ivith the Flow of Electricity in a Plane. 377 



straight flow-lines which enclose any angle but a submultiple 

 of 7r, unless the poles are put infinitely close together, forming 

 a complete circular ring whose centre is at the intersection of 

 the two lines ; and in this case every straight line drawn in the 

 sheet from this centre is a flow-line. 



Another way of expressing the facts is to say that both sides 

 of every flow-line are to be regarded as mirrors, and its images 

 as always infinite in number though often coincident with one 

 another, and that the problem of reducing the bounded case 

 to that of the infinite sheet becomes impossible whenever the 

 images reflected back into the sheet itself do not coincide both 

 in sign and position with the images already there. Similar 

 statements are true of the images in a pair of intersecting equi- 

 potential lines. 



Another distinction between optical and electrical reflection 

 is, that when two optical images coincide their light-emitting 

 powers are added, but, in the electrical case, coincidence of one 

 image with another does not increase its strength. These dif- 

 ferences appear to forbid our imagining that there is any phy- 

 sical meaning in electrical reflection at boundaries analogous 

 to the physical reflection of the waves of light in a mirror. 



§ 5. Consider now the images of a point inside a polygon of 

 n sides. We have seen that, without silvering both faces of the 

 boundaries and thereby introducing an unconscionable number 

 of images, some of them real, we can only treat angles that are 

 submultiples of it by the method of images. Hence obtuse angles 

 less than it are not allowable. This condition at once excludes all 

 polygons of more than four sides; for as the largest interior angle 



n 2 



of a polygon of n sides cannot be less than tt, and as it 



it n 



must be less than or equal to ~-, it is necessary that n shall be 



equal to or less than 4. And if ?i=4, the polygon must be 

 equiangular, that is, must be a rectangle. If ?i= 3, the tri- 

 angles whose angles are submultiples of it are the equiangular, 

 the isosceles right angled, and the one whose angles are 90°, 

 60°, and 30°. The pseudo-triangle whose angles are 90°, 90°, 

 and 0° is also a possible case, and is treated below as a special 

 case of the circular sector. 



I have not mentioned the case when it itself is one of the 

 angles of a polygon, because this merely reduces the polygon 

 to one of n — 1 sides ; but there is the case of the equiangular 

 polygon with all its angles 7r, viz. the circle, about which a 

 little shall be said later. 



The images of a point in a rectangle occur in groups of four 

 surrounding the vertices of rectangles whose linear dimensions 



