46 G. PLARR ON THE SOLUTION OF THE EQUATION Vp¢p=0. 
the second of the results gives us the means of forming expressions which give 
the vector p, as for example, by an expression like 
pC=by=V(o —g)a(e'—g)B, 
C being a certain scalar, whose value will have to be determined as the 
case may be. 
We now divide the subject into two parts. 
In the first part (which will be the longer one), we shall consider the con- 
ditions which ¢ (and its scalar elements) will have to fulfil, in order that the 
three roots of the equation (1) be all three real ; and in the second part, we shall 
examine the properties of the expressions giving p, for which the condition of 
parallelism with ¢p is realised. 
The results which we have tried to establish in the first part may be 
summarised briefly as follows :— 
Supposing that the condition of reality of the three roots (1) be expressed 
by 
> Oe 
I’ being a known function of the scalar coefficients of equation (1), we 
transform the expression of I’ by introducing into it several auxiliary vectors, 
amongst which the vector commonly designated by e, according to the definition 
(p—)( )=2Ve(_ ), will play a prominent rdle ; | 
and with the help of these vectors, we find for I’ an expression in function of 
the squares of the auxiliary vectors, and of scalars of some of their products. 
We then reduce the utter indetermination of the question in treating the 
general case following :—We assume that the auxiliary vectors which consti- 
tute the self-conjugate part of ¢ remain invariable, and that e, and all that 
depends on it, alone varies, and we construe the surface defined by 
(2) r=0, 
in which the vector of the surface is to be the value and direction of « 
answering to the condition [=0. 
We will be able to show that this surface, besides some particularities 
interesting in themselves, enjoys the property of embracing’a closed and 
contiguous space round the origin of its vector, which is central. 
The condition, then, [>0, will be realised for every value of e which has 
its origin at the centre, and its extremity contained within the inside of the 
surface (2); and consequently for all these values of «, the equation (1) will 
have three real and different roots g. 
For the values of « answering to the surface (2) itself, the equation (1) will 
have three real roots also, two of them, if not all three, bemg equal to one 
