86 G. PLARR ON THE SOLUTION OF THE EQUATION Vp¢p=0. 
Replacing G , H, and ordaining in respect to g , we get : 
Sapa= Tis. ae 29m,—39° ; | 
which is nothing but 
—f’g , namely of we . Therefore 
LSapa=Wava=—/'g. 
Then as to the vectors 
ZV .faa=2[—HV. gaat+V. gaa], 
as — ZV . daa=2Vaha=2e 
rV . Paa= —TVaGa= + 2e—2mz€ , 
and as generally for any integer x : 
2(Va¢g"a+ Vag"a)=0, 
we get with ZV (ha. a)= +H. 2e+ 2He—2Zmz€ 
namely : 
LV (a. a) = +2(9—g)e= —TVV(ba’. a) 
We have therefore ; 
/TSUpUp’ = —S’9 
J/TVUpUp’ = +2(¢—g)e. 
From this, as T7?UpUp =1, we get: 
© =(SF'9"—A(—9) ef - 
This shows, as for example, that when two of the roots g are equal, and 
consequently for them (/’g)=0, then the two directions p and p’, correspond- 
ing to that root, will be at right angles to one another (always provided that « 
be not zero). 
Also, when we treat the vector of the product pp’ by SVe¢e(_) we get 
SVegeVpp' =0, 
and this whatever be the root g ; so that it follows that the planes which p and 
p determine, in the case of each of the three roots g (three planes), these three 
planes when drawn through the origin are all three cutting each other in the 
direction of the vector Vege. 
We have now the means of expressing the tensor of pw , by the help of 
Aa+ BB+ Cy=’, 
