The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 41 

 d^t = ^^^^dt['d?^'l'-d?-^d?-^'^d^d?-^^'t'-d? 



1 X— - — ^^P°') X 4- ^ — 

 2h dt 2h. dr h, dr • 



(8) 



We must now eliminate -5^ , — ,^, &c. For this purpose we take 



d-r dr- 



d<j) dxtf" d^(j> 



dr * dr ^ dT 



which squared, neglecting terms above the third order, gives 



X^.iTrf.2,Tr^ = 



dW 



dr 



dW 

 - W + 2 TT'.Y + I TF^ + 20Tr ^ . 



d^_dX ^d^d^ (P^ _ 

 d-r^~ dr^^'d^ l^^'^l? ~ 



dr ^ dr ^ dr- 



Substituting these values in (8), it becomes 



U dt 2h^ dr ^ ^ A, d-T ■ 

 Let 



Y-. _ i,p-. , J„X- Jixf -i» -^ ,.Y., + ?^ If *. (9, 



By means of this the integral of the preceding is 



^t = »r.*(f.JT.^^.J*'^.r. (,0, 



VOL. XXII. G 



