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since the other will then be satisfied by the exclusion of the odd powers of y from $. The equation (A) gives, since here Ary-B,

Octobe..... (when y= B) .......(a). Similarly, if 2 -% = 0 is the equation of the bottom of the

Orient ..... (when = = ry]............. (6). If moreover 8-50.. = 0. be the equation of the upper surface, Ouvre un mese

... (when : = 5)... (). But here p=0); .. also by (2 Substituting from (3) in (5) we get

0=4,+ 4,9+&c. - any content contre 1 +&c.}'; or neglecting quantities of the order ro,

0=4,+4,4-e mer........... (6%), Similarly (a) becomes

0=948 - ...................... (a), and (c) becomes, since & is of the order of the disturbance, 0 = ,

when z = 5
or neglecting (disturbance) 2 = 0

di da

provided as above we neglect (disturbance)".

Again, the condition (2) gives by equating separately the coefficients of powers and products of y and %,

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It now only remains to integrate this equation,

For this we shall suppose B and y functions of a which vary very slowly, so that if written in their proper form we should have

B=th (w), where w is a very small quantity. Then,

a cargo (eor), = *%"(x), &c. Hence if we allow ourselves to omit quantities of the order w', and assume, to satisfy (4),

$. = Af (0+X), where A is a function of x of the same kind as ß and y, we have,

omitting (A),

dan - Af",

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Substituting these in (4), and still neglecting quantities of the order w', we get

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equating now separately the coefficients of f' and f", we get


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The first, integrated, gives

and the second

Hence if we neglect the superfluous constant kvg, the gene ral integral of (4) is, (: A=Bby t),

therefore, by (c'),

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and the actual velocity of the Huid particles in the direction of the axis of æ, is

neglecting quantities which are of the order (w) compared with those retained.

If the initial values of 5 and u are given, we may then determine f' and F', and we thus see that a single wave, like a pulse of sound, divides into two, propagated in opposite directions. Considering, therefore, only that which proceeds in the direc tion of a positive, we have

................. (5).

sashay (e - Some


Suppose now the value of F'(x) = 0, except from x= a to a = a +a, and dr to be the corresponding length of the wave, we have

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Hence the variable length of the wave is

&x=a.1gy ........................... (7). Lastly, for any particular phase of the wave, we have

- Someone = const :

de = V gry ...

.......... (8) is the velocity with which the wave, or more strictly speaking the particular phase in question, progresses.

From (5), (6), (7), and (8) we see that if ß represent the variable breadth of the canal and yy its depth,

$= height of the wave ce byt,
u = actual velocity of the fluid particles ce phy

dx = length of the wave ceny
and = velocity of the wave's motion = Ngy.




From the Transactions of the Cambridge Philosophical Society, 1838.

(Read Dec. 11, 1837).

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