378 Mr. 0. J. Lodge on some Problems connected 



are twice those of the original one, and which cover all the 

 plane. 



The images of a point in an equilateral triangle occur in 

 groups of six surrounding the vertices of triangles \/3 times 

 the linear dimensions of the original one, and covering all the 

 plane. 



Application of images to certain cases of bounded plates. 



§ 6. Now although the potential of any point is really deter- 

 mined by (A), § 2, as soon as we know the positions of all the 

 images in the sheet, the expression is not in a practicable form ; 

 and it would not in the majority of cases be readily possible to 

 bring it to such a form. The way to proceed in such cases is 

 to consider what is required of the potential, and then to write 

 down a function of x + iy, 



W=- Y + TJi, 



such that the real part satisfies all the conditions required of 

 the potential function, viz. to remain together with its deriva- 

 tives continuous and finite all over the sheet except at the poles 

 and their images, where it is to become infinite like q log r, 

 to fulfil certain given conditions at the boundary, and to 

 satisfy the equation 



da? + ~dj? 



This being done, V= const, will represent the equipotential 



lines, and U= const, the stream-lines, and the components 



dY dV 



-z—9 -j- of the flow at any point can be obtained. In this way 



Dr. E. Jochmann * has expressed all the conditions of the flow 

 in a rectangle, bounded either by stream-lines or by equipoten- 

 tial lines, or by both. 



§ 7. But though the potential expression (A) is often of 

 impracticable form, the expression for the resistance of a con- 

 ductor, which follows from it, is simpler, and admits of being 

 reduced to a useful form with much ease in many cases. Ex- 

 pressions so obtained may serve to verify the results of 

 more abstruse and general processes ; but the principal reason 

 for entering into the subject is that the expressions themselves 

 and several of the continued products which lead to them seem 

 to be of some little mathematical interest. 



* "Ueber einige Aufgaben, welche die Theorie des logarithmischen 

 Potentials und den Durchgang eines constanten elektrischen Stroms durch 

 eine Ebene betreffen," von Dr. E. Jocbmann in Berlin. Schldmilch, Zeit- 

 schriftfiir Mathematik, 1865, p. 48. 



