Makch 23, 1906.] 



SCIENCE. 



453 



are determined empirically for the date of 

 an observation near the middle of the avail- 

 able data. Being accurately determined 

 from the disturbed geocentric coordinates, 

 these six quantities correspond to the oscu- 

 lating elements. 



In the ease of a distant satellite of 

 Jupiter disturbed by the sun, let be: 



p, a, S, a = p cos S, ?, ri, f = geocentric coordi- 

 nates of the satellite. 



r, a, 6, s = r cos 6, «, y, s = Jovicentric coordi- 

 nates of the satellite. 

 ' W, [a], [6], W = M cos [6], [X], [2/], [2:] = 

 heliocentric coordinates of the satellite. 



(p), (a), (5), (<r) = {p) cos (5), (|), (7,), 

 (f) = geocentric coordinates of Jupiter. 



(r), {a), (6), (s) = (r) cos (5), {x) , (y) , 

 («) = heliocentric coordinates of Jupiter. 



R, A, D, S = B cos D, X, Y, 2 = geocentric 

 coordinates of the sun. 



(k)^-^mlc', where to = mass of Jupiter. 



Then from the equation of motion of the 

 satellite referred to Jupiter's center as 

 origin : 



d% 



+ [kY 





and the corresponding equations in y and z, 

 three equations are deduced which give 

 p, p, p", in tei'ms of known coordinates, 

 velocities and accelerations, and in terms 

 of the unknowns (r) and [r]. The com- 

 plete expression for p is 



A, B and C being expressed, as indicated 

 above, in terms of known quantities. The 

 solution of this equation together with the 

 equations 



[r]= = 7J=-l-p= — 2i?p cos W 

 1^= (p)' + p^ — 2{p)p cos (./') 



gives all the values of p corresponding to 

 the various possible solutions. In practise, 

 the solution of these equations is very sim- 

 ple. From p the value of p' is derived. 

 The remaining steps leading to the oscu- 

 lating elements are similar to those of the 



short method. The short method, itself, is 

 a special case of that under discussion, 

 terms depending upon the mass of Jupiter 

 dropping out and the motion being referred 

 to the center of the sun. 



Among other special cases of this analyt- 

 ical method is that obtained by omitting 

 the terms depending upon the solar attrac- 

 tion in these formute. The resulting 

 formulas will then be those for the deter- 

 mination of the undisturbed motion of the 

 satellite. The general method is applicable 

 to material points moving under the attrac- 

 tion of any number of masses. The method 

 presents the advantage that it is free from 

 all arbitrary assumptions, whether intro- 

 duced with the assistance of graphical con- 

 struction or otherwise, and that it reveals 

 all possible sets of osculating elements 

 which will represent the disturbed geocen- 

 tric positions given by observation. 



Planetary Inversion: William H. Pick- 

 ering. 



This paper was illustrated with a gyro- 

 scope. The sub,iect has already been 

 treated theoretically in other places, and 

 the present paper is therefore purely ex- 

 perimental. The gyroscope was arranged 

 so that it could turn about an axis passing 

 through the plane of the wheel, and also 

 about a vertical shaft which supported it. 

 According to the nebular hypothesis, 

 when the rings or spirals surrounding a 

 central sun broke up into planets, each 

 planet should, by Kepler's third law, have 

 a retrograde rotation. The fact that nearly 

 all planets rotate in the opposite direction 

 has always been held a serious objection to 

 the hypothesis, and several different sug- 

 gestions have been offered to explain the 

 discrepancy. None of these even attempted, 

 however, to explain the rotation of Uranus, 

 which lies in a plane practically at right 

 angles to the orbit. 



The gyroscope was set spinning in a 



