since the other will then be satisfied by the exclusion of the odd powers of y from 4. The equation (A) gives, since here Ay-8, Similarly, if y=0 is the equation of the bottom of the z canal. If moreover -.,0. be the equation of the upper surface, .t and (c) becomes, since is of the order of the disturbance, provided as above we neglect (disturbance)". Again, the condition (2) gives by equating separately the coefficients of powers and products of y and z, equating now separately the coefficients of f' and f", we get 0 = () 1 gy dA dB dy dx +2 Adx+ Bdx * ydə° √gy k = A2ßy = A2 Hence if we neglect the superfluous constant k√g, the general integral of (4) is, (: A=ß3y1), gry and the actual velocity of the tuid particles in the direction of the axis of x, is neglecting quantities which are of the order (w) compared with those retained. If the initial values of and u are given, we may then determine ƒ' 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 direction of a positive, we have By dx Suppose now the value of F" (x) = 0, except from x = a to x=a+a, and dx to be the corresponding length of the wave, we have Lastly, for any particular phase of the wave, we have (7). 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 represent the variable breadth of the canal and y its depth, and height of the wave, u = actual velocity of the fluid particles . dx = length of the wave y1, |