G. PLARR ON THE SOLUTION OF THE EQUATION Vpop=0. 51 
which gives 
(16) 3 Q=Snn, +Sa,7+S7,0. 
Finally, (11) gives : 
—2 P’= 2Saa'(e’a) = — Saw" aSaa’a + BSBa'a + ySya'a | 
= — [Say,San, + Syo,Syz7, +$67,8Bo, | 
(17) —2P?=7} + 2Soy7, . 
§ 3. We consider now the expression 
(18) P=(91—-92)"(9s—1)'(Y2—-Js)” 
91, 92, 93, being the three roots of the equation fg) = 0; and if we put 
Nn=M; +h, J=Ms+ he, Js=mM;+}s ; 
bis be, Bs , becoming thus the roots of the equation F(b) = 0, we will have 
h-J=lh—, &c., and consequently we have also 
r= (bi er be)*(bs— bi) "(be re bs)” : 
Now we borrow from algebra the knowledge that we have 
(19) | T=—4P°—27 Q?, 
and we will now express I by the help of P, Q, and their transformations 
(15), (16), (17). 
For this we put I’ under the form 
(20) T= (2P) (—2P”)—3(3Q)’, 
and we get : 
TP =(y? +2807) (7; + 2807) —88°(yy, tot +710) . 
Here we meet with a difficulty for further transformation, because the factor 
3 is not in evidence in the first term of the second member. 
We will not lengthen this already long deduction by indicating step by step — 
the circuitous route which has led us to the discovery of a vector « whose 
introduction in the stead of y,, according to the relation : 
(21) K=m—5PS, 
where 
