of Edinburgh, Session 1881 - 82 . 
343 
directions which the function is susceptible of representing are 
limited to those which may he comprised in a given plane. In this 
case we may assume the expressions 
(4) d>p — aScqp + /JS/^p , 
(5) (f)'p = cqSap + /JyS/^p . 
In virtue of these definitions the coefficient m in the cubic, whose 
definition is 
mSXpiv = S ft pL(f> v , 
will vanish, because the factors , <f>[ y , <£V are all three co- 
planar. 
The value m = 0 characterises in a more general way the class of 
functions <f)p which we have in view. But in the case m = 0 the 
equations (2) and (3) lose their definite signification because we do 
not know a priori what the result 
mc{> 1 p 
will become when one of its factors vanishes. 
If we eliminate the unknown term 
mfi ~ 1 VAp, 
from the relations (3) and (2) (this last being written then for 
YXpi) we get the relation 
Y — (<£ 2 — m 2 (fi + mf)V\pL , 
and there cannot he a doubt as to the validity of this equation 
when <hp is constituted so as to he susceptible of representing any 
direction in space. But when <£p is of the more particular class, 
represented by (4), then it seems to us that the last written equation 
requires a verification, and the present paper effects this verification 
by two methods, adding a few applications. 
We remark that the expressions (4) and (5) may he looked upon 
as theoretical and not actually given. The true data for the class 
of functions which we consider will be the vectors y and y 1? 
namely : 
(6) Ya(3 = y , and Ya l /3 1 — y 1 , 
to which the directions represented respectively by </>p and by </>'p 
will he perpendicular. 
