Make 



86 The Rev. Brice Bron win's Differential Equations 

 i ^(g 2 ) _ Jt + JL * o. 



2 c?/ 2 g a 



? r = g + _p, and to abridge g = /x( J; for « and 



a , A and h may not be respectively equal, but may differ by 

 quantities of the order of the disturbing force. This value of 



j ^2 / r 2\ „ „ 



r being substituted in — — j-~ — + — + P = 0, making 



djp _ dp dv _ h dp d*p _ h^ d?p 2h dgdp 

 ~dt~~dvTt~Y' J*' lH*~~g T ~dv* ~~f~dtTv 



or making v the independent variable ; in virtue of the assumed 



equations we find 





■&+p + h<P + ,) + r,-3ir + a. 



f,%P +pJ hJ^ v + «c. —u. 



"0 n % n 5 



Or if — = u ; 

 ? 



d*/> 1 /T1 . 1 



i?+p ■ 



Q l u A v ' u dv \ dv/ 



(1.) 







V 1 + up 



The two last terms are of the order of the square of the 

 disturbing force. There will be a term of the form A cos (»— it) 

 arising from the disturbing force, and the quantity e or a is to 

 be so determined as to take it away. 



The equation r 2 dv — dt (k — Q) will become by this sub- 

 stitution 



dv(h-Q) . „ s-i ( h 1 r d ^ dv \,n\ 



dv = mtzsr** +up) ■ U -gj-.ii.?} (2 ° 



From this we shall find v = bv +f{v), b being a constant, 

 andy'(v) a function periodic of v; and we shall have {b — ■ 1) v 

 for the progression of the apse. We might have supposed the 



central force — s H — *• instead of -h>, and have assumed —^ 

 Q l o A p 2 dt 1 



h 2 — <? 

 ^—3 1 — 2" = 0j g Q d v = h Q d t, and have found p = v 



+ y ( v ) » Dut I think the development will be more simple with 

 the assumption above made. 



To find the latitude, let s = sin. lat., i= inclination, 3= long, 

 of the node on the plane of the orbit, having a fixed origin on 



