342 The Rev. Brice Bronwin on the Theory of the Tides, 



When «=1, 



4 cos^ 6 

 ^ 1 — 4) cos^ 9 



The first value of wzg is infinite negative at the pole, and the 



second is so likewise when cos 6= -, or in the latitude of 30°. 



2 



If, therefore, r be less than unity, r'"2 is infinite at both places; 

 and if r be greater than unity, it is nothing. This is inad- 

 missible, since we can give to r any numerical value we please. 

 We must therefore have C2=0, D2 = 0, or the constant arbi- 

 traries by which the values of these quantities are multiplied 

 must be nothing. Thus we have 



A=Cir'«i, B=:Dir'«i; 

 and therefore 



C, or C=Ar-'««, D, or D = Br-'"i. 



These values, substituted in the first of (21.), give 



(m + 4.)zB — (2m + 4) cos 9 A = 0. 

 If by means of this we eliminate- B from the equations 



^(sineA)-/B = 0, 2^(sineB)-(e2-2sin2d)A = 0, 



given in the first paper, we find 



. , rfA m . . 



sm fl -rr cos 9 A = 0, 



^^^.4(sinecos9A)-(22-2sin2a)A=0. 

 w + 4 </9 ^ ' ^ ' 



Let /=2, m= — 3 ; and both these reduce to 



f /I c?A . . 



sinfl-jT- +3cos6A = 0, 



which, integrated, gives 



A_A - ^2 



This will be found upon trial to be a particular integral of (7.). 



rfA 

 Let 2 = 1, w=0; and both the above reduce to -j^ =0, 



which gives A = Ai=ai, which is also a particular integral of 

 (7.) in this case. These results serve to confirm the legiti- 

 macy of the process j and since each of the above equations 

 leads to the same result, the confirmation is in a manner 

 doubled. 



