228 Royal Astronomical Society. 



which, in respect of the existing means of pointing its telescope in a 

 given direction, and reading the divisions, is by no means insensible. 



" Now^, as it is necessary to be able to determine this influence to 

 such a degree of approximation as to be in a situation to judge whe- 

 ther it is possible to adopt any mode of using the instrument by 

 which the results shall be freed from it, I have been led to undertake 

 the solution of the following statical problem : — ' To determine the 

 figure of equilibrium of a circle placed in the vertical plane.' This 

 problem is manifestly considerably complicated. For a circle having 

 m radii, and each pair connected together, not only by the circular 

 rim, but also by a direct connexion, there is (in respect of both cir- 

 cles) an aggregate of 4 m elastic lines, both expansible and flexible, 

 for the determination of which 3 x 4 m = 1 2 m constants are neces- 

 sary ; besides which, 6 m unknown quantities also come into consi- 

 deration ; namely the coordinates of each of the particular points 

 in which two or more of the lines are united, and the directions of 

 the lines with respect to the same ; consequently, three unknown 

 quantities for each of the 2 m particular points. The problem, there- 

 fore, includes 18m unknown quantities; and, in the case of Rep- 

 sold's circles, which have 10 radii, 180. The equations necessary 

 for their determination will be obtained from the condition that the 

 sum of the forces which act not only upon the particular points, but 

 also upon every point of a line, arising from gravity and the con- 

 nexion of the difi^erent parts, shall be in equilibrium. 



" The general solution of this problem, which is limited neither 

 by the assumption of a symmetry of figure, mass, or elasticity, nor 

 by that of the absence of primitive tension between the different 

 parts of the whole, may be reduced, as I find, to the solution of 3 tw 

 linear equations, and, on the supposition of symmetry, to three such 

 equations. But this last supposition very probably corresponds to 

 no actual case. It is also probable that in no case can we obtain a 

 knowledge of the deviations from actual symmetry, which it is ne- 

 cessary to possess, in order to obtain a result which shall give the 

 required influence in numbers. I find the following general propo- 

 sition respecting the law of this influence, which is restricted by no 

 particular supposition : — 



" If the angle between the initial radius of the circle and the ver- 

 tical be denoted by m, the angle between the same radius and a point 

 of the division by v, then the variation of this angle arising from 

 gravity, in the vertical plane of the circle, is expressed by fv . cos u 

 +/'v . sin M, where fv,f v are functions of v, independent of the 

 position of the initial radius, and dependent only on the construction 

 of the circle. 



" This proposition is of considerable importance for practical as- 

 tronomy. "We readily infer from it, that the influence of gravity 

 upon a zenith distance entirely vanishes when the latter is deter- 

 mined by the mean of four observations, namely, observations of the 

 object itself and of its image reflected from a horizontal mirror, re- 

 peated in reversed positions of the axis of the instrument. From the 

 same proposition we also easily deduce the mode of arranging the 



