Prof. Challis on the Principles of Hydrodynamics. 101 



these substitutions and performing the integrations, it will be 

 found that 



■< sin -^(R— /ca^ + c)H-sin— (E, + ««^ + c) r 

 < cos— (R— /c«^ + c) — cos— - {U + Kat-^-c) >. 



27rR2 ■ 



kruKK 



^ 27rR2 





R« 



It will suffice to remark, that the previous general solution 

 accords with this solution on giving a particular form to the 

 arbitrary functions, and that the change of the relation between 

 the velocity and condensation in passing from free to constrained 

 motion is confirmed. 



If the fluid were put in motion by a smooth solid sphere 

 moving in it in a given manner, the velocity impressed at each 

 point of the surface at any instant would be known. The direc- 

 tions of the impressed velocities determine the directions of the 

 initial propagations, and as we found in the general considera- 

 tion of the mutual action of the parts of the fluid no other than 

 rectilinear propagation, it follows that the initial direction of 

 propagation continues to all distances from the centre. The 

 motion will therefore be the same for an indefinitely small portion 

 of a wave, as when the disturbance is made at a given distance 

 from a fixed centre ; but the direction of propagation at a given 

 point will be continually changing, and the motion of a given 

 particle will be curvilinear, being directed to or from positions 

 which the centre of the sphere had in successive instants. I see, 

 therefore, no reason from these new researches to alter the solu- 

 tion of the problem of resistance to a small sphere in an elastic 

 meditim, which I have given in the paper already referred to, 

 excepting that Ka must be substituted for «. The effect of this 

 change will be to diminish the condensation, and therefore the 

 pressure, for a given velocity, in the ratio of 1 to /c ; and the 

 coefficient of buoyancy and resistance, instead of being 2, will 



be 1 -t- -, or 1*844, which accords well with observation. 



K 



The same principles would suffice to determine the motion 

 when the disturbance is in the directions of the normals of any 

 continuous surface, and is a function of the distance from the 

 surface ; and also to ascertain the effect of the motion of such a 

 surface in a given manner in the fluid. 



I have now completed the explanation of my hydrodynamical 

 theory sufficiently to enable any mathematician who may regard 

 it with favour, and who may have more time for these researches 



