OF SYMBOLICAL GEOMETRY AND MECHANICS. 507 



and from (7), we have 



y=^fDu.U.dt (9). 



(8) and (9) give h and y separately. 



We may observe respecting these formulEe for h and 7, that iS(i. (J expresses the resolved 

 part of the disturbing force U in the direction /3, i. e. perpendicular to the radius vector and in 

 the plane of the orbit; and Du . U is the .symbol representing in magnitude and direction the 

 moment of the Couple which transfers the force U from the point ji to the origin. (See former 

 Paper.) 



39. To integrate the equation (6) directly as we did the equation (1), we have only to tai<e 

 the same steps, (observing that h is now variable,) as follows, (6) becomes 



d^u IX dS 



— = - — + ^7, 



dt- h dt 



hence, (integratnig 7 -p by parts), we have 

 n, dt 



dt h^ J ft' dt^ ■' 



and therefore by (8), 



^^ = l^+ f'^(Afi.U)^dt + fUdt (10). 



This is the symbolical expression for the velocity of the disturbed body. To find the parallax, 

 put in (10), 



du dr h 



dt ~ Ift" "^ r^' 

 and then, performing the operation A/3 on both sides, we find 



which determines the parallax. 



40. To determine the eccentricity and position of the axis major. 

 We have seen, that when there is no disturbance, 



du ,j. 



assuming this equation to be still true, e and e being now variables, and comparing it with (10), 

 we find 



ee= |fe(A/3.f7)/3 + u\dt (11); 



or^) = g:(A/3.^7)/3+f7 (.'.). 



Now e is the eccentricity, and e is the direction unit of a line at right angles to tiie axi.s 

 major in the plane of the orbit: hence, (12) or (11) determines at the same time the eccentricity 

 and the position of the axis major. 



The Dynamical investigations just given are good instances of the naUni- of the Symbolical 

 method here proposed. 



M. O'BIIIEN. 



UppKR Noiiwooji, Sukukv. 

 January, 1047. 



