Colloquium publications.

34

THE CAMBRIDGE COLLOQUIUM.

Ela = a, E13 = 0, E1 r = 0,
which we obtain by considering its action on a, 3, y themselves. These
equations are however satisfied by the dyadic E1 = aa'. Hence the formulae:
(7) E = aa', E2 = A', E3 = 77'.
Moreover, since any vector will be the sum of the three components obtained
by resolving it in three non-coplanar directions, we have the identity
(8) p = (E, + E2 + Ea)-p = (aa' + 33' + 7'') p,
which is merely another way of saying that the dyadic in parenthesis is the
idemfactor I.
23. The Condition of Isogeneity. Let r be the unit vector in
the direction of the curve C at P, and let a, 3, r be the reciprocal
system to V1, V2, T. From (7) and (8) we have
(7') E1 = Va, E2= V,
(8') Wi = (Via + V20l). W,, i = 1, 2.
From (2) we have
W2/2 W't Wl W2'
IV 21 [Vil ' V2- I Vl'
whence
W2r" W'V2 w ' - tw" I Iv.l
Making use of the fact that E1. W1 is a vector in the direction of
V1, and that therefore its algebraic magnitude is given by the
formula
_/ V1. El. W1
W V- IVl '
we may rewrite the first of the above equations in the form:
W V1l. E l W1
V21 V —V V2
whence, since E2. W2 is a vector in the direction V2, and with
reference to the direction of V2 of algebraic magnitude W2",
we get:
2 2 V1 E -V1

By (7') this-reduces at once to the value E2' W2 = (a. W)V2.