44 The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 



I shall give some theorems here similar to (6) of the first section, without, 

 however, making any use of them. 



We have u = sini sin (u — ^) = sin i (sin v cos ^ — cos v sin b), -j- = sin i 



cos (v — ^)-r=—, sin i cos (v — ^). Therefore, -j- ^- = sin i cos (v - ^) = 

 ^ at r- h at ' 



sin i (cos v cos S + sin « sin ^). From these we easily find 



r- da ... 



0- sm u + -r —r cos V = sm i cos S. 

 h at 



r' da . • . • a 



— (T cos V + -r T- sm w = Sin t sm 5. 

 A at 



Again, let *• be the same function of t and the variable elements that a is 

 of t and the same elements ; so that 



*• — sin i sin (X — S), -r- = -; sin i cos (\ — S); 



and, as before, we shall have 



. ^ p- dK ^ . . . 

 K sm X + ~- -r- cos \ = sm * cos 3-, 

 h rtT 



p^ dK . ^ .... 

 — A- cos \ + V ^- sm A = sin ? sm 3. 

 h dr 



Equating the two sets of values of sin i cos 3- and sin i sin 5, 



(T sm y + -^ -r cos u = *• sm A + y- ^- cos A, 

 h dt II (IT 



r' da . ^ P' die . 



a cos V — -, — - sm V = K cos A — ^ -5- sin A. 

 h at n ciT 



By multiplying these by sin v, cos v, sin \, cos \, and adding and subtracting 

 the products, we shall have no difficulty in deducing 



p- dK 



a = K cos {\ — v)-. — -7- sin (\ — v). 



r' da . /^ . p' dK ^^ , 



^ ^- = A- sin {X — v) + -r -r- cos (A — v). 

 h dt ^ ' h dr ^ ' 



