324 Prof. Challis on a Mathematical Theory 



dY 

 always in the direction of the motion, we shall have -t7=0, and 



f{t) a constant which is given for a given line of motion. Thus 

 the equation becomes 



\^ 

 K^a^ Nap. log p + -^ = C. 



Conceive a straight line to be drawn parallel to the direction of 

 the stream through the centre of the sphere, cutting its surface 

 in the points A and A', and let the motion of the stream be in 

 the direction from A to A'. Then it is evident that at these two 

 points the velocity is zero, and that every line of motion along 

 the surface of the sphere passes through them. Hence, since 

 the above equation proves that the density is the same on the 

 same line of motion wherever the velocity is the same, it follows 

 that the density is the same at A' as at A. Again, the motion 

 of the fluid being by supposition small compared with Ka the 

 velocity of propagation, the velocity at any point P of the sur- 

 face of the sphere may be assumed to be compounded of the 

 velocitv of the stream and the velocity impressed on the fluid by 

 the reaction of the surface of tlie sphere. If the velocity of the 

 stream be "\V, and the radius PC make an angle 6 with CA, this 

 impressed velocity is W cos 6, the surface of the sphere being 

 supposed to be perfectly smooth. Hence compounding this 

 velocity with the velocity of the stream, the actual velocity of 

 any particle in contact with the surface is Wsin^. At the 

 points corresponding to ^ = 90°, the velocity is the same as that 

 of the stream, there being no reaction of the surface at these 

 points. Consequently if p, be the density of the stream where 

 it is undisturbed, we shall have 



K^a^ Nap. log p, + -y- = C, 



and in general 



Nap. log — = -— cos^ 6. 

 ' ^Pi a 



This equation gives the density and pressure at every point of 

 the hemispherical surface on which the stream is directly inci- 

 dent, and shows that there is a maximum of pressure at the 

 point A. The pressure on the other half of the surface is deter- 

 mined by the following considerations. 



A given particle of the fluid, setting out from A or its imme- 

 diate neighbourhood, and moving in contact with the surface of 

 the sphere, will acquire the velocity W on reaching a point B 

 distant 90° from A, as may be thus shown : — Let v be the velo- 

 city of the particle at any time. Then the force accelerating it 



