12 THEOKBITOPNEPTUNE. 



Substituting these expressions for the differential coefficients in the values of 



clR , clB 



-^— and "^- , we have 



a^ du 



dR m , • »T f / V , -A X , / • , -N 1 , '" f^^* 



-j-,=z—li sm x' sm N { (^' +j') cot y + (t -\-j ) cosec j/ } — \— -j-coslycos x' cos iV! 



1 dR m,-, , • TiTf/v , -A X , /• , -N 1 ,?nc77i . . 



^T— ^^^ =— -/tcos;i'siniV{(4'+^)coty+(i+j)cosecy}— i- ^cos^j^smz'cosiV: 



Let us now put 



It may be remarked that i will then be the coefficient of the longitude of the 

 common node of the orbits in the usual development of the perturbative function. 

 The above equations may then be put into the form 



-^-j = — — ih cosec y sm / sin N — — [i +7 ) h tan \ y sm / sin iV — i — -7- cos J y cos / cos N. 



1 dR m , / • ,T , ™ ^ .. , ■,^ V , , t ■ ^T ^mdli , . , 



-: ryr- = —rth cosec y COS z sm iV 4- -r (i' 4- /) ft tan i y cos z' sm iV — -l -7 "^ cos J >- sm x cos iV. 



Bin 55 afi' a ' ' a' ^ ' -^ -^ ~ ' ~ a' du ~ ' 



Substituting these expressions in (6), and integrating, we shall have the values 

 of ^6' and h^', the perturbations of the inclination and node. 



For the perturbations of the latitude, counted in the direction perpendicular to 

 the plane of the orbit, we shall have 



h^' zz 5^)' sin [v' — 6') — sin ^W cos (v' — Q') 



= mv sec il{T^I] sin (i\^+ Y) (7) 



+ mv sec i! { T—I] sin \n— 7) 



Where 



Putting 



dh 

 T-=i\^CQ)^\y; Izz \h{i cosec y + {i -\-j') tan \y] 



Frr true distance of planet from common node. 

 B,-T+I; B, = T—I, 



and developing Fin terms of /I and/ to terms of the second order with respect 

 to the eccentricity, we shall have 



(1 — e'')sin(iy^+;i) \ /(I — e'-)sin(iV^— ;i) 



+ e- sin(i\r+a+ /) ) \+ e' sin(iy^— X— /) 



^l3 = mvB, {— e' sin(iV'+;i— /) \ +mvB, — e' sin(i^— ;i+ /) ) (8) 

 + I e'^ sin (i\r+ ;i + 2/) i / + § e'^ sin (iV^— ;i— 2/) 



— I e'2 sin (iV^+ a— 2/) I \ — 1 e'- sin (iV^— X + 2/) 



