,; JBKTRIO INDUCTION'. 



Hence equations (6) become, by introducing the stream-function <f> from (l), 



V > = ~ ~I'A T (P "^" P ) ~J~ 



ay '<'.'/ WB 



^ ^ " i ~ * 

 t**c 



A solution of these equations is 



d_. p 



Substituting the value of <f> in terms of P, as given in equation (4), 



i 



i 



The quantity is evidently a velocity ; let us therefore for conciseness 

 call it R, then 



24. Let P.' be the value of P, at the time t r, and at a point on the 

 negative side of the sheet, whose co-ordinates are x, y, (z Rr), and let 



<?= ( P.'dr (13). 



J o 



At the upper limit when T is infinite P/ vanishes. Hence at the lower limit, 

 when r = and P ' = P., we must have 



but by equation (12) 



~dt~ "dt 

 Hence the equation will be satisfied if we make 



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 



