1881.] 371 [Chase. 



gave conclusive evidence of the importance, in all physical investigations, 

 of studying elastic reaction as well as centripetal or centrifugal action. 

 An important fact, in connection with such comparative study, is the varia- 

 tion of elastic density in inverse geometrical ratio, when distance varies in 

 arithmetical ratio. We may, therefore, look for exponential harmonic 

 relations between planetary positions, of a character analogous to the simple 

 harmonic relations between wave lengths. Let /fc^ represent the kinetic 

 projectile ratio of Neptune, Sun's semi-diameter being the unit ; ko, the 

 mean projectile ratio of the nebular centre of planetary inertia ; k , the pro- 

 jectile ratio of the centre of spherical gyration of the centre of condensa- 

 tion ; X^, X^, etc., the corresponding wave-lengths ; tx, the ratio of orbital 

 time ; t^, the ratio of photodynamic time ; v^, orbital velocity at the rup- 

 turing locus of mean planetary inertia ; «/, orbital velocity at the sub- 

 sidence locus of mean planetary inertia; n, the quotient of Eartli's semi- 

 axis major by Sun's semi-diameter. Then we have : 



log. k^ : log. A;^ : : 4 : ;.^ (1) 



log. k^: log. k^:: X^: X^ (2) 



X^- X^ :: 3^: 4* (3) 



t-^ '• ^A : •■ «A : «^ (4j 



These four proportions furnish data for estimating the four photodynamic 

 values which are least known, by substituting the six which are best known. 

 In equation (1), the estimates of k^ range between 30.034 n (Stockwell) and 

 30.070 n (Newcomb), the estimates of n varying from 214.3 to 214.9. As 

 the uncertainty of n is the greater, I adopt it as one of the four unknown 

 quantities, and use Leverrier's intermediate coefficient, 30.067. For X^ I take 

 the wave-length of the Fraunhofer line A, 7612 ten millionths of a milli- 

 meter. In equation (2), in the sethereal sphere of which the centre of con- 

 densation is the nucleus, the mean projection of spherical gyration is repre- 

 sented by .4 n, and the radius of solar reaction {k ) by .6 n. In equation 

 (3), the numbers 3 and 4 are linear representatives of the exponential radii 

 of synchronous nucleal and atmospheric nebular variation ; 3^ is the pen- 

 dulum ratio of varying central force; 4* is the ratio of variation in the cen- 

 tripetal energy of nodal collisions in paraboloidal aggregation. In equa- 

 tion (4), ty. =(30.037 nY ; t^ =the solar modulus of liglit divided by Sun's 



w; v^ -^ vx 



/ 31558149 \- 



semi-diameter = I , I = 101790260 



V2 TT W2 X 497.827/ 



/I + e\h 



( ^ ) , e being the maximum secular eccentricity of the planetarj^ 



aggregation (Saturn) at the nebular centre of planetary inertia. Making 

 these substitutions, we get, 



log. 30.037 » : log. k^ :: 7612 : X^. (l)i 



log. 30.037 n : log. .6 w : : 7612 : X^ (2)i 



(30.037 n)i : 101790260 -^ n : : (\ ^ e)^ : (.1 — ef (4)i 



PROC. AMER. PHILOS. SOC. XIX. 108. 2u. PRINTED MAY 14, 1881. 



