Neill. — On the Deflection of the Plumb-line. 



579 



amount of deflection and its direction that a plummet will deviate from 

 the vertical line, or normal, at its point of suspension, in the plane of the 

 meridian, for any length of the suspending cord, due to the spheroidal form 

 of the figure of the earth. 



Statics of the Problem op a Plummet suspended from a Fixed Point. 



Let PEPjEj, fig. 1, represent the meridian ellipse of the earth, 

 PPj the axis of rotation or polar axis, and EE X the equatorial axis. 



We can infer from consideration of the symmetry of the figure that a 

 plumb-line suspended at the Poles or at the Equator does not deviate from 

 the vertical, and that at all other latitudes there is a deviation. 



Let A be a point at the mean level of the sea, the latitude of which is 

 <f>', and suppose p a plummet suspended from the point A. Such a sus- 

 pension can be actually obtained by means of a mine-shaft for small depths, 

 but for the statement of the problem the existence of an opening in the 

 earth need not be considered, for the solution of the problem gives the 

 direction that a shaft takes when sunk by means of the plumb-line. 



Let C be the centre of the earth, F and Fj the foci. 



The forces acting on p are the attraction of the earth and the tension of 

 the suspending cord, the weight of the cord being neglected. 



Through p let a confocal ellipse be described, and let it generate a 

 spheroid by revolving about the axis PP X . It is a well-known theorem in 

 attractions that the potential within a spherical or ellipsoidal shell is 

 constant, and therefore the force acting on p is the attraction of the 

 above-generated spheroid, the surface of which is equipotential (Routh's 

 " Analytical Statics," ii, p. 104). 



A second theorem in attraction is : If there be any resultant force, due 

 to any attracting mass, this force acts along the normal to that equi- 

 potential surface on which the point lies (Pierce's " Newtonian Potential 

 Function," p. 38). 



Since the tension of the cord is the resultant force, it is normal to the 

 confocal ellipse, and therefore bisects the angle FpF x . 



Thus for all positions of p the direction of the resultant is towards the 

 point A, and the angle FpF x is bisected by kp producsd. 



19* 



