274 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
not lamellar unless adx+jSdy+ydz is the differential of a function of three independent 
variables. 
Let -4/ be a variable which has a certain value for each of the series of surfaces by 
which the magnet may be divided into magnetic shells ; so that, if be considered 
as a function of x, y, z, any one of these surfaces will be represented by the equation 
■4y{x,y,z) = n (a); 
and the entire series will be obtained by giving the parameter, 11, successively a series 
of values each greater than that which precedes it by an infinitely small amount. 
According to the definition of a magnetic shell (§ 67-), the lines of magnetization 
must cut these surfaces orthogonally ; and hence, since a, j3, y denote quantities 
proportional to the direction cosines of the magnetization at any point, we must have 
d'^ d'\) 
dx dy dz 
(b). 
Let us consider the magnetic shell between two of the consecutive surfaces correspond- 
ing to values of the parameter of which the infinitely small difference is w. The 
thickness of this shell at any point (x, y, z) will be 
'UT 
\ dx^ dif dz'^ ) 
Now the product of the intensity of magnetization, into the thickness of the shell, 
must be constant for all points of the same shell ; and hence, since ts- is constant, and 
since a, |3 , 7 denote quantities such that is the intensity of magnetization 
at any point, we must have 
djd 
\dx^'^ dy^'^ dz^J 
=m) 
(c), 
where F(-4/) denotes a quantity which is constant when \}yis constant. This equation, 
and the two equations (b), express all the conditions required to make the given 
distribution lamellar. By combining them we obtain the following three, which are 
equivalent to them : — 
r=FW)f; 
and hence, if fF(\p)d'>l/ be denoted by we have 
d<p d<^ d<p 
^ dx’ ^ dy’ ^ dz ’ 
( 11 .), 
where is some function of x, y and z. Hence the condition that a magnetic distri- 
bution (a, |3, y) may be lamellar, is simply that a.dx-\-^dy-\-ydz must be the differen- 
tial of a function of three independent variables. The equations to express this are 
