28 



THE I'EXDULUM AND GYROSCOPE 



lowest point will bo, sensibly, i/Jgl {l—cos d'i), and hence the moment at that point 

 will be I sin 6, ^Ogi(^\—cos%)=^C=^—nl' sin X siir Q.,, from which deducing the 

 value of sin 0^ imd substituting iu (//), we have 

 (/ j (^'— )t sin X ^ 1 — ^sin= 02 j< 



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

 l)eing implied, of course, that the amplitude of oscillation is preserved unimpaired. 

 Though small, this retardation would be sensible, especially if Q, 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 Qo, 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 G 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 fio 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 ])eriod of the experiment, the value of Oj and Q^. ^^^d the direction of the 

 orbital motion, the coefficient of t in the formula (//) will give the theoretical rate 

 of aziniutlial motion at that instant. 



If the orbital motion is retrograde and 



4 • . I W 



{m) sin 01= n sin 1 I— - 



3 \//(l 



iil(\ — cos 62) 



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

 X=00004:717, witli a pendulum of 2G feet in length, and a value of O2 of 6°, this 

 would give Oi=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 te sufficiently accurate to substitute for 1— cos 0„, i sin^ a,, 

 by wliich formula (//j would become more simply, 



('•') <?'=[« sin X+ '^-^j sin 01 sin Q^ 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 Q^ and Q, 

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

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

 through which 0. and e, are incessantly changing, and it is not improbable that the 

 duuuje o/.7>ape of the orbit may, in itself, cause some variation of direction of the 

 luie ot apsides: a matter which cannot be decided until the problem of the spheri- 

 cal pendulum is solved with the resistance taken into account. 



ce-m^lfl'tlr'T °',"" l^""';" '■ '"'" '' "" P'"''' ^"'1'*!--^' "^"tion of a body attracted l,y a 

 ceut..l foue p.o,.onu,„al ,u the distance, iu a n.ediu.u whieh resists either directly or as the square 



