G. PLARR ON THE SOLUTION OF THE EQUATION Vp¢p=0. 59 
41) BT STV (ye 200) 4 29T' Ln). 
This result is essentially positive. Of course its use is only theoretical ; there 
is no question of expanding it again by making use of the properties of the 
symbols T and V, &c.; that operation would lead to other expressions, and 
there can be no doubt that the expressions of hy which have been established by 
other methods (as for example, G. Bavgr’s, “ Crelle,” vol. Ixxi.) might be estab- 
lished with the help of formula (41). 
The value of I’ being that of [ for the case when «=0, we have therefore 
a demonstration, by the quaternion method, of the theorem, that when in the 
expression of the scalar 
S(¢—g)a(e—-J)B(9—-9)y =, 
we replace ¢ by its self-conjugate part ¢, then the equation 
S(G-9 al F—-I)BE-J)y =hI =0 
is satisfied by three veal roots g’. 
§ 7. Let us now return to the originary question, the condition namely that 
I’ should be positive, and limit the question in the following manner : 
Let us suppose that the elements of the self-conjugate part of ¢, and conse- 
quently of w, remain invariable, and given by the vectors 7, ¢, «’, ¢ as 
defined in (34), and let us admit that the elements of «, as defined by (27) or 
(33), namely by : 
<= 52aS@(sy—a’y) 
are alone variable, we varying in consequence. 
Then under this supposition, let us determine the limits of the values of ¢ 
which satisfy to the condition 
T>0. 
The question will be solved when we can assign a surface limiting the space 
for the one side of which all points, having « as vector, will give to e, and con- 
sequently to I’, a value satisfying the above inequality. 
The equation of the surface will be 
(42) r= 0, or 4P* + 27Q? = 0, 
wherein ¢e (drawn from an origin, of course common for all points), will repre- 
sent now the vector of a point of the surface. When the surface has been 
constructed then we will be able to show that it is the znside of the surface 
