52 The Rev. Brice Bronwin on the Theori/ 0/ Planetary Disturbance. 



It must be observed, that ynt is the uniform regression of the node, and 7 is to 

 be so determined as to take away the constant terms from Q. 



Now 3/0 = sin (jr — &o) the value when the disturbing force is made nothing . 

 therefore, integrating, 



y = sin {x-^a) +1^^^'- 



For a first approximation, we make --^ = — cos {x — S-q) ; and after tlie in- 

 tegration is performed, we may make x anything we please. We shall make 

 X = v + Jilt = 11^ + ^nt + jnt. Then y — a, and we shall have 



<, = sin {v + jnt - &„) + | ^ Qdt. (16) 



After this substitution has been made for .r, and v replaced by v^+^nt, «^, being 

 a given function of r, may be allowed to remain, or may be developed in terms 

 of z, and z may be developed in terms of f. 



We may make y ~ sin i sin (,« — 3- — jni) = sin i cos (.& + jnt) sin x — sin i 

 sin (5 + jnt) cos x =p sin x — q cos x. 



p = sin i cos (3- + jnt), q ~ sin « sin (3- + 771^). 



rf» . , „ . rfi . . . , „ , (d^ \ 



~ = cos / cos (3 + 7?iQ -^ — sm j sm (3 + 7?z<) ( — + 7?^ J . 



Or, 



dq ■ • /„ ,(//.. ,. . /(/•& , 



' = cos I sm (3- + 771?) y + sm « cos (3 -(- 7?i<) ( -77 + Y'* )• 



(/» cos i di /rf3 \ p ds /dB- 



do cos I di /<^3 \ q ds fd^ 



dt = '^^ldt + P[-dF + ^V'^^dt+P[di + ^'\ 



ds rf3 



We may substitute in this, for -j- and -=-, their values from (14) ; but as I pre- 

 fer the former method, I shall not pursue this further ; nor is it necessary, since 

 any one who is desirous of doing it may easily carry it through. We may ob- 



