464 Messrs. G. C. Foster and 0. J. Lodge on the Flow 



34. As already observed (§ 1), both the general equations 



2 (Q0) = const, and 2(Q log r) = const. 



were given thirty years ago by Kirchhoff, who obtained the 

 former as a purely mathematical consequence of the latter. In 

 fact, as the systems of lines represented by them are orthogonal 

 to each other, it follows that their first derived functions differ 



tin doc 



only in one having -j- where the other has — j-; hence dif- 

 ferentiation and subsequent integration, after making this sub- 

 stitution, convert one expression into the other *, This rela- 

 tion between the two equations is of importance, since the 

 form of the equipotential lines can be readily determined ex- 

 perimentally, whereas no practicable method exists for ascer- 

 taining experimentally the course of the lines of flow. 



35. Before going further we may point out some general pro- 

 perties of lines of flow and equipotential lines. Physically consi- 

 dered, a line drawn from any one pole to any other, so that no 

 electricity crosses it at any point, may be regarded as an inde- 

 pendent line of flow, as was done in treating the case of two 

 equal and opposite poles. The lines, however, given by the 

 equation 2(Q0) = const, are continuous curves each of which 

 passes through every pole; and, in a mathematical sense, the 

 whole of the curve given by any one value of the constant must 

 be regarded as a single line of flow. When flow-lines are spoken 

 of in the sequel, it is to be understood that they are complete in 

 the above sense. If all the poles are of equal strength, every 

 flow-line passes once through each of them ; hence, if they are 

 unequal, every line passes through each of them respectively as 



* Let the coordinates of the several poles be (a-fi^), (a 2 b 2 ), &c. The 

 distances of a point from each will be r 1 —W(x — a 1 ) 2 -\-(y — b 1 ) 2 ,r 2 = &c, 

 and the angles which the joining lines make with the axis of x will be 



■h 

 ae — a x 



1 2(Qlog r)=S . Q a log V0*-a) a +(y -bf 



given by tan X = y - — - 1 , tan0 2 = &c. Then 



q/ x ~ a 4. y-t> dy\ 



\{.x-ay+{y-b)^(x-a) 2 +(y-bf ' dx i 



<x-aY+{y-by ' {x-af+iy-b) 

 ake the substitul 



tegrate, 



Now make the substitution, equivalent to writing </>+ J for $, and in- 



J 2 O / y-6 _ x-a dy\i 



\{x-ay+(y-b) 2 {x-ay+iy-by'diJ 



=2.Qtan-iL*=2(Q0). 

 x—a 



