Chase.] 428 ^Oct. 6, 



in the theories of sound, light, heat, etc." Nearly all the results of my 

 physical investigations go to confirm the truth of this statement. The 

 velocity which is involved in the time-integral of projection against a con- 

 stant gravitating retardation, is measured by gt. The theory of the ballistic 

 pendulum assumes [op. cit, § 298) "that the ball and pendulum are mov- 

 ing on as one mass before the jiendidum has been sensibly deflected from the 

 vertical. This is the essential peculiarity of the apparatus. A sufficiently 

 great force might move it far from the vertical in a small fraction of its time 

 of vibration . But in order that the time-integral may have its simplest 

 application to such a case, the direction of the force would have continu- 

 ally to change so as to be always the same as that of the motion of the 

 block." 



This is precisely the case in the identity of the foregoing note, accord- 

 ing to LeSage's hypothesis. The doctrine of correlation of force leads us 

 to look for the simplest forms ot harmonic motion at the centres of stellar 

 systems. The simplest value of t, in a harmony of luminous undulation 

 and stellar rotation, is that of a single oscillation of half-rotation. We 

 have no means of knowing whether the identity, vX:=vs = vy, holds for 

 any system except our own, but its verification by our sun and the variet}' 

 of ways in which photodynamic harmonies are deduced are very signifi- 

 cant. 



283. A Secondary Time-Integral. 



The harmony between Sun's constrained rotation and luminous undur 

 lation warrants an expectation of subordinate harmonies between sola- 

 and planetary motions. We may naturally look for the simplest relation 

 in some harmonic motions of Sun and Jupiter. Jupiter is at the nebular 

 centre of the system, on a diameter which is bounded by mean loci of 

 Neptune and Uranus, and the velocity which is involved in its time-inte- 

 gral of rotarj' oscillation, {gt), is nearly, and perhaps exactly, the same as 

 the limit of planetary velocity in a circular orbit {\/gr at Sun's surface). 

 Although planetary revolution at Sun's surface is impossible, the influences 

 which tend to produce it are continuous, and any wave motion which 

 may be thus produced is propagated with uniform velocity through the 

 medium in which the waves originate. The uncertainties in regard to tlie 

 exact values of Earth's semiaxis major and the apparent semidiameter of 

 Sun and Jupiter, introduce a range of uncertainty into the velocity of Ju- 

 piter's time integral, which amounts to about five per cent. Values may 

 be taken which are very near the mean values and which make the accord- 

 ance exact. This accordance may, perhaps, lead to a special extension 

 and modification of George H. Darwin's beautiful investigations. 



283. A Third Time-Integral. 



The centre which seems to hold the third rank in point of cosmical im- 

 portance, in the solar system, is the centre of the belt of greatest conden- 

 sation, which is represented by Earth's orbit. The velocity which is in- 



