190 The Rev. Brice Bronwin on the Tlieory of the Tides. 



terms of which will be of the form A'a and B'g, A' and B' 

 being functions of 5 only, and « and § functions of t only. 

 Therefore the values of m and v will be sums of terms of the 

 form 



A'« sin /f + B'§cos if = C(a sin /(p + ^ cos /<p) 



+ D(« sin /<p — 5 cos if)y 

 if 



C=i(A' + B'), D=1(A'-B'). 



But the two terms in the second number of this equation may 

 be put under either of the forms Aa sin i{f — ^, or Aa cos /(<p — §), 

 which are therefore the most general possible. We may 



therefore make u and v^ and also -7-, -r-, equal to the sum of 



at at ^ 



terms of this form ; for the first general assumption applies to 

 the differential coefficients equally with the quantities them- 

 selves. 

 Let then 



•^ =sSA«sin /(<p— §), sinfl^ =SB/3cos/((p— g). 



These values substituted in (4.) and (5.) will give results of 

 the form 



M sin /(f — g) + N cos i{<^ — g) = 0. 



"Whence M=0, N=0, will be equations for determining A 

 and B. 



From (4.) we get 



(sin fl -^ + cos dA ja— 2B/3 = 0. 



As fl and t are independent of each other, this equation cannot 



subsist unless one of them divide out. We must therefore 



have 



dA. 

 ^=a, sin6 -^ + cosdA = /B. . . . (a.) 



And there is no ambiguity ; for whichever of these equations 

 we assume, it leads to the other. 



If we had made /3 = ^a, we should have had /rB everywhere 

 in the place of B ; therefore making ^B=B', the final result 

 would not be altered. 



Making the same substitution in (5.) which has been made 

 in (4.), we find, putting 



rfvj/ d^ 



dt dt 



—pn 



