of determining Stream Lines and Equipotentials. 267 
of magnetic induction in soft iron, taking into account the 
variation of the permeability with the force. But these again 
will be left to those who need the results. 
VI. Note on Boundary Conditions, 
It may be convenient to the reader i£ we bring together 
certain well-known facts concerning boundary conditions. 
Let us regard V simply as a function of position, not 
necessarily satisfying \7 2 V = or any other equation; and, 
as always, let contours be drawn at small intervals of V each 
equal to k. Then the first space-rate of V in any direction 
at a point is inversely as the intercept cut off from a line in 
that direction by two contours of V one on each side of the 
point, and is directly as K. Suppose, further, that the whole 
distribution of V can be represented by a single graph. 
1. If we have to make V such that the magnitude and 
direction of its maximum first space-rate, the Hamiltonian 
vector V^j satisfies given values over a boundary of a given 
shape. Then it is easy to set off the ends of the contours of 
V with a ruler and scale, for their directions are known and 
also the distance apart of successive pairs. 
2. If we are not given y V over the boundary but only 
the first space-rate of V in a given direction. Then there are 
an indefinite number of ways in which the contours of V may 
cut the boundary ; and as it will not generally be possible to 
say which of these is consistent with the internal conditions, 
they must be drawn and modified freehand as the approxi- 
mation to the internal conditions proceeds. This is usually 
not difficult. 
3. To make V continuous at any surface cutting the distri- 
bution^ all that is necessary is that the ends of the contours 
of Y approaching from the two sides should meet one 
another at this surface. Whether they meet at an angle or 
not does not matter. 
4. To make the first space-rates of V in every direction con- 
tinuous at any surface where Y is continuous, not only must 
the contours of Y meet one another, but they must pass 
smoothly into one another without making an angle. For if 
they made an angle and a straight line were drawn tangent 
to one branch of the contours at the angle, then the ratio of 
successive intercepts of this line by the contours of Y would 
not become unity when the contours were drawn at indefi- 
nitely small intervals of V, so that the second space-rate along 
this straight would be indefinite at the angle. 
5. Suppose next that a non-divergent vector is normal to 
