Frederick Guthrie on Stationary Liquid Waves. 381 



The work done and potential at maximum excursion vary di- 

 rectly with displacement. What is true for a plane is of course 

 true for a rectangular trough. It can also be shown to bs true 

 for disturbances in circular troughs from fig. 3, § 11, and fol- 

 lows indeed without further proof from the equal tetrahedron 

 and wedge having been generated by the rotation of a triangle 

 balanced about one of its lines of gravity. The fact that the 

 surface of both wave-systems is curved, and that this curve in- 

 volves the cohesion and viscosity of the liquid, puts the exact 

 solution of the question beyond the present power of analysis. 



§ 24. Absolute comparison of circular waves with pendulum. — ■ 

 It appears that both the binodal waves in circular troughs 

 and the binodal waves in rectangular troughs follow the pendu- 

 lar law, and that the mononodal waves in rectangular and 

 circular troughs do so nearly. It follows from § 23, where it is 

 shown that no dead weight has to be stirred, that, if the liquid 

 has no viscosity, not only should the pendular or torsion law 

 hold good, but the actual rate of undulation in the binodal 

 system in the unhampered or circular wave should be the same 

 as that of a pendulum of radius equal to half the wave-length — 

 that is, to the diameter of the cylinder. If a pendulum of length 

 I be suspended over a circular trough of radius /, it keeps time 

 very nearly with the fundamental undulations in the trough. 

 This can be beautifully shown in an experiment where the pen- 

 dulum-length can be altered till it is isochronous with the pul- 

 sations. But it appears best from the data already given. 

 Taking the length of the seconds-pendulum at London at 994 

 millims., the number of oscillations which pendula of the radii 

 of the circular troughs give is shown in the following Table, 



*=6oy^ 





(1) 

 Radius of 

 trough, r. 



Number of 



liquid 

 undulations. 



(n<). 

 Number of oscil- 

 lations of pendu* 

 lum of length r. 



A 



297-5 



228 



183 



157 



150 



106-9 

 122-4 

 136-7 

 1490 

 151 



108-6 

 125-3 

 139-4 

 1509 

 154-4 



B 



i C 



D 



E 





The pendulum-oscillations exceed the liquid ones, on an 

 average, 2'8; on the corrected average, under the omission of 

 E, 2-05. The mean per-cent. deficit is about 1*5. Consi- 

 dering that the liquid undulations do not, even in the largest 

 troughs, sustain themselves for more than 20', while a pendulum 



