128 Astronomical and Nautical Collections, 



magnitude and the situation of one of them, one might not destroy 

 the deviatory action of the whole system, with respect to a hori- 

 zontal compass placed at that point ; setting aside its reaction on 

 the different bodies of the system. 



The formulae of my memoir show immediately that this action 

 can never vanish for all the directions of the magnetizing force ; 

 consequently the duration of the oscillations of the needle will al- 

 ways be affected, even when there is no alteration in its direction. 

 In order that the horizontal compass may be free from deviation, 

 it is sufficient that the horizontal results of the action of the earth, 

 and of the action of the system of magnetized bodies, should coin- 

 cide with each other for all directions of the terrestrial magnetism. 

 Now we find that this coincidence is only possible when a certain 

 quantity, depending on the magnitude of the given spheres, on their 

 mutual differences, and on their distances from the compass, and from 

 the horizontal plane which contains it, is either positive or evanes- 

 cent, and when another quantity, depending on the same elements, 

 vanishes ; and reciprocally, when these two conditions are fulfilled, 

 we may produce tlie required coincidence, by adding a single 

 sphere to the given system. Its magnitude, and the place of its 

 centre, may still remain undetermined ; but the direction of one or 

 more right lines drawn through the centre of the compass, on which 

 it must be placed, will be given ; and its distance from the needle 

 will depend on the diameter assigned to the sphere, and will be 

 proportional to it. This latitude of the solution of the problem . 

 depends on the identity of the action of a sphere on a given point, 

 while its centre moves on a line passing through the point, and at 

 the same time its diameter increases or diminishes in the same 

 proportion as the distance from the point. If the given system of 

 spheres consists of a single one only, its centre, and that of the 

 added sphere, must be situated in the horizontal plane which con- 

 tains the needle, and the right lines drawn from the two centres, to 

 its middle point, must be at right angles to each other ; and it will 

 be necessary that the lengths of the lines should be proportional 

 to the diameters of the spheres; in this case the compass will con- 

 stantly retain its natural direction, when the system of the two 



