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

 In like manner we shall have in the value of Im the terms > 



(PAo , „ . . , d^u\ ^ • • (2-) 

 . — V- Q.n sm fl cos fl — r— > 



{ 



sin2 d -^-2- + 2n sin 6 cos 6 , , 



In these formulae Am and At? are the parts of « and v 

 depending on the terms of the second order. To the last of 

 them must be added the terms depending on the powers and 

 products of u and v. 



It will be convenient to develope the formula (F) No. 32 

 of the chapter before referred to anew. For this purpose let 



p = r + Sf (p = 9 + w, v|/ = 'n7 + u; 

 then we have 

 a:=pcos<p, i/=p sin <p cos {nt+^), z=f) sin (p sin {nt-^ 4')' 



These values being substituted in (F), neglecting ^, ^, 



and 8p where it is multiplied by the displacements, these quan- 

 tities being exceedingly small compared with those retained, 

 we find 



dv\ o^, r • ^^ « • d<p 



— sm<pcos<p-^ >+p^tfvI/<C sm^(p-^ + 2wsm(pcos(p^ 



+ 



2 sm -P cos <p ^ -^ j- - - 8(p2 sm2 ,p). 



At the surface in the state of equilibrium the last term of 

 the preceding equation becomes 



^^8(r2sin2,p), 



which is to be subtracted, as we shall perceive, if we consider 

 the mode of procedure with reference to terms of the first 

 order only. But p = r + h, p^—r^=2rh, neglecting h^as being 

 insensible. We may therefore make 



J 8(^(p2_^2) gin2 i^^=:n^^rh sin^ ^) = 0, 



since rn^h is very small compared with gh. And thus, if we 

 make p=r=l as heretofore, we have 



. . rd^<p ^ . d^ . d^n 



^ r . „ d^; . d<t) . d(i>d^~\ 



+ crI/-( sm^f -^ +2»sm(pcos<p^ +2 sm<p cosf-^ ^ j-. 



