720 PROCEEDINGS OF THE AMERICAN ACADEMY. 



9x 2 9t 2 ' 



and the boundary conditions and Krigar-Menzel's law are obviously 

 satisfied. To show that Young's law is also satisfied, it is necessary to 

 get the Fourier development of the curve u = <f> (f) corresponding to 

 some incommensurable value of a:. Krigar-Menzel has proved 20 that 

 any continuous periodic curve, whose period is made up of n straight lines, 

 can be developed in the form 



1 °° ■ OP . 



A n i\~p . n-n-t ST* _, . mrt 



M = y + > A cos ^- + > j5„ sin — , 



with 



A 



HTTTq 



+ ( ^° dl } i sin f T 



m-2 m-1' ( s i n ) T 



where 2 T is the period, t , r v . . . t _ 1 and r (= r + 2 T) are the 

 abscissae of the various corners, and ^q 1 , 6 x 2 , . . . fJ '_ l are the slopes of 

 the /x lines. Let this formula be applied to a section of any integral sur- 

 face along a line x — e, where c is a small incommensurable quantity. 

 The left hand edge of an integral surface is always made up of the diago- 

 nals of a number of its constituent facets, aud each of these planes 

 would be cut by the plane x = c in a horizontal line. Therefore, the 

 desired section, for a case p/g, is always g long horizontal lines, at 

 various heights, connected by g short oblique lines of various slopes. 21 

 Beginning with one of the oblique lines, the abscissae of the corners 

 will have one or the other of the following sets of values, according as 

 p is odd or even. 



20 "VVied. Ann., 49, 645 (1893). 



21 The stair-like forms already described in section 3 are special cases of this 

 general type of figure. 



