182 Prof. Challis on the Hydrodynamical Theory 



contribute nothing to the condensation at the surface of the 

 sphere if the plane cuts the second hemispherical surface, be- 

 cause in that case their axes are intercepted by the first 

 hemispherical surface ; whereas the corresponding components 

 will be fully effective, as respects their condensation, at the 

 first half-surface, because, although they are reflected at this 

 surface, the effect of the condensation accompanying this re- 

 flection may, by what is said in art. 13, be left out of con- 

 sideration. It appears from this argument that to find the 

 whole accelerative action on the sphere we have only to find 

 the accelerative action, on the first hemispherical surface, of 

 so much of the incident waves as is composed of the intercepted 

 vibrations, inasmuch as this action is not opposed by corre- 

 sponding condensations on the second hemispherical surface. 



15. The investigation will be sufficiently approximate for 

 our purpose after making the following simplifications of the 



e\ 2 



expressions for u\ co, and a in art 11. Since /c 2 = lH s-, and 



1 7T 2 



k? — l=-r, it follows that e= —ttt„ and that the value off 



(art. 10) becomes 1 ^p. As, for each set of component 



vibrations, we shall have to take account only of distances r from 

 its axis which are less than the diameter of the sphere, and the 

 ratio of this diameter to \ is by hypothesis very small, it will 

 be seen that the second term of the expression for f will always 

 be excessively small compared with the first, and in the present 



investigation may be neglected. Thus we have/=l, -~ = 0, 



and consequently 



. y 2fc 5 m 2 

 w=*m sm^tj 5 — cos zqg, 



OCl 



o) = 0, 



V 



*~ 2fc 4 a/ 



/civ mf . o „ 

 °"= — + o^5- sm & 



Km . y m?/2ie e 1 \ „ , 



4/ra 



o'' 



the value of w being substituted in the expression for a. Ac- 

 cordingly these values of w, co, and a relative to a given axis 

 apply to all points of the portion of the transverse plane con- 

 tained between the spherical and cylindrical surfaces. Taking 

 C to be the position of the centre of the sphere and P a point 

 on its surface, let the line PC make an angle with the line 

 CO drawn from C to the origin of the ordinates z in the di- 



