190 Mr CHALLIS's RESEARCHES IN THE THEORY 



because when the fluid is of unlimited extent, there is no cause* to 

 produce motion at any point of the line, but the impression made at P, 

 which is transmitted instantaneously, varying at different distances 

 from O according to the law of the inverse square. Hence if ^ be a 



point in OP produced, and OR = r, the velocity at R — —, = ^ — . 



Let ADE be the axis of x, a vertical through A the axis of z reckoned 

 positive downwards, and a line through A perpendicular to the plane 

 of these two the axis of y. Then if the co-ordinates of R be x, y, %, 



we shall have r- = (:r — a)" -I- y^ + (s + 7)' ; and cos0= . Therefore 



the velocity {v) at R, 



•A-A I'? 



VoH^X-a) 



Ka;-ay^ + y' + (» + 7)-}5" 



And 



Hence 



dv dv da re rr :i 4. ..k 



~j- = 7- • 77 > (lor ^ and 7 are constant), 



_ F«^(3cos-e-l) (la 

 r" ' dt 



rV(3cos*^-l) 



/^rf,=/(o-g5:(3cos'e-i). 



Therefore, gravity being the only force acting on the fluid, the pressure 

 ip) at R, 



* This cannot be said of the parts of the fluid adjacent to the radii produced which pass 

 through the circle in which the surface of the water meets the surface of the sphere, because 

 the water outside of the conical surface formed by these radii must be put in motion by that 

 within by reason of the difference of pressure occasioned by the motion. On account of the 

 difficulty of estimating this effect, it is left out of consideration in our solution, which can 

 therefore be only considered approximate. 



