CH. Ill] GEOMETRICAL RECREATIONS 61 



forks, and so on; let w be the number of watercourses, then 

 there will be also w watersheds. Hence, by the theorems of 

 Cauchy and Euler, 



h = l+p 1 + 2p 1 + ..., 

 d = l+f 1 +2f i +..., 

 and w=2(p 1 +f 1 ) + S(p i +f i ) + .... 



These results can be extended to the case of a multiply- 

 connected closed surface. 



ADDENDUM. 



Note. Page 52. The required rotation of the lamina can be effected 

 thus. Suppose that the result is to be equivalent to turning it through 

 a right angle about a point 0. Describe on the lamina a square OABG. 

 Rotate the lamina successively through two right angles about the 

 diagonal OB as axis and through two right angles about the side OA as 

 axis, and the required result will be attained. 



