362 E.. Loomis on vibrating Water-falls. 
minute very satisfactorily. The descending sheet of water was 
very uneven, being marked by numerous ridges, probably caused 
by the flash boards having been only partially carried away. 
spent two days in almost constant attempts to count the vibra- 
tions, and my result was an average of 148-68 vibrations per 
minute, corresponding to a depth of 441 feet water by the 
gauge. If we reduce this depth according to the rule determined 
at Holyoke, it would give a depth of 3°68 feet on the crest of 
the dam; but according to Mr. Coolidge’s measurements, it 
would indicate a depth of exactly 3 feet on the crest of the dam. 
Comparing all these observations, they seem to favor the con- 
clusion that the time of one vibration increases as the depth of 
water on the dam increases; but the discrepancies are so great 
that even this conclusion might be called in question, if it were 
not confirmed by the observations at South Natick. If we take 
an indiscriminate mean of all the observations, we find the 
average time of one vibration to be 0-409, corresponding to a 
depth of water on the crest of the dam amounting to 4 feet, and 
a temperature of the water 44°. : 
column of air 805 feet in length, 30 feet high, and 7 feet 
wide, according to the principles stated on page 359, would make 
one vibration in 05-766; or if it vibrated in two portions, the 
time would be 0383. A heavy body would fall through a 
space of 4 feet in 08-499. The observed time is intermediate 
between the times computed by these two methods. 
t seems impossible to explain all the preceding observations 
upon the supposition that the air behind the sheet of water is 
the sole vibrating body, following the laws which have been 
established in the case of organ pipes, The observations at 
South Natick naturally suggest the idea that the vibrations — 
originate in the sheet of water itself ; the time of one vibration 
being nearly proportional to the square root of the depth of 
water on the dam. 
8 there any known principle which will enable us to explain 
vibrations of this description 2 : 
It has been suggested that such vibrations may originate in the 
unequal velocity of the upper and lower strata of the sheet of wa- 
ter which cag over the dam. If there were no friction, the lower 
stratum should have a velocity equal to that which a heavy 
body would acquire in falling through a space equal to the height 
of the water above the dam. The upper stratum has only the — : 
velocity of the stream, combined with that which is imparted 
could be computed from the depth of water on the dam. But — 
