G. PLARR ON THE SOLUTION OF THE EQUATION Vpgp =0 ; 67 
The single root gives : 
: a 
ON + hohs = qu = g(ho+hs)’, 
res. 
therefore = 4 (h2—hs)’ ; 
and always supposing 
hi heh... 
we have the positive values : 
h,—h h,—h h,—h 
a=OS4, gat, a= 4. 
i ; ‘ 
The double root w= —5 cannot give us any real values for ¢ corresponding 
to E=h,, neither any for E=/;, because we have seen a prior? that there are 
no double values possible unless E be comprised between h, and E,. Let us 
therefore take E=h, and w= = This gives : 
ao 
et+hsh, 2 
@+hsh,+2h,=0. Owing to 2h,=0, we have 
hh, + 2h5=Nsh, + 2(hi + 2hyhs+h;) ; 
= 2hi + Bhihs + 2h; ; 
= (2h, +h,)(hy + 2hs) = — 6. 
But 2h, + hs = (hy — hz) 9 h, + 2h; = (hs —h,) ° 
Therefore 
€, = J (hy —hz)(h2—hs),_ which is real. 
We remark that owing to ,>h,>h; we have e,< a=; (hi —hs) . 
For E=0, the equation Fe?=0 gives 
O=@ + PY + 2 Q 
ee RE ONS 
@ =—P’-(7Q" . 
Now we have 4P°+ 27 Q?®= —I’, which gives —P*?= ribs + zt OF 
therefore 
o=[Gr+Ze- Gey]. 
