10 LECTURES ON THE LUNAR THEORY. [LECT. 



Hence with the time as independent variable we have the equations of 

 motion 



p 



df T \dt~ 



rdt dt 

 d' 2 , 



Or we may write these with as independent variable ; let 



t de 



r dt= H > 



so that ,. = Hu". 



dt 



TK Tf dH T 



Then H ^ = - t , 



d"r _ z d"n n du 

 = * ~ <Jl d0 



d , 



= - u > 



,du 



TT> * u 



whence H ' U + U + H d 



i \ Tj-i d*s d*u\ ^dH I ds du 

 again, ( rs ^ H uu dffi~ S d^) + H dO U ( U d6~ S 



JT, T r ds n 



whence Ji-u 3 \-j^,+ s } + H~ r ^u 3 ^ 7 . = Ps S ; 



\cia~ ] at> cW 



or the equations of motion may be written 



ud0' 



