and an integration of each of these functions produces
μ 3 =^1
2
xz +f(y, z )
μ 3 =−
1
2 yz +g(x, z)
μ 3 =^12 (x−y)z+h(x, y ),
where f(y, z), g (x, z), h (x, y )are treated as constants of integration during the integra-
tion of partial derivatives. One finds that by selecting
f=−^1
2
yz, g =^1
2
xz, h = 0
there results the family of surfaces
μ 3 =^12 (x−y)z=c 3
At first glance it appears that μ 3 =c 3 is a solution family different from μ 1 =c 1
and μ 2 =c 2 .However, from μ 1 =c 1 there results
z=x+y
c 1
which can be substituted into μ 3 to produce
μ 3 =
1
2 c 1 (x
(^2) −y (^2) ).
Hence the solution μ 3 =Constant reduces to the solution μ 2 =Constant. When
one of the surfaces μi=ci, (i = 1 or 2) has been obtained, this known solution
may be used to determine the second surface. The known solution can be used to
eliminate one of the variables in the differential system and thereby reduce it to a
two-dimensional equation which theoretically can be solved. Three-dimensional field
lines are in general more difficult to obtain and illustrate than their two-dimensional
counterparts.
Solid Angles
A cone is described as a family of intersecting lines. A right circular cone is an
example which is easily recognized, however, this is only one special kind of a cone.
