333 Prof. Challis on a Mathematical Theory 



These values of V and U' make \'rd6 + \]'dr an exact differential. 

 The value of V, applied to the motion of fluid in contact with a 

 sphere, is of the same form with respect to 6, as the term added, in 

 the previous part of the reasoning, to W sin 6, to express com- 

 pletely, to the first order of small quantities, the velocity of the 

 fluid along the hemispherical surface opposite to that on which 

 the waves are directly incident. That term is thus proved to be 

 consistent with the hydrodynamical equations. I proceed now 

 to deduce from the complete expression for the velocity the pres- 

 sure on the surface. 



Resuming the value of V, viz. 



dW 

 Y =^N ?xn 6 — q —jT shi6cos0, 



in which W represents m sin {bt + c), and 9 is a veiy small quan- 

 tity, and substituting in the equation 



b^mpAosp+(^-^cde+^ = m, 

 we have by integration, 



rlW en d- W V^ 



i^Nap.logp-c^cos^+|-^i^cos2^+-^ = F(0. 



Where 0= -^i V is equal to W, and p to p^. Hence 



*'Nap.Iogp,-f..5%:|'=FW. 

 Consequently, eliminating F(0> 



Pi dt 3 at~ A A 



Expanding to terms of the second order, putting 1 + s for p, and 



W 



remembering that p, = 1 + o- = 1 + -r, the following result will 



be found : — 



2/ . c d^ a cq d^W „. , W2 2_ 

 «^(^— ) = ;^--^cos^-^,.^^cos^^+2^,cos^^ 



The terms in this equation, which are entirely periodic, may be 

 omitted, having no application in the present inquiry ; and for 

 the sake of brevity, the terms involving q^ will not be retained. 



