134 



NATURE • 



[August 2, 1888 



(8) If 



d¥ 



d_G 



dt' 



and — 



dt 



relate to an external system and its 



magnetic screen on S, we have 



d_ d¥ _ d dG 

 dt dy dt dx 

 whence it follows that 



_ d¥_ _ dty dG 

 dt dx dt 



If, therefore, 



;, &c, within S, 



dy 



= -r . C 



&c. 



_cl¥ _ dG _ d¥L 

 dt ' dt ' dt 

 are the components of an electromotive force within S, there will 

 form on S a distribution of statical electricity having potential 

 i/>, and forming a complete electric screen to the external system. 



(9) Of Self inductive Systems of Currents on a Surface. — 

 If any system of currents in a conducting shell be left to decay 

 by resistance, uninfluenced by any external induction, it may be 

 the case that they decay proportionally ; so that, if U , V , W 

 denote the initial values of the component currents, their values 

 at time t are U = U e ~ w , V = V e _x ', W = W e ~ xt , and 



-j— = - AU, &c, where A is a constant proportional to the 



specific resistance, and inversely proportional to the abso- 

 lute thickness. If this be the case, the system is defined to 

 be self inductive. 



(10) By Ohm's law we have, whether the system be self- 

 inductive or not, — 



d¥ dii o 

 au = - I , &c, 



dt dx 



where u is the component current per unit of area, and a the 



specific resistance. 



If h be the thickness of the shell, au — — U, and the equations 



h 

 may be written — 



_ d¥ _ dty _ dG _ dty 



a _ dt dx dt dy , &c. 



h~ U ~ V 



/F ,'U'fi A ^ 



-y- = AF, &c, and - f- = 

 at dx 



Ax, where % ls the associated function to F, G, and H, and if' to 



d¥ dG , dll 

 — — , , and 



dt dt - dt 



G + <* H + ** 



dz dy dz 



If the system be self-inductive 



+ 1* 



Therefore- 



h 



= A 



= A 



V 



W 



(il) Now if we assume as current function on S any arbitrary 

 function, <j>, we thereby determine U, V, W, and therefore also 

 F, G, H, and x, at all points on S. It will not be generally 

 true that — 



dx 



G + 



II + £ 



dy _ dz. 



U 



w 



These equations constitute a condition which the current func- 

 tion </> must satisfy in order that the system may be capable of 

 being made self-inductive. Their geometrical interpretation is 

 that the tangential component of vector potential of the currents 

 in the sheet coincide with the current at every point. 



If (j> be chosen to satisfy that condition, then by the equation — 



AQ (suppose) 



we determine h, the thickness of the shell at every point, 

 necessary to make the shell self-inductive, i.e. h = — - =-. 



(12) Examples of Self-inductive Systems. — 1. S a sphere, and 

 <p any spherical surface harmonic of one order. Here h is a 

 constant. 



2. S a surface of revolution about the axis of z, and <p a 

 function of z only. 



3. Any surface, with <f> a function of z only, if x is independ- 



ent of z. Example : an ellipsoid whose axes are the axes of 

 co-ordinates, and <p = Az. it is found in this case that \p cc xy, 

 and therefore the necessary condition for a self-inductive system 

 is satisfied ; and also, to make it self-inductive, h varies as the 

 perpendicular from the centre on the tangent plane at the point. 



(13) Co-existence of Self-inductive Systems. — If any number of 

 self-inductive systems be created in the same shell, each decays 

 according to its own law, unaffected by the others. If all have 

 the same value of A, then, as the effect of resistance apart from 

 induction, we have 



Q + Afi = O, 

 dt 



where fi is the magnetic potential of the whole system. 



(14) General Property of Self inductive Systems. — If an ex- 

 ternal system so vary as that the system of currents in the shell 

 S, induced at any instant, shall always be self-inductive, and 

 with the same value of A, we have, to determine the currents in 

 the shell at any instant, the equation — 



dn n 



+ 



da 



dt 



+ An = o, 



from which fi can be found, if 



dn 



is given. 



Example 1. — Let 



dt 



= C, 



a constant. 



In this case we find 



C 



n = - 



A 



If C be very great, and / very small, this 



approximates to the ideal case of an impulsive force, and £1 be- 

 comes equal to Ct, and is independent of the resistance. If, on 



Q 

 the other hand, \t be very great, we have fi = - , and fl varies 



A 



inversely as the resistance. 



Example 2. — Let £1 = A cos kt, where k is constant, and A 

 independent of the time, but a function of position. This leads 



to the result — 



fi = - A sin a sin kt - a, 



!1 + fi = A cos a cos kt - a, 



at all internal points. Here, o is the retardation of phase, and 



is equal to cot -1 



H Gfik 



For instance, if S is a sphere of radius a, and $ = A cos /'/, 



Q 



2>l + 



4ira 



— , and the result obtained agrees with that given by 



Prof. Larmor in Phil. Mag., January 1 1 

 (15) If the shell be infinitely thin — 



Qhk 

 a — sin o = - — , 



the same phase is reached in the inner field at a time later by 



£, that is, -^-, than in the outer field. The ratio which in 

 k a 



the limit h bears to this difference of time is — , and is, in case 



of a solid conductor, the initial velocity with which the currents 

 penetrate the solid. 



(16) If S be any homogeneous function of positive degree in 

 x, y, and z, the space within S = o may be conceived as divided 

 into a number of concentric similar and similarly situated shells, 

 each between two surfaces of the type S = c and S = c + dc. 

 Let <f> be a function, which, as current function, gives a self- 

 inductive system of currents in each shell of the series, if made 

 a conductor. Let an outer shell of the series be described on S, 

 and an inner shell of the series on S'. Let currents of the type <f> 

 be generated in the shell S. Let u, v, za be the functions — 



dS dd> 

 u = — — T 



dS dip „ 

 dy dz 



[dz dy 



Then u, v, w may be the components per unit of area of a 

 system of currents in the shell S. And since this system is 

 self-inductive, — 



djC 



dx 



Now v 2 F ="0, and V"X = o at all points within S. 

 If, therefore, -y~u = o at all points within S, 



dx' 

 dx 



au = A ( F + 



on S. 



au = A'( F + -? ) at all points within S. 



