Probable Form of Magnetic Core Potential. 
11 
which he calls one of the first kind, and which in stretched bars is known 
to be related to Young’s Modulus,* the equation V 2 V 2 ^=o is satisfied by 
one of the form 
/f f f '(<*, P, r ■>) R, da, dp, dy 
which he calls a potential of the second kind. This latter biquadratic 
differential equation is the subject of special mention by Emile Mathieu 
in his treatise on potentials. j* 
*For example, if Young’s Modulus is called ^ then the above integral 
is denoted by F. C is related to the usual elastic coefficients X and ju, as 
follows: For a cylinder with a force F per unit of area acting on its ends 
C=F 
/j,-\-X 
ju(3X + 2jay 
therefore 
F 3X+2n _ 
—fx - —e 
C X+ju 
or Young’s Modulus. 
(See Rieman's Partial Differential Equations and elsewhere.) 
j* Emile Mathieu gives the equation /7%=o and shows that its solution 
is u=gd + v where 
and 
00 —^ prdd 
, r^fP-dd. 
(“Theory of the potential,” Vol. I., pp. 80-84.) 
p and p' are functions of the co-ordinates of each point of a surface 6 
and r is the distance of x , y , z, from dd. v and go are first and second 
potentials of the two layers of matter which cover that surface. Re¬ 
duced to the form 
dHc dhi d i u_ 
dx i+ dx 2 dy 2 ^~ dy*~ 
it has for solution the sum of two integrals 
/ ( r2 log 7 + ‘V‘) f lo %^ds-, 
cp and being two functions of a co-ordinate proper to determine a 
point of a contour S, and r the distance of the point a?, y, from the ele¬ 
ment ds. 
Every continuous function which in the interior of a curve S satisfies 
d 4 u d A u dhi _ 
dx‘ 1 dy' 2+ dy* 
is the sum of a first potential of a layer covering the curve S, and of the 
second potential of another layer on the same contour; likewise, also, 
the derivatives of this function of the first three orders. 
