(b.) 



The Rev. Brice Bronwin on the Theory of the Tides. 191 

 ip ^ (sin eB)- (i>-2 sin2fl)A=o"' 



{|(si„m)-.'A}|=0 J • • 



The first of these has been divided by Uf which multiplied 

 all the terms. We cannot make in the second 



-TT- (sinSB)— 2A=0; 



for this would reduce the first to 2sin^fiA=0, which cannot 



be. We must therefore have -r- =0, and 



at 



u=a, a constant • (c.) 



Then, since A and B are not functions of ^, the first of (b.) 

 cannot subsist unless p be constant. We must therefore 

 make 



dS d^/ / J \ 



s- + s-="'' (''•' 



where c is a constant, and v is the mean motion of the planet, 

 the first member being of this order. 



y , 



If now we divide the first of (b.) by p=l — c-, neglectmg 

 the powers of- above the first, we have 



. d 



(sin6B)-(i2-2sin2fi-.2c-sin2AA = 0. 



And if in this we neglect the very small quantity 2c - sin* d A, 

 we have 



«4(sin9B)-(i2-2sin2e)A=0. . . . (e.) 



From (a.), (c.) and (d.), we have 



^ d^ d^ ,^ V 



Eliminating B from (e.) by the second of (a.), there results 



sin2d^+3sin6cosd^+(l-»2)A=0. . (7.) 

 Particular integrals of this are, when i—% 



k^^a^-\-lQ.o^^-\ 



sm- 



cos^ 



