88 G. PLARR ON THE SOLUTION OF THE EQUATION Vpop=0 : 
but Tp, Tp’ are not susceptible to be annulled, because the original solutions 
of (¢—g) p = 0, (¢’—g) p’ = 9, are dependent only on Up, Up’ respectively. 
There remains the factor ¢, which however will be easier discussed when 
looked upon as implicitly contained in the expression of €, namely in 
@ = (fg) + 4T(G—g)e. 
By the introduction of }, a, by 
g=m+h, ¢=mt+oa 
we get also 
© = (Fh)? + 41°(w—h)e, 
where 
ab = b+ Ph+Q 
Fh = 3h + P. 
Incidentally we will state also that by the same new variable we have : 
Yo = (P+ h+ho+a°)o. 
Now @ cannot vanish unless both of its terms vanish. 
We omit, as being a particular case only, the case when « =0, and 
two of the roots i,, hs, 4;, are equal to one another (in which case € may be 
annulled), because our hypothesis about the data in the present question is, 
that the elements of the self-conjugate part of ¢ are THE data, and « alone is 
left free to be disposed of in tensor and direction. 
So we suppose /,, h., hz different from one another generally, and we sup- 
pose e to take any value and direction at will. 
Then we observe that none of the two terms of @, in the three values 
C1, ©, Ts, corresponding to the three roots hy, he, bs (or at least those corre- 
sponding to the real values of h, roots of #)=0), can be vanishing, unless it 
be for a value of € corresponding to a point on the surface '=0. 
In this case we have 7 }=0, because [=0 is the expression of the condi- 
tion of the presence of equal roots of 7p=0. 
Let h, bz be the two equal roots. We find by 7h=0 that 
h=h=W, h=—2W. 
Thus we get 
{ o=,0=(a—W)(e+2W)e 
0 = (a—W)’*o 
@.=@.,=4T'(w—W)e 
@s=9W’'?+4T'(wt+2W)e. 
