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a velocity proportional to the sharpness of the curvature. So 
that not only does the ring continually change its shape, but 
one part is continually falling behind, and then overtaking the 
other. 
These are some of the forms of fluid motion which imagina- 
tion or reason had failed to show us, but which have been 
revealed by the simple process of colouring the water. 
Now that we can see what we are about, mathematics can be 
most usefully applied ; and it is expected that when these facts 
come to be considered by those best able to do so, the theory of 
fluid motion will be placed on the same footing as the other 
branches of applied mechanics. 
