Prof. Challis on Hydrodynamics, 269 



part of the value of qr. In this case, as I have on several occa- 

 sions argued, the reaction of the hemispherical surface on which 

 the waves are immediately incident is such that the condensation 

 at every point of it is quam proocime equal to the condensation 

 (ct^) of the waves at incidence. The equations apply exclusively 

 to the condensations at points for which the angle 6 is greater 



IT 



than 7^, and to those parts of the velocities at such points which 



depend on the variations of condensation due to the mutual 

 action of the parts of the fluid. The following process of rea- 

 soning is the same, mutatis mutandis, as that employed with 

 respect to the former set of equations. In the first place, since 

 U=0 where r = c for all values of 6 and at all times, by putting 

 c for r in the general value of U, we have 



3(/i-Pi) 3(/-F) 4(/'-F) f«-V< ,.. 



? J— + — 3~c 8~ ~ °- * (X) 



Again, introducing, for the same reasons as before, the condition 

 that the ratio of a to <r 1 is at each point a function of r and 6, 

 and giving to /, F, and cr l like expressions to those in the first 

 case, the ratio of a to cr 1 will be found to be 



1 + H? {(? " I) cos b <r + ^ + J sin %+ c 'i)}<» s2 e > 



o 

 b being put, for brevity, for — . To obtain this result, — m\ 



was substituted for m! 2 , and c\ for d r The same substitutions 

 being made in the equation (X), there results for determining c\, 



tan^ + ^H-^.^^. 



The condensation at any point of the second hemispherical 

 surface can now be found by putting c for r in the general value 

 <r, and eliminating c\ by means of the last equation. These 

 operations give the following exact result : 



ar 7n' l Kab b c' 2 cos 2 6 



3m(81 + 96 V - 26 V + b 6 c 6 ) * 



Since be is by hypothesis a very small quantity, we have very 

 nearly 



cr __ -. 3277-Wj icac 1 cos 2 6 



V x ' 27m\ 6 



By the same argument as before, in order to include terms of the 

 second order, S must be put in the place of <r 1 . Then the total 

 pressure on the second hemispherical surface not counteracted by 



