Tlie Rev. Brice Bronwin oti the Theory of Planetary Didnrbance. 37 



We now turn to the equations (1) of this section. The first contains botli 

 ^ and ^ ; the second only 0. This last, therefore, is preferable ; and as we 

 do not want them both, we shall take this, which reduces to 



drdt ^ "^ drdt \dT dr' J ^ It dr^ y ~ '■^ T 



Por the first and second power of the disturbing force only, 



-^ = 5 + ^ ^'^ 4. ^ ^"^ _ c , o "'</' dtp <l'4, 

 drdt dr drdt ^ dt dr' ~ ^ '^ '^ d^ '^ Vt dr' ' 



Putting this value in the second member of the above, we have 



drdt ' ' dr "^ dt d-^ It Th d7 ' \ 



3. The values of and ^ would be very troublesome to find ; for they 

 would contain a great number of terms having r-Hn their coefficients, and 

 which in to and /3 would vanish ; and many which, when t is changed into t 

 would unite with others. If, therefore, we could get rid of these ^antities,' 

 we should greatly diminish the labour of integration. This, happily, we are 

 able to effect. To accomplish it we will develope S relative to these quantities. 



Equation (2 ) of section ( 1 ) gives ^ = -{\ j^ ^YL ^Vi _ i ^ , 3 ^'^ 



/'*V drj /,i\ '^dT^^dr'J- 



Also, '^^ = ^ = r ^^ + /-^^^ - £^ ^^^P'^ ^^ 4. ^ ^(^''> ^' '^(^:) 



KB') 

 P''\^~ • These values are to be substituted in that of S, after we have 



changed p into p,^, v into v^ + Qnt, and \ into \ + Qnt. Make, therefore, 



^^^\and 

 A, dr 



