Royal Society. 69 



characters of the electricity of the Torpedo, the purposes it appears 

 to serve, and the varieties exhibited by different individuals, ac- 

 cording to the age, the sex, and other circumstances. 



The Meetings of the Society were then adjourned over Easter to 

 the third of May. 



May 3. — The following Report, drawn up by the Rev. William 

 Whewell, M.A. F.R.S., the Rev. George Peacock, M.A. F.R.S., and 

 the Rev. Henry Coddington, M.A. F.R.S., on Mr. Lubbock's Paper, 

 read before the Royal Society Feb. 9, 183S?, and entitled, "Re- 

 searches in Physical Astronomy," — was read. 

 Report. 



The method of the variation of parameters as applied to the in- 

 vestigation of the perturbations of the solar system has been suc- 

 cessively developed in modern times. This method gives the vari- 

 ations of the elements of the elliptical orbit in terms of the differentials 

 of a certain function R of these elements, and of the disturbing forces. 

 Euler, Lagrange (1783), Lagrange and Laplace (1808) obtained the 

 formulae for d a, d e,dvj,d p, dq where p = tan (p sin 8, j = tan ^ sin 9. 

 Poisson first gave the expression for d s. Pontecoulant, p. 330, has 

 introduced rfi and dv instead of rf^ and dq; but those develop- 

 ments gave expressions neglecting the square of the disturbing force. 

 Mr.Lubbock has published (in a Paper in the Phil. Trans. April 1830,) 

 expressions which include the effect of any power of the disturbing 

 force. This method has been principally applied to the secular in- 

 equalities; but it is susceptible of being applied with no less strictness 

 to periodical inequalities, all of which may be represented by certain 

 changes in the elements of the elliptical orbit. 



But the same problems may also be approximately solved di- 

 rectly ; for we obtain a differential equation involving the radius 

 vector and the time. In this equation there occurs the same func- 

 tion R of which we have already spoken; and this function is ex- 

 panded according to terms involving cosines of the mean motions 

 of the disturbing and disturbed planet, and cosines of the difference 

 of certain multiples of these motions. This expression has been 

 treated of by various authors, and among others Mr. Lubbock has 

 himself (in memoirs read May 19 and June 9, 1831,) given the ex- 

 pansion of R in a form suited to his present object. 



The coeflicients of the terms in this expansion are arranged, as 

 usual, according to the order of the excentiicities, their powers and 

 products, and to the power of the sin^ of half the inclination. These 

 coefHcients involve also certain quantities b ■ where n and i have a 

 variety of values ; and these quantities depend on the ratio of the 

 mean distances of the disturbing and disturbed bodies from the sun. 



Solving the differential equation which involves r, by the equating 

 of coefficients, Mr. Lubbock finds a value for the reciprocal of r in 

 such terms as have been mentioned. By certain algebraical trans- 

 formations of the fractional coefHcients in which i occurs, (and by 

 certain equations of condition between h., . ,, 6.,., h.,., ,, and 



' 0,1— I ^ o,l ' nl-\- 1 ' 



between similar quantities,) the exprcBsion for the reciprocal of r is 

 transformed and rcdiitcti, tlic arcs remaining as they were. 



