Chase.] ^^^ [June 17, 



g.^ = 32.088 ft. = .0060773 miles "3 .7837087 



G„ — V -^ ^'o — .0000003909446 r„ T.5921152 



T„= V„ -^ G„ =^ 1101875 seconds 6.0421324 



L,r= Vo T„ = (V„ ^ v,y r, = 474657 r„ 5.6763800 



M„= V„ ^ «t = r„5 (?o ^ ri d; = 328438 to^ 5.5164402 



Substituting the value of r^ we find, 



?„ = 15,681,230 miles 7.1953800 



V„ =^ 185,752 miles 5.2689342 



v^ = 269.615 miles 2.4307442 



G„ = 890.1 ft. r±= .16858 miles T.2268018 



L(, == 2213.37/J3 == 73.67 Neptune's semi-axis major = 



204,675,900,000 miles 11.3110666 



76. Vis viva of Wave Propagation. 



In 1857, Professor Stephen Alexander announced to the American Asso- 

 ciation the approximate equality of md? in the two planets, Jupiter and 

 Saturn, which make more than |f of the aggregate planetary mass in our 

 system. He also showed that the mean distance of Saturn is nearly equiva- 

 lent to f of Jupiter's mean distance, and that the ratio | has been largely 

 influential in planetary distribution. In 1872, I showed* that the same 

 ratio may be deduced from the thermal energy of chemical combination, 

 representing the ratio of mean vis vica\ of oscillating particles to vis viva 

 of wave propagation. In 1877, Maxwell and Preston:); published the same 

 ratio, without seeming to have known that I had already deduced it from 

 the laws of kinetic oscillation in elastic media. Its importance in discus- 

 sions of kinetic unity, is evident from the fact that it is alike operative in 

 the establishment of cosmicalg and of molecular harmonies. 



77. Conical Pendulums. 



The tendencies to concentration, through the mutual attractions of par- 

 ticles which are subjected to a simultaneous linear projection, and other 

 combinations of tendencies to uniform and to variable velocities, lead to 

 many oscillations which follow the laws of conical pendulums. It is 

 therefore important, in investigations which involve cyclical elastic action, 

 to inquire what forms of pendulum-vibration will best fulfil the require- 

 ments of Fourier's theorem. The general formula for the velocity of a 



conical pendulum is, 



V = -[/gr sin d tan d. 



Hence, since t = 27rr sin ^ -r- «, we find 



i = 2;r sj-T- tan 0. 



* Proc, Am. Phil. Soc, xii, 392-4. 

 t lb. foot note. 

 J P. Mag. [.5], ill, 453; Iv, 209. 

 2 Proc. A. P. S., xii, 403-5, et seq. 



