il|48 Tloyal Astronomical Society. 



" This theorem, however, fails totally when the minor axis of 

 the oval is small. It is then found that the velocity of progress of 

 the apses is nearly proportional to tlic minor axis. But, although 

 the movement of the j)endulum in this case may be defined to any 

 degree of accuracy by infinite series, it does not appear that it can 

 be expressed in finite terms of any ordinary function of the time. 

 This is to be expected, inasmuch as, when the problem is reduced 

 to its utmost state of simplicity by making the minor axis = 0, the 

 motion of the pendulum can be expressed only by series. The, 

 utmost, therefore, for which we can hope is, to determine the ge- 

 neral form of the curve and the rate of progress of its apses, on the 

 supposition that the minor axis is small, in series proceeding by 

 powers of the major axis. This might be so extended as to include 

 higher powers of the minor axis, if it were judged desirable. 



" I have thought that an exhibition of the first steps of solution 

 (carried so far as to include the principal multiplier of the first 

 power of the minor axis) might be acceptable to this Society, not 

 purely as a mechanical problem, but more particularly because it 

 bears upon every astronomical or cosmical experiment in which the 

 movement of a pendulum is concerned. The difficulty of starting 

 a free pendulum so as to make it vibrate at first in a plane is ex- 

 tremely great ; and every experimenter ought to be prepared to 

 judge how much of the apparent torsion of its plane of vibration is 

 really a progression of apses due to its oval motion." 



After a careful analysis of the problem, when the pendulum de- 

 scribes an extremely elongated ellipse, the Astronomer Royal ar- 

 rives at the following conclusion, which is the principal object of 

 his present investigation. If the length of the pendulum be a, 

 the semi-major axis of the ellipse described by the pendulum-bob 

 be b, and the semi-minor axis be c, then the line of the apses of the 

 ellipse will perform a complete revolution in the time of a complete 

 double vibration {i. e. the time of describing the ellipse) multiplied 



by 5^. 

 ^ Z be 



" Thus If a pendulum, 52 feet long (which performs its double 

 vibration in 8 seconds), vibrates in an ellipse whose major axis is 

 52 inches and minor axis 6 inches, the line of apses will perform 

 a complete revolution /rem this cause in 30 hours nearly. 



"If a common seconds pendulum (which performs its double 

 vibration in 2 seconds) vibrates in an ellipse whose major axis is 

 4 inches and minor axis -^ inch, the line of apses will perform a 

 complete revolution from this cause in 30 hours nearly. 



" The direction of rotation of the line of apses is the same as the 

 direction of revolution in the ellipse. 



'• It is worthy of remark, that the expression which is thus found 

 for the progression of the apse on the supposition that the minor 

 axis is much smaller than the major, will, if we make in it c very 

 nearly equal to b, correspond exactly to the formula cited in the 

 beginning of this paper, as found by an accurate investigation when 

 the ellipse approaches very near to a circle. It appears, therefore. 



