TIIEOUBITOFURANUS. 29 



dd) 'In , . dt) 



— = cos T , + sin T , - 

 di dt ' dt 



— cosO , — sm0 , 

 dt dt 



dO ( . dq . dp\ ,c,A\ 



-=cosec^[-smr^-+co.r~f^-) (34) 



-l-cosec(^( sin0-~ cos6 f ) 



If we clifFerentiate (33) and snbstitnte tliese values of -^ and , , we shall have 

 ^ ^ dt dt 



dL dv dD , , / dp . dq . ^ dp' . . dq \ „-. 



If we consider only quantities of the first order with respect to the disturbing 

 forces, we may, in integrating, suppose t and equal and constant, and (p constant. 

 The integral will then be 



L = v^D^ tan i <?> | cos {hh + W) — sin {hn + hr!) \ (36) 



In the case of Uranus, tan ^ is so small that this equation will be sufficient for 

 a long time before and after our epoch. 



In the application of the method to otlicr planets the mode of operation must 

 depend on the circumstances of each particular case. The differential equations 

 (34) between 0, <r, and ^ are rigorous, and their integrals may be approximated to 

 in various ways, out of which that best applicable to the particular case must be 

 selected. 



Expressions for the latitude. 



If the position of the orbital plane and of the ecliptic were each determined by 

 the preceding formulae, there would be no perturbations of the latitude, the lati- 

 tude itself being given rigorously by the equation 



sin ^ = sin <^ sin (v — t). 



= sin ^ cos T sin v — sin ^ sin r cos v. 



But the instantaneous values of ^ and r, or of sin ^ cos t and sin ^ sin r, are 

 troublesome to tabulate ; it will therefore, in practice, be foTlnd more convenient to 

 use only their mean values, and to consider their changes from this mean as per- 

 turbations of the latitude. Representing by the sign 5 the deviations from the 

 mean values, which are of course arbitrary, we have 



cos ^SjS = cos (p sin {v — t) ^^ — sin ^ cos (v — t) St. 



Let us substitute for S^ and hr their values given by the integration of (34) to 



