Gl Prof. Challis's second Reply to Mr. Airy 



Mr. Airy's reasoning consists of two parts. The object in 

 the first is to show, that the motion of a fluid, when directed 

 to or from a centre, must be the same at the same time at all 

 equal distances from the centre. In the other it is argued 

 that motion resulting from two components, one in the direc- 

 tion of a straight line from a centre, the other in a direction 

 perpendicular to this, is inconsistent with the hydro-dynamical 

 equations. 



The mathematical reasoning under the first head is the 

 same as if the problem to be solved had been thus enunciated : — 

 To determine the motion of a fluid, assuming the whole velo- 

 city to be directed to or from ajixed point : in other words, 

 the motion of a given particle is assumed to be rectilinear. It 

 will be readily admitted that in this instance the motion is al- 

 ways the same at the same distance from the centre, because 

 if the motions of two contiguous elements at the same distance 

 were at any instant different, their densities would be different, 

 and their motions would cease to be rectilinear. In the case 

 of motion under discussion, viz. that of fluid put in movement 

 by an oscillating sphere, the motion of a given particle of the 

 fluid is obviously not rectilinear, and it is not therefore ne- 

 cessary for me to show why a solution which I propose for 

 this case is not included in the more restricted one of recti- 

 linear motion. 



My concern is rather with the second part of the argument. 

 The reasoning here is not brought to a conclusion. By fol- 

 lowing out the investigation I find that not only the three 

 equations Mr. Airy adduces, but three others also, which 

 have equal claims for consideration, are exactly verified by 

 the kind of motion which Mr. Airy considers to be impos-^ 

 sible. I proceed to give the mathematical reasoning. 



The letter P being for shortness' sake substituted for the 

 Napierian logarithm of g, the four hydro-dynamical equations, 

 sufficiently approximate for the proposed instance of motion, 

 become 



dF j du 



1^ +i -d7 = M 



rfP » dv ... 



'dJ + t -Tf = ° W 



d P , d 10 , . 



U +! '--dT = ° ( c -> 



l* + |£ + *1 + i?L = o (d.) 



dt dx dy dz 



