13 
1921 - 22 .] On Tubes of Electromagnetic Force. 
Introducing homogeneous co-ordinates by writing x = xjx^, y = xjx^, 
z = xjx^, t = xjx^, the equations of the plane ^ become 
1 ^2'^2 ^3^3 ^4^4 “t 9 
( + ’>?2^2 + '^4^4 ~ 
and therefore the equations of the line in which the plane ^ meets the 
hyperplane at infinity are 
J ^1^1 f 2^2 ^3^3 ^4^4 — 9 
1 yJiX^ + >72^2 '^4^4 — 9' 
Similarly, the equations of the line in which the plane meets the hyper- 
plane at infinity are 
j 'T j + ^2 ^2 ^3 '^3 ^4 ^4 ~ 9 
+ '^2 ^2 Vi ^4 — 9. 
The condition for half-parallelism is that these lines should meet in a point, 
which evidently requires 
k. 
^2 
Vi 
V2 
Vs 
Vi 
^3 
^i 
Vl' 
vf 
Vs' 
vf 
Now if a plane in hyperspace is defined as the intersection of two 
hyperplanes whose equations are 
-|- ^2V ^3^ ^4^ ”t ^5 = 9 
vi^ + v^y + ^3^ + v^f + % = 9, 
then the six quantities 
) 
^3 
J 
^1 ^4 ’ 
^2 ^4 
t 
^3 
Vl V-2 
Vl Vs 
Vl Vi ! 
CO 
V2 Vi 
Vs V4 
(or any quantities proportional to them) are called the six direction-ratios 
of the plane. They are evidently independent of the choice of the two 
particular hyperplanes used to define the plane. Denoting the direction- 
ratios of the plane ^ by a, /3, y, S, e, and the direction-ratios of the other 
plane 7^5' by a, /3', y', S', e, and expanding the determinant in (9) by 
Laplace’s formula in terms of minors selected from the first two and last 
two rows, we obtain 
a^' -{- a'^ -f yS' + y'8 — e_j8' — e'/3 = 9 . . . . (19) 
This is the condition that the two planes should he half -parallel. 
Next, let us find the condition for half-perpendicularity. 
Let two straight lines have the equations 
' ^-^0 _ y-Vo 
l m 
Z-Zc, 
n 
