The Rev, Brice Bronwin ow the Theory of the Tides. 347 



the integration being relative to their variation equally with 

 the variation of S'. Now, making 



^-pd?'=k, ^-fh\dV>^1c,, &c., 



we have 



V 



y=— —k-\-k. sin^u+ &c. 



for the height of the tide resulting from the disturbing force 

 of the sea, which therefore introduces no new terms, but only 

 makes an alteration in the constant coefficients. 

 Now let 



^=5F.cos?(<p-/3j 



be the sum of the equations of short periods. Make 



/3,=^ + ^, 

 k being constant, and A variable and very small. Then 

 3/=2F^.cos i(<p— /^)+SF^X sin i{<^—Jc) 



= S(cos ik— sin ikK)Y^ cos if + 2(sin ik + cos ikX)F. sin i(p. 

 But 



/being constant, and x variable and small ; we may therefore 

 write 



j/ = S( A + Bjx) cos «<p + S(C + Dv) sin «f , 



ju, and V being variable or functions of t. Consequently 



y = 2(A' + BV) cos «y + S(C' + DV) sin «>', 

 where 



(p' = n^ -f. ot' — \I/ = w^ + ■57 — vj; + (■or' — ot) = (p 4- (■sy' — -nr) . 



With this value of <p' we have 



y = S{(A'+ By) cos i('sr'-'5y) + (C' + DV) sin ?(w'-'5r)}cosi(p 



+ 2{-(A'+By)sin«(^-a;) + (C' + D'v')cosi(t!r'-t;j) s in/. 



By changing the values of A', B', &c., and also those of/*' 

 and v'j this result may still be written, 



y = 2( A' + By) cos /f + 2(0' + DV) sin 2>, 



which gives 



V=S(/AW+/ByjF)cosi> + 2(/C'^F+/D'vViP') 



sin /(p. 



To perform the integrations we must put 



fi! =m+m^ s'm^ v+ Slc, 



y'=:w + ?tj sin^t;+ &c. 

 2 A 2 



