110 BIRKHOFF AND LANGEK. 



means that every tt sided polygon with vertices at the points yj(xo\ 

 j =1, 2, . . .n, a ^ Xq^ b, at most expands or contracts about X = 

 as .ro is allowed to vary. 



If the lines along which jj(.v),j = 1,2,...//, vary are all distinct, 

 the conditions of the system are regular provided W a and W b are 

 suitably chosen. If, however, one or more sides of any of the polygons 

 mentioned lie on a line through the point X = 0, we have the irregular 

 ease discussed in section VII. In particular all the sides may lie on 

 such a line, (for instance the quantities yj(x) may he all real) and it is 

 upon this configuration that the irregular case in question bases its 

 chief claim for interest. Since the condition (96 iv) is automatically 

 fulfilled in this case the functional dependence of jj(x) upon .r is far 

 less restricted than when the points y,(x) form the vertices of actual 

 polygons. 



Substituting in formula (86) the value G(x, t) = ± Y(x) Z(t) we 

 have 



G(x,t,\) = Y(x) | ± U + hA^\U\,Y«i) - W b Y(b)}} Z(t), 



the upper sign holding for t < x and the lower sign for t > x. It is 

 desirable in the following work to express G{x, f, X) in a somewhat 

 different form. 

 Upon setting 



+ § I + 1 A~ ] { W a Y(a) - W b Y(b)} = (5* y ) + U, 



multiplication by A yields 



I W a Y{a) + i W h Y{b) + I W a Y{a) - \ W b Y(b) = 



[W a Y(a)+W b Y(b)\ (4-)+Atf, 



which, in view of the relation (8$) + (5 ,y ) = /, reduces to 



W a Y{a) (S**) - }V b Y{b)(8*j) = AU. 

 Hence 



U = A- 1 W a Y(a) (5?) - A- 1 W b YQ>) («J). 



Now setting 



-i/ + ^A-'{ir n }>) - w b Y(b)) = -(5**) + r, 



