1872.] Prof. J. C. Maxwell on Electric Induction. 



167 



but by equation ( 1 2) 



dP _ dP dP , . 



Hence tlie equation will be satisfied if we make 



p — ; • • • < 16 > 



25. This, then, is a solution of the problem. Any other solution must 

 differ from this by a system of closed currents, depending on the initial 

 state of the sheet, not due to any external cause, and which therefore 

 must decay rapidly. Hence, since we assume an eternity of past time, 

 this is the only solution of the problem. 



This solution expresses P, a function due to the action of the induced 

 current, in terms of P ', and through this of P , a function of the same 

 kind due to the external magnetic system. By differentiating P and 

 P with respect to z, we obtain the magnetic potential, and by differentiating 

 them with respect to t, we obtain, by equation (10), the current-func- 

 tion. Hence the relation between P airi P , as expressed by equa- 

 tion (16), is similar to the relation between the external system and its 

 trail of images as expressed in the description of these images in the 

 first part of this paper (§§ 6, 7, 8, 9), which is simply an explanation of 

 the meaning of equation (16) combined wi-.h the definition of P ' in § 24. 



Note to the preceding Paper. 



A.t the time when this paper was written, I was not able to refer to two 

 papers by Prof. Felici, in Tortolini's 'Annali di Scienze' for 1853 and 

 1854, in which he discusses the induction of currents in solid homogeneous 

 conductors and in a plane conducting sheet, and to two papers by E. 

 Jochmann in Crelle's Journal for 1864, and one in Poggendorff's ' Annalen ' 

 for 1864, on the currents induced in a rotating conductor by a magnet. 



Neither of these writers have attempted to take into account the induc- 

 tive action of the currents on each other, though both have recognized 

 the existence of such an action, and given equations expressing it. M. 

 Felici considers the case of a magnetic pole placed almost in contact with 

 a rotating disk. E. Jochmann solves the case in which the pole is at a 

 finite distance from the plane of the disk. He has also drawn the forms 

 of the current-lines and of the equipotential lines, in the case of a single 

 pole, and in the case of two poles of opposite name at equal distances 

 from the axis of the disk, but on opposite sides of it, and has pointed out 

 why the current-lines are not, as Matteucci at first supposed, perpendicular 

 to the equipotential lines, which he traced experimentally. 



I am not aware that the principle of images, as described in the paper 

 presented to the Royal Society, has been previously applied to the phe- 

 nomena of induced currents, or that the problem of the induction of 



YOL. XX. O 



