278 Prof. Challis on the Effects produced by Fog and 



number of these small globules. The problem of determining 

 the velocity and condensation of the air due to the reaction of a 

 single small sphere on which such vibrations are incident, I have 

 already discussed and applied on several occasions. (See the 

 Number of the Philosophical Magazine for June 1864, pp. 457 

 & 458, that for October 1865, pp. 262 & 263, and the solution of 

 Example VI. in pp. 279-287 of < The Principles of Mathematics 

 and Physics/) Suppose the velocity at any point due to the 

 reaction of the sphere to be resolved along, and perpendicularly 

 to, the radius vector drawn to the point from the sphere's centre, 

 and let the former resolved part be U, and be reckoned positive 

 in the direction from the centre, and the other be W, and be 

 reckoned positive when it is in the same direction as that of the 

 incidence ; also let a be the condensation at the same time at 

 the same point. Then if r and 6 be polar coordinates of the given 

 point, the origin being at the centre of the sphere and 6 being 

 the angle which the radius vector makes with an axis drawn in 

 the direction of incidence from the origin, and if b be the radius 

 of the sphere, and T the velocity of the incident vibration at the 

 time t in a plane through the origin transverse to the axis, the 

 solution of the above-mentioned problem gives 



TT Tb s a w Tb 3 . p 8 dT b s a 



Now auy reflection of the incident waves, caused by the reaction 

 of the sphere, will be proportional to a<r, and must be extremely 



small on account of the smallness of the factor - • -7-. If, for 



a dt 



instance, we suppose that T=m sin-^— , A, being the breadth of 



A, 



the waves, the maximum value of that factor becomes — - — , and 



A. 



therefore, by reason of the ratio -, is excessively small for all 



such values of m and X as those which are applicable to the ex- 

 periments. 



b* 

 Again, on account of the small size of the globule, the factor ^-^ 



&r 



will become excessively small at small distances from its centre, 



because such distances may still be many multiples of b. Thus, 



by reason of the two factors combined, the reflection from a single 



globule may be so extremely minute at all sensible distances 



from its centre, that the sum of the reflections from a very large 



number of globules contained within a given finite space might 



only generate reflex waves of moderate magnitude and coming 



