion of the 

 ponderous ball. 



" It is proposed to remedy these disadvantages by regulating the 

 descent of the ball, so that its motion may be uniform, and causing 

 it to pass through three or five horizontal hoops. The motion may 

 be so regulated that the ball may pass through the distance between 

 one hoop and another in a determinate interval, as about 20 s ; and 



