112 HEPOKT — 1882. 



outline. Hence the discontinnous 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 — 



± 1 ;r — i a = sin + 1 sin 20 + ^ sin 39 + 



_ 1 e = — sin + i sin 29 — ^ sin 36 + .... 



-^ ,r q^ = - I cos + ^2 cos39 + i cos 59 + ... j 



The upper sign being taken for values of 9 between the infinitely small 

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

 negative and — tt. 



Adding these three series together we have — 



2 I -I sin 29 + :i sin 49 + . . j + i | cos 9 + .J^ cos 39 + -^^cos 59+ . . j 



■equal to t — 29 from 9 = to + tt, and equal to zero from 9 = to — tt. 

 Hence the required expansion of the discontinuous function is — 



-^^' |isin29 + i sin49+ . . .j 



- ^ ( cos + J, cos 39 + i-, cos 59 + . .1 (6) 



Avliere ^ ^ f (^) 



For it vanishes from z=: — I to 0, and is equal to — /i(l — 2r:/l) from 

 s = to + Z. 



Now looking back to the analysis of the preceding section we see that 

 if the equation to the mountains and valleys had been x = — /isin(;j/5), 

 a would have had the same form as in (2) but of course with sine for 

 cosine, and y would have changed its sign and a cosine would have stood 

 for the sine. Applying then the solution (2) to each term of our expan- 

 sion separately, and only writing down the solution for the surface at 

 which X =■ 0, we have at once that y = 0, and 



a = 0^l\lsm2d+l^sini9+ ;^. sin G9 + . . .] 



TTV TT L 2'' 4'' 0- J 



+ ^2!^ . ?J ( COS0 + l. COS 30 + ^3 COS 50 + ... 1 (8) 



The slope of the surface is — '* or f " ; thus 

 ^ dz I do 



■di 

 dz 



a^^gwk 1 1 cos 20 + i cos 40 + I cos 69 + . . .1 



_ ^i 2 f j^g _^ 1 j^ gg ^ 1 i^ 59 ^ _ . . j (9) 



TTU TT L 3 6" J 



The formulse (8) and (9) are the required expressions for the vertical 

 depression of the surface and for the slope. 



