The Rev. Brice Bronwin on the Theory of the Tides. 341 



«>;• - .- — 2r cos fl -i 1- 2i{p + 1 )B— 4 cos d A = 0. - ! ^^• 



or 



y 2rCOS fl — r- 



(Ir dr 



We shall neglect v, then 7)= 1, and the last becomes 



/r-j— — 2rcos6 V-+4'«'B— 'l*cosfiA = 0. . (19.) 



ar dr ^ ' 



To make' Iw a complete variation relative to r and 5, we 

 must have 



Substituting for -7-5-, &c., and dividing by n cos /(ip— 5), we 

 find 



^{ipr^k - 2r2 cos AbI- = - 2/- ^ (sin (3B,) 

 or 



/>r-T7 — 2rcosfl -^ |-2z>A— 4 cos dB + 2 ^(sin 6B)=0. 



Making ^= 1, and putting for -rr (sin SB) its value from {e\ 

 there results 

 //•-^ 2r cos fl -^ + (4/— ^ sin^ n A-4 cos flB = 0. (20.) 



Make ■*»'''^^ 



A=XCr-, B = 2Dr-. ,,t,j,. 



By the substitution of these values in (19.) and (20.), we find 



(w + 4)/D — (2w + 4)cos6C = 0, j 



(2w + 4) cos fl/D - (i2(TO + 4) - 4 sln^ 5) C = 0. J 



Eliminate p- between these, and there results 



i^{m + 4)^ — (2w + 4)2 cos^ fl — (w + 4)4 sin^ fi = 0. ./ 



Thus (19.) and (20.) are rendered identical. Let the roots 

 of the last equation be m^ and ;«2> ^"^^ ^^ have 



A = Cir"'i + Cg/-'''^ B = Dir"'! + D^r^'s. ^ g < r,.' ^. 



When /=2, we find 



4 

 m,= — .% w,,= r-5-7. 



