TEE LA WS OF FLUID RESISTANCE. 95 



proceed to show that this is also the case whatever may be 

 the outline of the pipe, provided that its beginning and 

 end are in the same straight line. 



Assume a pipe bent into a complete circular ring with 

 its end joined, and the fluid within it running with velocity 

 round the circle. The inertia of this fluid, by centrifugal 

 force, exercises a uniform outward pressure on every part 

 of the uniform curve ; and this is the only force the fluid 

 can exert. This outward pressure tends to enlarge or 

 stretch the ring, and thus causes a uniform circumferential 

 tension on each part of the ring. 



Now take a ring of twice the diameter and suppose the 

 fluid to be running round it with the same linear velocity 

 as before. The diameter of the curve being doubled, and 

 the speed being the same, the outward pressure due to cen- 

 trifugal force on each linear inch of the ring will be halved ; 

 but since the diameter is doubled, the number of linear 

 inches in the circumference of the ring will be doubled. 

 Since, then, we have twice the number of inches acting, 

 each with half the force, the total force tending to enlarge 

 the ring will be unaltered, and the circumferential tension 

 on the ring caused by the centrifugal force of the fluid will 

 be just the same as before. 



In the same way we can prove that in any number of 

 rings of any diameters, if the linear velocity of the fluid 

 in each is the same, the circumferential tension caused by 

 the centrifugal force of the fluid will also be the same in 

 each. 



Now let us take each of these rings and cut out a piece, 

 and then join all these pieces together so as to form a con- 

 tinuous pipe, as in Fig. 4, and suppose the stream of fluid 

 flowing through the combined pipe, with the same linear 

 velocity as that with which it was before flowing round 

 each of the rings. The fluid in each of the segments will 

 now be in precisely the same condition as when the seg- 

 ment formed part of a complete ring, and will subject each 

 piece of ring to the same strains as before, namely, to a 

 longitudinal tension or strain, and to that only. And 

 since we have already seen that the tension is the same in 

 amount in each ring, the tension will be the same at every 

 point in the combined pipe. 



This being so, if we imagine the pipe to be flexible (but 



