Law of the Induction of an Electric Current upon itself. 25 
__ The inductive forces at the centre and circumference are there- 
| fore as 7:2, or as the semi-cireumference of a circle to its diam- 
| eter.. ‘This would lead us to expect the initial development of | 
+ an electric current in a wire at the first instant of the application 
_* of an electro-motive force, acting as is usually the case with 
| equal intensity through all parts of the cross section, to be more 
rapid at the surface than at the centre, for the inductive force 
7 must be exactly sufficient at every point to balance the force 
’ applied. Z 
Equilibrium of the initial inductive forces within the mass 0 
a straight cylindrical conductor, or law of initial development 
of the current through different parts of its cross section. * 
12. Let abd, fig. 6, be the cross section of 
a cylindrical conductor of great length, and - 
let a current be developed in it by the appli- 
cation of an electro-motive force acting uni- 
formly through all parts of the cross section. 
_. ‘Take any point ¢ in the diameter ad (whose 
Be distance fe from the centre we will call z), 
_ and draw the chord kel, making with ad any | 
angle acl=90°+ 9. Draw the perpendicular’ 
fg, and in kl take any point h and draw the 
Straight line fh producing it to mand n and de- 
hote gh by x and the angle gfl by g’. Following the motion of 
the lines fg and fl as kel makes its motion of 180°, this angle 
si 
ERS: 
d 
ie, 
: is to be taken greater or less than 90° according as ¢ is greater or é 2 
= _less than 90°. Putting R for the radius of the circle, we shall ~* a 
have kg or gl=R sing’. The rate of development an the 
point h may be safely assumed to depend on the distarice fh from 
the centre, which distance determines the value of the rectangle 
: dQ 
mh. hn and consequently that of kh. hi or R? sin? ¢’—2?; ~> 
must therefore be some function of this latter quantity, or 
dQ : 
a? = FY R? sin?q/— zx? ). 
Substituting this value in formula (F’) and also that of dr=dz 
nd then replacing the limits of r by those of z, we have 
i, 7 +R sin g’ my 
alee aaice es F(R? sin?9' - 2) de \dp, 
“g 
~ Rsin gq’ 
hie Our problem requires that such a form be given to the function i 
F + $in* 9’ —2?), which we may call simply F, that the second _ 
Segms, Vol. XI, No. 31—Jan, 1851. 
