ON THE HYPOTHESIS OF UNDULATIONS. 373 



/Ic d s Sn \/k . . / k oc d s 



have to the same approximation, s = .S' cos « X/ -r ; -r- = ^^ . sin w 'V t »" • - ; and -— 



'^^ 2 dx y/2 2 r dy 



— sin n sj - r . - . Consequently substituting in (17), and putting the arc for the sine, 



v/2 2 r 



dcj? da^ Sr^k I da-, dtrA , ^ 



d,r^ dy^ '3. \ dx dy I 



Such a value of ixi is now to be found as will satisfy the equations (18) and (20) for small values 

 of ,1- and y. The equation <jx = mcoi,(gx + hy) will be found to answer. For substituting in 

 (18) we get the equation of condition, 



g' + li' - n^k = (21). 



and by substituting in (20) we have, 



to' (g' + A') sin' {g.v + hy) {gx + hy) sin {gx + hy) = 0. 



Hence, putting the arc for the sine, 



. . Sn k 

 m(g' + A') — = (22). 



g 

 Comparing this equation with (21), it follows that m = — , and consequently that 



s s 



cr, = — cos (g.v + hy). Similarly we should find that o-j = — cos (g'x + fi'y). But since s = cti + ct;, 

 we must have. 



A A S , , . S 



;n\' - ■ 



S cos n \/ - r = — cos (gx + hy) -\ cos (g'x + h'y). 



Expanding to terms involving the squares of the small quantities, 

 n^kr^ = {gx + hyY + {g'x + h'y)' 



= {^ + g") x' + {k' + h'"~) f + 'i{gh^ g'h') xy. 

 This equation accords with (21) if ^' = A and h' = - g. Thus we have 



S S 



cr, = - cos {gx + hy), (T2 = - cos {hx - gy). 



It hence appears that co becomes identical with cr, by changing the directions of the axes of 

 co-ordinates through 90". Since ^ + A' = n^k, we may assume that g = n \^k cos G, and 

 h = n y/k sin 0. Then, 



S — 



cTj = — cos \n \/k i^x cos Q + y sin 0)}, 



and (Tj = — cos j n \/k {x sii\Q - y cos G) \ . 



V — S — 



As 9 is quite arbitrary, let it equal 90". Then ct, = —co%7i\/ky, and a = — con n\/kx. The 



axes of ,r and y are now evidently in the planes of symmetry, and these last values of <t and 



