due to Elasticity of the Earth's Surface. 417 



mountains and valleys whose outline is given bjx= —h(l — 2z/l) 

 from z=0 to I, and x = from z = to — /. To utilize the 

 analysis of the last section, it is necessary that the mountains 

 and valleys should present a simple-harmonic outline. Hence 

 the discontinuous function must be expanded by Fourier's 

 method. Known results of that method render it unnecessary 

 to have recourse to the theorem itself. It is known that 

 ±i 7r —xg =z sin + 1 sin 20 + ^ sin 30 + ... 

 -10=- sin + 1 sin 20-^ sin 30 + ... 



1tt + 0=- J COS0+ q2 cos 30+ ^cos50 + ... V, 



the upper sign being taken for values of between the infi- 

 nitely small positive and + ir, and the lower for values between 

 the infinitely small negative and — ir. 



Adding these three series together, we have 



2{isin20 + ^sin40 + ..} + -{cos0 + icos30 + J2Cos50 + ..} 



IT I O" J 



equal to ir— 20 from = to +7r, and equal to zero from 

 = to — ir. Hence the required expansion of the disconti- 

 nuous function is 



27? 

 - — Hsin20 + isin40+...[, 1 



Ik 1 1 * (6) 

 2 ) cos + ^ cos 30 + ^2 cos 50 + . . . j- J J 



where 0——. (T) 



for it vanishes from z — — I to 0, and is equal to — A(l — 2zjV) 

 from z=0 to +/. 



Now, looking back to the analysis of the preceding section, 

 we see that, if the equation to the mountains and valleys had 

 been x— —A sin (z/b), a. would have had the same form as in 

 (2), but of course with sine for cosine, and 7 would have 

 changed its sign and a cosine would have stood for the sine. 

 Applying then the solution (2) to each term of our expansion 

 separately, and only writing down the solution for the surface 

 at which # = 0, we have at once that 7 = 0, and 



a= ^Ml in2 + i 0+i + ...},] 

 7tv tt(2 2 4 2 6 2 J • I 



, 9 wh 2/ i /)_l. 1 tta^ 1 ^ L \ \ 

 + y - 5- « cos + -ko cos 30 + ^ cos 50 + . . . . 



