i73] STREAM LINES 51 



It is therefore true, also, for the equipotential lines, as is 

 otherwise obvious. 1 



The general nature of the stream lines will be different, 

 according as the number of sinks is or is not equal to the 

 number of sources. In the former case, 2(0) = is satisfied 

 at all points infinitely distant, the radii being all parallel, 

 and the positive and negative angles equal in number. 

 Hence one stream line has the straight line at infinity as a 

 branch, or intersects the straight line at infinity at right 

 angles, and therefore has an asymptote. This stream line 

 will, in general, be of the n - I th degree. In some cases it 

 may be of a lower degree ; as, for example, when the 

 conic at infinity is its other branch. A case of this sort 

 will be given below. The other stream lines of the system 

 cannot meet the line at infinity, and cannot have asymp 

 totes. However far they run out, they must therefore 

 loop and return. 



When there are more sources than sinks, 20 becomes 

 indeterminate at an infinite distance, as might have been 

 anticipated from the fact, that in this case there is a 

 constant flow of electricity outwards, implying a sink at an 

 infinite distance. The line at infinity is not in this case 

 a stream line, and will be cut by all the stream lines, 

 which do not loop except at finite distances, and have all 

 asymptotes. 



The asymptotes, in this case, may be easily constructed 

 by the aid of equations (6) and (8). 



At the infinitely distant point of contact the velocities 

 due to all sources are in the same direction, or the asymp 

 tote must be parallel to the radii. 



If there are m sources and n-m sinks, the stream line 

 whose asymptote makes an angle a with the initial line is 

 obviously 



1 I have since found that this result has been already proved for 

 plane curves by Professor Rankine and Professor Stokes (Proc. R.S., 

 1867), and for spherical harmonics by Sir W. Thomson and Professor 

 Tait, in their treatise on Natural Philosophy. 



