22 
Proceedings of the Royal Society of Edinburgh. [Sess. 
It will be remembered that the electropotential surfaces were defined 
by the covariant set of total differential equations (4), and the magneto- 
potential surfaces were defined by the covariant set of total differential 
equations (8); and these equations do not define families of surfaces at all 
unless the determinants of the coefficients, namely, 
0 
K 
-hy 
^x 
and 
0 
dz 
-dy 
-K 
-K 
0 
dy 
-dz 
0 
dx 
~~ h X 
0 
dz 
d y 
— dx 
0 
-K 
~dx 
^'V 
-d. 
0 
^^X 
h y 
K 
0 
are zero: this condition was satisfied by virtue of the relation dxhx + dyhy 
-\-dzhz — 0, which we assumed to hold, but it is no longer satisfied when the 
electric and magnetic vectors are not everywhere at right angles. We can, 
however, form a linear combination of these two invariant sets of total 
differential equations, namely, 
(kh^ 4- jxdz)dy - {Xhy + jxdy)dz -f {Xd^ - fxhx)dt = 0 
— (A ./?2 fxd^dx 4- {Xh ^ 4" fxdx)dz 4- {Xdy — yliy^dt = 0 
{Xhy 4- fxdy)dx — {Xhy. 4- jxdx)dy 4- {Xd^^ - fjLhg)dt = 0 
^ — {Xdy. — ixhy.)ifx — {Xdy ~ fxl} y) dj/ — {Xdz - ixh^)dz = 0 
and this set will be equivalent to only two equations provided the 
ratio X : /a is such that the determinant of the coefficients of the differ- 
entials is zero : that is, provided 
or 
0 
XJi^ jxd^ 
Xhy 4” y'd y 
— Xdy.-\- fxhy. 
Xh^ 4~ yd ^ 
0 
— Xhx — yd X 
'xdy 4“ yJi'y 
Xhy ydy 
xhy. + ydx 
0 
— Xd^ 4- yh^ 
X(^ y. yhx 
Xdy yhy 
Xc'g - yhy 
0 
= 0 
{dxhx 4- dyhy 4- dJi^)X^ 4- {d^^ 4- dy"^ 4- d^- - _ 
h y‘^ h^^Xy {dxhx~\~dyhy-\-ddi^yd‘ — 0. 
This is a quadratic equation in the ratio X : /x, and therefore defines 
in general two ratios X^ : and y 2 - define one family of 
geometrical forms in our four-dimensional hyperspace by the set of total 
differential equations 
(X^iz + yyK)dy — {X-^iy 4- jx-^dy)dz 4- {X-yiy. — yd^^)dt = 0 
- {X.Ji^ 4- y-^dy)dx 4- iX-^^x + yy^x)^dz 4- {X^dy - yd^.y)dt = 0 
{X-Jiy 4" y-yiy^dx — {X^ix y\d^dy 4- {X-^d^ — yd^'^)dt = 0 
^ - {X-^dx — yihx)(fx - {X^dy - ydiy)dy — {Xyi^ — y^iz)dz - 0 
This device was suggested to me by the recollection of the method published in 1876 
by Hamburger for integrating systems of simultaneous partial differential equations. 
