i8 7 3] STREAM LINES 47 



The equation of the lines of flow follows at once from 

 the equation of continuity. Across any element ds of a 

 stream line subtending angles d0 1 d0 2 etc. at the sources, 

 and d9^ d0 2 r etc. at sinks, no fluid must flow. But the 



quantity of fluid per second reaching ds from P n is E. 



2vr 

 dO 



The quantity withdrawn by P n is - E. Hence the 



27T 



differential equation of the stream line is 



Integrating, 2(9 - 20 = const. 



where and are the angles between radii vec tores and 

 any fixed lines. If we agree to reckon in opposite 

 directions for sources and sinks, the equation becomes 



^0 = a . . . (6). 



The following are elementary consequences of this 

 equation : 



(a) When we have one source P and one equal sink P , 

 the stream line through any point Q has for its equation 



2&amp;lt;9 - QPF + QFP =- - PQF - . 



Hence the locus of Q is a circle through P and P , which 

 is Kirchhoff s case. The orthogonals are circles whose 

 centres (R) lie in PP produced, and whose radii 



(b) If we have two equal sources 

 and no sinks, or what is the same 

 thing, sinks at an infinite distance, 

 the stream lines are rectangular 

 hyperbolas. For in this case, 



QPN + QP N = a = QN* if we make P QN-QPN. 

 Also QN touches the circle through PP Q, therefore 



