50 LECTURES AND ESSAYS [1869- 



(9). 



The nature of the intersection of the branches 

 of a stream line at a multiple point is easily deter 

 mined. 



At an w-point, the angles at which the branches cut the 

 axis of x are the roots of the equation 



10. 



Where, since Jx* = ~d* 



d m u d m u d m u 



d m u d m u 



dx m ~ l dy 

 Whence 10 becomes 



. t 



dx m \ 1.2 



We can choose the axes so that j~ = &amp;gt; and reduce the 



equation to 



&amp;gt;+...=0 (n), 



or tan m&amp;lt;f&amp;gt; = . . (12), 



&amp;lt;f, = , where / is any integer from i to m. 

 m 



Thus the branches make equal angles with each other. 

 This proposition depends solely on the relation 



