very probable that, while h is moderatel}'- small, the expression for 

 the progression of the apses is true for all values of c up to h. 



" Although the principal object of this paper, as mentioned in 

 the beginning, was to point out how far an apparent rotation of 

 the plane of a pendulum's vibration may depend on causes which 

 would exist if the suspension were perfect, and if the point of 

 suspension were unmoved and the direction of gravity invariable, 

 still it may not be uninteresting to point out liow an effect, in some 

 respects similar, may be produced by a fault in the suspension. If 

 a pendulum be suspended by a wire passing through a hole in a 

 solid plate of metal, the orifice of that hole may be oval. If the 

 wire be part of a thicker rod tapering to the size of the wire, it 

 may taper unequally on different sides. In either case there will 

 be two planes of vibration, at right angles to each other, in which, 

 if the pendulum is vibrating, it will continue to vibrate, and in one 

 of which the time of vibration is greater, and in the other less, than 

 in any other plane ; and, the amplitude of vibration being very 

 small, the coniiplete motion may be found by compounding the vi- 

 brations corresponding to these two planes." 



After investigating the effect of these causes of error, the Astro- 

 nomer Royal arrives at the following conclusion : — " It appears, 

 therefore, that the effect of faulty suspension may be sensibly eli- 

 minated between two experiments in which the azimuths of the 

 first vibration differ by 45° ; and it may be prudent, in making any 

 important experiment, thus to change the commencement-azimuth 

 in successive trials." 



ROYAL SOCIETY. 



[Continued from p. 80.] 



June 19, 1851. — The Earl of Rosse, President, in the Chair. 



The following papers were read : — 



1. " Resoarehes in Symbolical Phj'sics. On the Translation of a 

 Directed Magnitude as Symbolised by a Product. The Principles 

 of Statics established symbolically." By the Rev, M. O'Brien, M.A., 

 late Fellow of Caius College, Cambridge, and Professor of Natural 

 Philosophy and Astronomy in King's College, London. Communi- 

 cated by W. A. Miller, M.D., F.R,S. &c. Received April 10, 1851. 



In this communication the author (starting from the well-known 

 theorem, that two sides of a triangle are equivalent to the third, when 

 direction, as well as magnitude, is taken into account) proposes an 

 elementary step in symbolization whicli consists in representing the 

 Translatio7i of a Directed Magnitude by a Product. Any magnitude 

 which is drawn or points in a particular direction, such as a force, a 

 velocity, a displacement, or any of those geometrical or physical 

 quantities which we exhibit on paper by arroics, he calls a directed 

 magnitude. By the translation of such a magnitude he means the 

 removal of it from one position in space to another tvitkout change 

 of direction. 



U representing any directed magnitude and u any distance, the 

 Phil Mag. S. 4. Vol. 3. No. 9. Aug. 1851 . M 



