1888. ] of Curves and of Surfaces. 239 
equal in all respects. Thus ON and PQ are of equal length, and 
make with each other an angle « equal to that made by the 
tangent at P to PR with the axis of « (see above figure). Thus 
cp 
(PQ) 
1.e. the ratio of the vectors (OM) and (PQ) is independent of the 
direction in which PQ is drawn. And we have 
=cosa+izsin a; 
hd&+th,dn _ ds, -ids, _ (ON) 
aay We hh, . aay hh, - (CQ) 
Thus we see that the fact that the expression 
hdé + th,dn 
dx + idy 
possesses a definite value which is independent of the ratio dy: da, 
depends upon the fact that the vector (PQ) may be brought into 
parallelism with the vector (OM) by a twist whose magnitude 
depends only upon the position of the point P, and not upon the 
direction of PQ. 
Now suppose that we have an orthogonal system of surfaces. 
Draw a pair of consecutive surfaces belonging to each family. 
These six surfaces will enclose an elementary rectangular paral- 
lelopiped. Let P and Q be two opposite vertices of this parallelo- 
piped, and let PR, RS, SQ be consecutive edges of it. Draw 
another parallelopiped having its edges OL, LM, MN parallel to 
the axes of co-ordinates, and respectively equal to PA, RBS, SQ. 
The two parallelopipeds will be equal in all respects, and OW will 
be equal in length to PQ. Now the directions of PR, RS, SQ 
depend only on the position of the point P, and not upon the 
direction of the line PQ. Thus, whatever the direction of the 
line PQ, a single definite twist about some determinate axis will 
bring the edges of one parallelopiped into parallelism with the 
corresponding edges of the other, and therefore PQ into paral- 
lelism with ON. This twist will, however, no longer be represented 
by the ratio of the vectors ON and PQ, since that ratio would be 
a quaternion having its axis in the direction of the normal to a 
plane drawn parallel to OV and PQ, and would therefore be a 
variable quantity. On the contrary, if we write PQ=dp and 
ON =dc, we shall have an equation of the form do =q.dp.q"’, or 
q.dp=do.q, where qg is a quaternion whose value depends only 
upon the position of the point P, and not upon the direction of 
