The Rev. Brice Bronwin on the Theory of Planetary DlsturhaiKe. 29 

 The equations (5) may be put under tlie following form : 



y {r sin «) — -/-(/' sin \) = -r (cos v — cos A). 



1- (r cos w) — -7- (p co^ ^) — T (^i"^ A — sin r). 



The first and second of (6) may be written thus : 



7 =p II -cos (\-v)\+ ^^ Ipsin (X-y)!; 



^=^-sm(X-t,) + ^J,cos(X-«)|. 



These, and the third and fourth of (6), are very elegant equations, and are 

 deserving of notice on the ground that, possibly, some use may be made of 

 them. I shall only employ the third of (6). Differentiating it relative to t, 

 there results, 



I dp r2/x 2/1 ,, ,"[dh (\d\ ,. . 1 dh . ,^ .]dr 

 /,' dt \ h^ h" ^ ' \ dt \h dt ^ ' /r dt ^ ' \ dt 



/I u.\d\ . ,, , fld'r h iJi\ . ,^ 

 + (7 - PJ^ "'^ (^ - "^ + Kh 1? -r^-'h?) ^''^ (^ - ^> 



dv d^r Ic 

 Eliminating j- by the third and fourth of (6), and substituting for -^ ^ 



+ -2 its value from (1), we find 



\dp IdpdX ri /x /I yu\ ,, ^ dh I . ,^ .dR 



7dt-hTrirt+[-p+]?-[-;-+j?)'''''^''-'nm-h''''^''-'^irr- 



Multiply this by h, put for -j- its value from {!), and for -^ in the first 



member its value -j- ; then making 



r={(U£.)cos(x-..)-(U^)}'§-..(^-.)f 



dR 

 dr 

 we have 



