1869-1873] STREAM LINES 45 



surfaces l in the first of the two ways we have indicated. 

 By an application of Ohm s law he expressed analytically 

 the conditions to be satisfied by v. When the electricity 

 enters and issues by a number of individual points, he 

 found (apparently by trial) that an integral of the form 

 2 (a log r), where r l r^ etc. are the distances of the point 

 (x, y) from the successive points of entrance and issue, 

 satisfies these conditions when the plate is infinite. For 

 a finite plate it is necessary that the boundary of the 

 plate should be orthogonal to the curves 



2(a log r) = const. . . (3). 



He was thus led to form the orthogonal curves, whose 

 equation he gives in the form 



2(a [r, R]) = const. . . (4), 



where [r, R] is the angle between r and a fixed line R. 

 These equations he applies to the case of a circular plate, 

 completely determining the curves when there is one exit 

 and one entrance point in the circumference, and showing 

 that in any case a proper number of subsidiary points 

 would make the equipotential lines determined by (3), 

 cut the circumference at right angles. KirchhorFs paper 

 is throughout properly busied with the function v, and 

 the stream lines are only dealt with incidentally. There 

 is no attempt to give a physical meaning to the equa 

 tion (4). 



In 1846 Thomson drew attention to the orthogonal 

 systems (3) and (4), as an example of Lame s theorem. 2 

 He showed that the rings and brushes of biaxal crystals 

 are a special case of these curves. They correspond, in 

 fact, as we shall see, to the equipotential lines and lines 

 of flow in an infinite plate with two equal sources of 

 electricity. 



Maxwell, in 1856, suggested the application to problems 



1 Poggendorff s Annalen, Bd. Ixiv. 



2 Camb. and Dub. Math. Journ. vol. i. p. 124. 



