10 
To deduce from this the impulsive pressures due to an 
impulsive change in the velocity of the sphere, we write r 
for the short time of action of the impulse, multiply equation 
(1) by dt and integrate from t~o to t—r. Let V' be the 
final value of Y. The last term in p will contribute nothing 
to the final equation, and we obtain merely 
-- i$(V'-V)eo S 0 (2) 
O 
Now, the motion in question having been produced from 
rest in an infinite liquid by the motion of the sphere, the 
motion is the same at every instant as that impulsively pro- 
duced from rest by giving instantaneously to the sphere its 
velocity Y (Thomson and Tait, vol. i., part 1, page 328. 
Kirchoff, Yorlesungen, chap xix., &c.). The value of the 
impulsive pressures due to the instantaneous production 
of the velocity Y of the sphere, and which produces the 
consequent velocities in the fluid is 
7T 
P 
= |^-Ye°s0 
( 3 ) 
7T 
where—-, of course, only differs from <j> by a constant. 
Now, there need be no difficulty in supposing a fluid 
capable of transmitting a tension, provided this tension 
tends to produce no discontinuity of motion or disruption 
between near parts, as in the present case. But at the 
surface of the sphere we get 
- = £aVcos0. 
p 
and thereupon a maximum of cohesion is needed of the 
order ap Y at the extreme rear of the sphere, which is double 
the mean value required.* Considering the coefficient of co- 
hesion as constant, all other things being alike, the velocity at 
# The pressure in front, or tension behind a sphere of radius 1 foot, 
started impulsively with unit of velocity in water, would be roughly 
represented by that due to a slab of granite i inch thick falling through 
3 inches* 
