1873] STREAM LINES 53 



curve in a double point, and has contact of the third order, 

 or x has three roots = oo . 



The condition for this is obviously 



2(fcA) = 0. . . (16), 



which since 2 ( k) = 0, does not depend on the point of 

 the asymptote from which h is reckoned. 



If (15) is satisfied identically, the asymptote meets the 

 curve in a triple point. Two of the branches belong to 

 the line at infinity, and the finite branch sinks to the 

 n-2 degree. 



In this case not only 2 ( * k) = 0, but 2 ( h) = 0. Hence 

 (16) no longer gives a fixed point on the asymptote, but 

 only fixes its direction. A further analytical condition is 

 easily found, but is unnecessary. For in this case the 

 centre of gravity of the sources coincides with the centre 

 of gravity of the sinks. The stream lines due to the 

 sources alone would have the same sets of asymptotes as 

 those due to sinks. One of these sets is necessarily 

 asymptotic in the complete system, which has always one 



line with real asymptotes. The set will consist of - rays, 



all passing through the common centre of gravity of the 

 sources and sinks, and equally inclined to one another. 



Rectilineal Branches are asymptotes coinciding with 

 their curves. 



Hence, in an incomplete system, all straight lines pass 

 through the centre of gravity of the system, and belong 

 to one stream line, unless the centre of gravity be a source. 

 In any case they are equally inclined to one another, for 

 if not branches of one stream line, they would be so for 

 the system got by removing the source at their inter 

 section. 



In a complete system there can be only one rectilineal 

 stream line, unless sinks and sources have a common centre 



A/I 



of gravity. In the latter case, there can be at most - 



