52 LECTURES AND ESSAYS [1869- 



20= (2m - n)a = tan C. . (13). 

 This equation has 2m - n roots 



7T 27T Q 



a-., a, + , cu H , CXC. 



2m -n : 2m -n 



So that each stream line has 2m - n asymptotes equally 

 inclined to one another. 



Transforming to rectangular co-ordinates, and choosing 

 the asymptote as axis of x, (8) reduces to 



(y^\(y^\, . . =0 . 



\x - h/i \x - h/s 

 When y = 0, x has two roots = oo if 



2(&) = . (14). 



Hence the asymptote is such that the algebraic sum of 

 the perpendiculars from the sources diminished by the 

 sum of the perpendiculars from the sinks is zero. It is 

 obvious without analysis that this condition is necessary, 

 that the velocity perpendicular to the asymptote, at its 

 point of contact with the curve, may be absolute zero. 

 If sinks weigh upward, all lines passing through the centre 

 of gravity of the system are asymptotes, and 2m- n of 

 these lines, equally inclined to each other, belong to one 

 stream line. The system must have a centre of gravity, 

 for by pairing sources and sinks we produce couples which 

 will always give a single resultant when compounded with 

 the weights of the extra sources. 



A complete system has no centre of gravity, but (14) 

 is satisfied for all lines perpendicular to the axis of the 

 resultant couple. If the axis of the couple formed by 

 pairing a source and sink at distance p m . makes an angle \j/ m 

 with the axis of the resultant couple, 



0. . (15), 



an equation with only one root to determine the direction 

 of the asymptote. In this case the asymptote meets the 



