84 G. PLARR ON THE SOLUTION OF THE EQUATION Vpop=0 : 
We treat these equations by Sa(_ ), SB( ), Sy( ), and we remark the 
relations : 
Saa, = Sa(g’—g)a = Sa(g—g)a = Saa; 
Sep, = Sa(¢’—g)B = SB(¢—g)a = SBa,, &e. 
So that we have the table of values : 
Saa,=Saa;, SaB, =SBa,, Say, =Syai 
SBa,=SaB, , SBBi=SB,, SB =SyB, 
Sya =Say,, SyB: =SBy,, Syy =Syy, 
Applying this to the foregoing equation we get, by the first in A, B, C: 
0=Sa’(Aa+BB+Cy) 
0=S6i(Aa+ BB+Cy) 
0=Sy,(Aa+ BB + Cy) 
Thus the vector Aa+B8+Cy must be perpendicular to the plane in which 
a,, 8, y, are situated, the coplanarity of these latter being expressed already 
by /g=Sa,Bry,=9 . 
But p, is also perpendicular to this plane. Therefore, 
Aa+BB+Cy=fp’. 
Likewise we conclude 
A’a+BB+Cy=tp. 
If we remember now the definitions of A, B, &c., namely 
ya=Ap, &ce., 
we deduce by multiplying by p the expression of fp’, and by p’ the expression 
of t’p we get : 
2.p~aa=tpp 
2.Waa=tpp 
Now the scalars of the first members are equal. For a demonstration, we will 
simply express by the help of the coefficients of 77. We have: 
pa= VBiyi= V(9'—9) B(' —9)y 
=V¢'Ro'y—gV(B9'y—y9'B) + 9'By . 
This gives, by known properties : 
, ya=(mg-'—gx + 9")a ; 
Or developing and putting ; 
G —m,—gm, +97, H=m—g, 
