TRANSACTIONS OF SECTION A. 343 
9. On the Interpretation of Signs in the Formule of Solid Geometry, 
By Professor R. W. Genesn, JA. 
Let the unit sphere about the origin O meet the rectangular axes of reference 
in X, Y, Z, and let OP, OQ, OR be any three radii the co-ordinates of whose 
extremities are (1,, 7, 1), (Ja) 2%, %)y (Ug, Mgy 2g). 
The equation to the plane OQR is 
oy) 2 \\=O0 
L234 Moy Ny 
dg, Mg, Ng 
Hence the determinants 
L, my, 2, LO; "O 
A= | l, ma, nm, | and L,= | 1, m,, ng 
13, Mz, 2g 15, Mg, Ng 
have the same or opposite signs according as P and X are on the same side or 
opposite sides of OQR. 
1. Let P and X be on the same side of OQR. 
Now L, is twice the area of the projection of OQR on the plane of yz, and is 
positive or negative according as rotation from OQ to OR viewed from X, and 
therefore also from P, is, or is not, in the same sense as rotation from OY to OZ 
viewed from X. Hence we have the same condition for the sign of A. 
2. Let P and X be on opposite sides of OQR. 
The signs of A and Ly are opposite, but the aspect of rotation OQ to OR 
from X is also the negative of that from P. Hence the conclusion is the same. 
It is easy to deduce from this the following rule for the sign of A, or of the 
tetrahedron OPQR, viz., It is positive if the aspect of the circuit P toQ to R 
from O is the same as that of X to Y to Z; negative if not. 
Or, again, in the special case in which OP, OQ, OR are at right angles, 
A= +1 is the condition that it should be possible to rotate the trirectangle 
OXYZ so that it may coincide with OPQR. 
With this condition we have also 
Thus 1, M,, N, are the direction cosines of the normal to OQR on that side Srom 
which the aspect of a rotation OQR is the same as that of OYZ seen from X. 
In the ordinary system this latter is clock-wise, so that L,, M,, N, determine the 
normal from which the rotation OQ to OR appears clock-wise.' 
As an illustration let us take Rodrigues’s problem to find the co-ordinates 
(’, y’, 2’) of a point P(r, y, 2) after, say, a right-handed rotation ¢ about an 
axis (d, m, 7). 
If PN be perpendicular to the axis, P’K to NP, we have 
ON=la+myt+nz.... C : ; - - (Ld) 
NP = ./(mz—ny)? + &e. 
and the direction cosines of NP are 
x—lON y—mON z—nON 
2INDD bw ss SND" 
The equation to the plane ONP is 
XYZ 
imn 
vys 
=O 
! One interesting reservation should be made, viz., that the rotation from OQ to 
OR must be taken the shortest way—i.c., that the rule requires the exclusion of reflex 
angles, | | | 
