G. PLARR ON THE SOLUTION OF THE EQUATION Vp¢p=0. 47 
another. Finally, the value of I‘ will be negative, and equation (1) will have 
only one real root when the extremity of € lies owtside the surface. (2). 
The case when e=0, and consequently when ¢ is self-conjugate, is included 
in the preceding. In this case, our expression of I becomes positive “a 
priori,” being then composed of the sum of the squares of two tensors. This 
result of [>0, when «=0, has been established by other methods, but not 
under the compact form under which it is given by the quaternion method. 
In the second part, we consider the connected problems of forming the 
expressions which satisfy respectively to 
Vp¢dp = 0, and to Vp ¢’p =0. 
It is known that those expressions, proportionate respectively to p, and to 
p, by a scalar factor, depend on an auxiliary vector, quite arbitrary in direc- 
tion, in such a way that the unit vector of the expression remains invariable, 
whereas the tensor of the expression varies when the direction of the auxiliary 
vector varies. 
The expressions which we establish for the tensors enable us to discern at 
once what the directions are of the arbitrary vector, to which correspond either 
the maximum of the tensor or its minimum (this latter being zero). 
Likewise we find for the product pp’ an expression of striking simplicity, 
both as to the scalar and the vector of the product, from which it is easy to 
deduce some properties belonging to the planes determined by the three 
Systems of vectors p, p , each system corresponding to one of the roots g, and 
some other properties relative to the angle between the vectors p, p’, of each 
system. 
Finally, a discussion of the cases in which the tensor of vanishes, gives 
us an example of certain solutions p, p’ (relating to a singular direction and 
value of «) becoming indeterminate. 
First Part. 
§ 1. We designate, for any value of g, by /g the scalar :— 
(3) S9=S(¢'—g)o(G' —g)B(¥—g)y, 
and having between a, 8, y, the relations ' 
aB=—Pa=y, Ga ba 2— —] 
we liken the function /g with 
(4) SG=P —M, GF? +M,g—M, 

