Variable Elements of a Disturbed Planet. 36 1 



and >j will satisfy 



Therefore from (e.) and (Jl) by addition, there results 



which are the same as if the plane of the orbit were fixed, the 

 result intimated at the beginning of this paper. 



To effect a further transformation of (F.), let v be the lon- 

 gitude on the orbit. Then ^ = r cos y, >j = r sin o ; 

 d^ dr . dv dfi . dr du 



rt = '''''Tt-'''''''dt^ Tt = '''''d-t-^''''''di'' 



dd^ ^dr . ddv . ^ dr . . dv . du 



-T-^ = cosy-7 rsmo— r- — smvbu-y,— sinvbr-^,—rcosvhv-:r: I 



dt dt dt dt dt dt 



%di\ . Idr Zdu ^ dr ^ du . . c?o 



-^=smy-5 — h rcosw-y— + cosy oy-T-+ cos oor -J-— rsmuSw-TT. 

 dt dt dt dt dt dt 



Also A^ = cosyAr — rsinyAy, A>j = sinyAr + rcosyAu. 



With these values we shall find 



A v^'^^ A ^^»3 A ^dr . ^ du 



a A ^du . ^ dr . ^ du 



-l-r^Ay-r- + r AySy-T- + r AySr^-. 



dt dt dt 



Making A and S change places, we have the expression of 



dt ^ ' dt ' 

 du _ 

 di^ r^ 



By these values, observing that -^ = -^, we shall find (F.) 



transformed into 



idr Adr 



Ar-j-—6r—j— ■\- r^ 



at at \ Mt- vui. , , /rT\ 



^^(Avir-Ar^u)+ARdt = 0. 



We shall suppose t to have such a value that y may be equal 

 to 71, the longitude of the apse. Then 



dr f^e . . Sr/r f^e , 



(-^^-"^")|. 



Also since 



Putting these values in (H.), it becomes 



du h ddu 1 J, T 2^5, 



