TRACT 38.

PROBLEMS.

351

Affair), for the time, we have t or —= i J— X ■■■ x —- :

the fluents of which ffive t = ' V — x arc to versed sine —

° 2 e a

and radius i, the general expression for the time. And

when x = a, it becomes t — -$pV— for the time of moving

^ l

from g to f, p being = 3 - 1416 ; and consequently ip\/~the time of a whole vibration from G to E, or from c to A.And which therefore is the same, whatever As is, the wholelength l remaining the same.

And the time of vibration is also equal to the time of thevibration of a pendulum whose length is 4 1 , or half the lengthof the axis of the fluid. So that, if the length l be 78^ inches,it will oscillate in 1 second.

Scholium. This reciprocation of the water in the canal, isnearly similar to the motion of the waves of the sea. Forthe time of vibration is the same, how'ever short the branchesare, provided the whole length be the same. So that whenthe height is small, in proportion to the length of the canal,the motion is similar to that of a wave, from the top to thebottom or hollow', and from the bottom to the top of thenext wave; being equal to two vibrations of the canal; thewhole length of a wave, from top to top, being double thelength of the canal. Hence the wave will move forward bya space nearly equal to its breadth, in the time of two vibra-tions of a pendulum whose length is (\l) half the length ofthe canal, or one-fourth of the breadth of a wave, or in thetime of one vibration of a pendulum whose length is thewhole breadth of the wave, since the times of vibration areas the square roots of their lengths. Consequently, waveswhose breadth is equal to 39-§ inches, or 3fA feet, will moveover 3|~|feet in a second, or 195|.feet in a minute, or nearly2 miles and a quarter in an hour. And the velocity of greateror less waves will be increased or diminished in the subdu-'plicate ratio of their breadths.