28 THE PENDULUM AND GYROSCOPE 



lowest point will be, sensibly, \/2gl(l cos 2 ), an( l hence the moment at that point 

 will be I sin 0j j/2^(leo* $j)=C=s P sin A sin 2 2 , from which deducing the 

 value of sin 6 1 and substituting in (//), wo have 



(1) $'=n sin*, (l |sin 2 O a }t 



The second term in brackets is the retardation from the true horary motion ; it 

 being implied, of course, that the amplitude of oscillation is preserved unimpaired. 

 Though small, this retardation would be sensible, especially if 2 had a considerable 

 magnitude, say eight or ten degrees, though inappreciable for very small oscillations. 



Practically, the resistance of the air constantly diminishes the value of 2 , and 

 failure to procure a perfect state of rest to the pendulum before it is set free, or 

 currents of air, may give quite different values to C and c, and determine the cha- 

 racter of the orbital motion to be progressive instead of retrograde ; and it is gene- 

 rally observed that the conjugate dimension of the orbit increases (probably owing 

 to the resistance of the air) as 2 diminishes. In this way the apsidal motion due 

 to the orbit may acquire a value quite considerable compared to the proper horary 

 motion, which will apparently be sensibly retarded or accelerated. By observing, 

 at any period of the experiment, the value of 1 and 6 2 and the direction of the 

 orbital motion, the coefficient of t in the formula (&') will give the theoretical rate 

 of azimuthal motion at that instant. 



If the orbital motion is retrograde and 



sn 1= n sn 



l_ _? 

 \<7(1 



"3 \0(1 COS 2 ) 



the line of the apsides would be stationary. For Columbia College, where n sin 

 A=00004747, with a pendulum of 26 feet in length, and a value of 2 of 6, this 

 would give 0j=:3',3, the actual semi-axes of the projection of the orbit being about 

 two feet nine inches and one-third of an inch. 



It would generally be sufficiently accurate to substitute for 1 cos 2 , i sin 2 2 , 

 by which formula (k') would become more simply, 



00 <'=[ sin X + J0 sin e i sin 2 It 



in which it is seen that the deviation from the proper horary motion is proportional 

 to the area of the projection of the orbit. 



The above, or (#), expresses the azimuthal motion as it would be were there no 

 other forces acting than those included in the investigation, in which case X and 2 

 would be invariable. In point of fact there are practically numerous disturbing 

 forces, of which, however, the resistance of the air is the most considerable, and 

 through which 0j and 2 are incessantly changing, and it is not improbable that the 

 change of shape of the orbit may, in itself, cause some variation of direction of the 

 line of apsides: 1 a matter which cannot be decided until the problem of the spheri- 

 cal pendulum is solved with the resistance taken into account. 



' An investigation of the simpler case of the plane elliptical motion of a body attracted by a 

 central force proportional to the distance, in a medium which resists either directly or as the square 



