358 The Rev. Brice Bronwin on the 



the whole vanishes when we differentiate A :r, 8;r, Ay, ly, &c. 

 for /. Also 



d^dx 8 d^x 



= -/^s(^)by(«.) = 0by(6.); 



dt^ d0- 



1,1 pdldy dldz . ,, , d/^dx „ 



and the same of -^, ~~dW' lastly, ,^ , &c. are 



multiplied by 8 a;- = 0, &c. The variation relative to t being 



nothing, it is independent of ^. 



di 3c d u 

 The quantities x, -r-.ty^ -fij &c. are composed of terms of 



the form A sin {i 7it + m), B cos {int + m); their variations 

 therefore will contain terms of the form Ats\n{ifit+ m), 

 B I cos (int + m)f multiplied by Sw. And by substitution 

 (B.) will be of the form 



M + Nif + P^2+ ARrf^ = 0. . . . (d.) 

 But we may make t anything or nothing without altering 

 the value of the three first terms of (d.) ; therefore N = 0, 

 P = 0. Hence if we make 1 = where it appears in the co- 

 efficients, or leave out those terms, the final result will not be 



affected. And hence if we use I ndt instead of nt for the 



mean longitude and consider n a function of the time, the result 

 will be the same; for the variations Ix, &c. will onlj^ differ 

 from their former values by wanting the terms containing t in 

 their coefficients. 



Let i be the inclination, 9 the longitude of the node on the 

 fixed plane, S^ the same longitude on the plane of the orbit, 

 having a fixed origin on it, in which case it is easy to see that 

 d^ = cos id$. Also let ^ and ij be the rectangular co-ordi- 

 nates on the plane of the orbit, the axis of ^ passing through 

 the origin of ^. Transforming by known methods from the 

 system {x^t/, z) to that in which x lies in the line of the nodes, 

 then to another in which y lies in the plane of the orbit, and 

 lastly to the system (^, r;), we find 



x = A^+Byi,7/=C^-{-Dri,z = E^+Fri, . (1.) 

 A = cos 9 cos S 4- cos /sin 9 sin ^, B=cos 9 sin ^— cos /sin $ cos d, 

 C = sin 9 cos 3^ — cos /cos 9 sin 3, D = sin 9 sin ^-f- cos /cos 9cosd, 

 E =s — sin / sin d, F = sin / cos <&. 



If we differentiate these last, making dd = cosidd, and 

 compare the values of rf A, c^E; dB,dF; rfC, </E; dD,dF; 

 we shall see that 



d A = tan / sin 9 ^i E, dB = tan / sin 9 c? F, 

 6? C = — tan / cos 9 c? E, <Z D = — • tan / cos 9 d 



F.}- • ^''-^ 



