ON THE HYPOTHESIS OP UNDULATIONS. 367 



also ^=0^ ^=0^ J^ = 0_^, 



d<r ^ dx dx- ^ dw^ dwdy ^ d.vdy' 



dxj, _ df d?y\, _ d\f d'^ _ d<p df 



di/ dy dy' dy'' dxdz dz dx' 



dz dz dz^ dz'' dydz dz' dy' 



Hence by substituting and reducing, it will be found that 



\R R'l \d.v' df*(j)-'dzV dy^'dx" dx^'dy" dxdy' dx' dy (p^' dz' [dw' dfj 



f d^(j) "^ ''^^^ "'^- 'ff"^ 



<t> 



d'(t> _^ d£\ idT df\ 



d? (j)'' dz') [da)' df) ' 



For our purpose we require an expression for r= + —, for any point on the axis of z. Now 



R R 



since by hypothesis u = and d = at all times for all points on the axis of z, it follows from 



df , df , df ^df 



dy dx dy 



foregoing equation gives, 



the equations u = <b -^ and v = d) - , that — ^ = and -— = o for these points. Hence the 

 ' dx ' dy dx dy 



R'^ R') (h' dz dx' dy' ^ ■'■ 



» 



(b dz dx' dy^ 

 But equation (5) becomes for motion along the axis of z, 



dz \R^ R') ^ ' ds^ 



Consequently putting a' (l + k) for a', and substituting from equation (6), 



d^(t> 1 Id'f d-f\ 



d?-rf[d'dp)^-' (^^- 



In this equation the coefficient of (p, not containing w and y, is a constant, and we may assume it 

 equal to — nr. It hence follows that 



d^ + '^'^=° f«)- 



I shall here take occasion to remark that since — -^ = when the velocity - is a maximum, it 



dz^ dz 



appears by equation (8) that = in the same case. Hence also ?< = 0, and v = 0. Consequently 



the assumption made heretofore that the arbitrary quantities c and c each = 0, was equivalent to 



assuming that the transverse velocity vanishes when the velocity is a maximum along the axis of z. 



It appears also from the expressions for u, v, and w, that, when the velocity -— = 0, and (p is 



consequently a maximum, u and v are each a maximum. 



At the same time that the equation (8) is true, we have by equation (3) neglecting the small 

 terms, and by wiiat has now been proved, 



?l---3- <»)■ 



