15 
1921-22.] 
On Tubes of Electromagnetic Force. 
Substituting from (14) in (11), we have 
■^(^ 1^1 ^ 2^2 ^ 3^3 ~ ^ 4^4 ) + "*■ ^ 2^2 ” ^ 4^74 ) ~ ^ 
+ V2^2 + “ Ia^a) + + V 2 V 2 + - VaVa) == ^• 
Eliminating the ratio X : /m between these equations, we have 
+ ^2V2 + 4^3' - ^4^4' = 
^ 1 ^ 1 ' + V 2 V 2 + V 3 V 3 - VaVa' 
that is to say, the product of the arrays 
^1 4 4 ^'^4 fi 4 4 ^^4 
^1 V2 Vs ^Vi Vi' V2 Vs O4' 
vanishes : but this product may be written 
^1^1' + 44' + 44' ~ 44' 
^l4' + V2^2 + Vs^s - Va^a 
-1- 
4 4 
4 4 
4- ... -1- ... - 
4 ^4 
y/ A 
% % 
Vi Vs 
Vf Vs' 
Vs Vi 
Vs Vi 
= 0 
or 
aa' + - yy' + 88' - ee' - = 0 . . . . (15) 
This is therefore the condition that two planes whose direction-ratios 
are (a, /3, y, 8, e, 0 P'’ 7’ 4 T) should be half-perj)e7idictdar 
in the hyperspace whose absolute has the equations 
xp + xJ 4- - X, 
2 = 0, Xr = 0. 
§ 10. The Partial Differential Equations of the Calamoids. 
We shall now proceed to find the partial differential equations whose 
solutions are represented by the tubes of force or calamoids. We shall 
derive them from the properties proved in §§ 7, 8, namely, that the calamoids 
are half-parallel and half-perpendicular to the electropotential surfaces. 
If the equations of a surface are given in the form 
x = xfu^ v), y^-y{u, v), z = z(u, v), t^t(u, v), 
it is easily seen that the direction-ratios of the tangent-plane to this 
surface at the point (u, v) are given by the proportion 
a : /I : y : 8 : e : ^ 
^d(pj) _ d(y, t) , 8(y, z) _ djx, t) _ _ dfopj) _ 8(rr, y) 
d{u, v) ’ d{u, v) ' d{u, v) ' v) ‘ 8(v/, v) ‘ dpi, v) 
Hence for an electropotential surface, by equation (7), we have 
a : yd : y : 8 : e : 8 
= //g I liy . . hx • dy . d^ , 
Therefore, if (a, y8, y, 8, e, f) now denote the direction-ratios of the tangent 
