326 Mr. Airy in reply to Prof. Challis 



that is, if in every different direction it is the same, or different 

 according to any arbitrary law ; but if in different points of 

 that direction, it is inversely as the square of the distance from 

 the centre. This shows that the same quantity of air passes 

 in every second of time in the same direction through every 

 section of each small solid angle. As we cannot in any prac- 

 tical case provide for the incessant supply of air which this 

 will require, we may lay aside this form of solution. I may 

 remark, that this solution is made possible only by the neglect 

 of terms of the second order of velocities ; for it would easily 

 be found, upon examining the forces corresponding to this 

 motion, that there are forces depending on the squares of ve- 

 locity which are inconsistent with this motion. 



3rd. Motion from the centre is possible without any other 



motion if the velocity[is expressed by - — ^^ — — — 5 



-\ — ; — — ^^-g , where ka^ ■=■ 1. This expression 



represents two series of waves, one rolling from the centre 

 and the other towards the centre : but, for each of these waves, 

 the phase and the intensity are the same in every direction 

 from the centre. 



The form of solution adopted' by Professor Challis is 



cos Q . y v—^ ) _ J 1.^—^ ; \ jf ^ jjg ^i^g g^j^jg £-j.Qj^ 



which Q is measured, cos = — , and this expression be- 



^'/'{^ — ^t) z.f(r—at) „!• • • 



comes —^ — -^ — — ^^^ • ^ 'I's expression is not 



included in any of the formulae already discussed ; and it 

 evidently will not satisfy the conditions found above ; and it 

 cannot therefore apply to the problem before us. 



II. But suppose (as an assumption of which the possibility 

 is to be tested), that the motion from the centre is represented 



by the formula -^^ — > as assumed by Professor Challis, and that 



this motion is accompanied by other motions normal to the 

 radius for every point. These motions may be represented 

 in the most general way by one velocity P in the plane pass- 

 ing through z and r, and another velocity Q perpendicular 

 to that plane. Resolving all the assumed velocities in the 

 directions of x, y^ and z, we have 



