100 Prof. Challis on the Principles of Hydrodynamics, 



number of normal vibrations be propagated equally in all direc- 

 tions, tbeir axes passing through the centre, the resulting velo- 

 city and condensation will be functions of the distance from the 

 centre, and the resulting motion will be in lines drawn through 

 the centre. The summation of the effects of all these vibrations 

 should, therefore, give a solution in accordance with that obtained 

 above. But it must be remarked, as before, that as this com- 

 position of the motion is an analytical conception, and not a 

 physical reality, the reasoning can only extend to cases in which 

 /=1 — cr*, and r is very small compared to \, This being pre- 

 mised, take any point P on a line drawn in a fixed direction 

 through the centre, and let ON be the direction of any axis 

 of vibration such that the zPON = a, and the plane PON makes 

 the angle 6 with a fixed plane passing through OP. Draw PN 

 perpendicular to ON, and let OP = R, ON=r, PN=r. The 

 condensation produced at P by a single set of vibrations is 



— • -J-, and the sum of the resolved parts of the direct and 



a az ^ 



transverse motions in the direction OP is 

 j,d^ , .df . 



The resolved parts perpendicular to OP destroy each other. The 

 number of axes that pass through the element zmnudO ,2du 

 varies as the element directly and as z'^ inversely, and may there- 

 fore be assumed equal to k sin a du dd. Consequently we have 

 to obtain the two integrals, 



k 11 If-r^ cos a H- <^ -^ sin a j sin a c?a d6, 



the integrations being taken from ^=0 to 6=27r, and from 

 a=0 to «=7r. Now let 



80 that 



, \m 2ir , ^ , . 



9=— ^cos— (z—Kat-\-c), 



d6 . 2'ir , , , 



-7^ =m sm —- [z—Kat -f c) . 



Since, from what was said above, the first order of approximation 

 is alone permitted, we must have /= 1, and 



df o ^r 



because eX^=4. Also R sin a=r, and R cos u=z. By making 



