42 The Rev. Bkice Brontvin on the Theory of Planetary Disturbance. 

 Since ( j^j = 777 + —r^ ; changing t into t, we have 



_ = (,r) -_' + «, ^_ + 4ir_j + iz^(^-^j+(F). (11) 



We have now got entirely quit of ^ and ; and t is contained only in p] and 

 X", which are the same functions of t and a^, e^, &c., as in an undisturbed orbit ; 

 and we may everywhere put them without the sign of integration, and then 

 change t into t. Thus we have virtually got rid of t, and the result may be 

 put under the same form as if it were obtained with only the ordinary time t. 



If we change z into i + «^, (3) of section (1) gives 



A _ dw Qnr^ 



hj'~ '^'dt'^H:^' 

 and 



-.7{-' 



dw Qnr^ , 



which will give /3 when ■^- is known. And we may change ji into e'', and take 

 the logarithm of both members, if we wish to have a result in the experimental 



form, whicli is that of M. Hansen. Or we may change ^ into 1 M — ; then 



we have 



r = r^ + f). 



But, changing f into t + 0, we may employ (2) of section (1), which gives 



A - 1 4. # 

 and 



If we wish to have a result in the experimental form, this is, perhaps, the 

 most convenient. And thus changing | into e*, and taking the logarithm of 

 both members, we have 



« = iiog(i)-i.og(.4f 



