G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 265 
Considering that buis the same as <jv, because v is a scalar, the con- 
ditions take also the form: 
Vale = V(a<v) 
(44) V<a6 = V(b dv) 
lL. Vay = V(y<v)- 
Let us consider the conditions first under their form (43.) 
We multiply the equations (43) respectively by a, 8, y, and sum up the 
results; and then we multiply them znto a, 8, y respectively, and sum up. 
Then remembering the definitions (12) of I, II, etc., we get the two results: 
Sadv.a— pvrad +1 —Il =0 
dqvla? — Sdabv.a + LII—IV=0. 
By formula (C) § 1, we have 
Ladv.a = <Iv, and as 2a? = — 38, 
and as Pv is identical with <jv, these equations become: 
dau + 1. — tt = 0 
—4dqv+TI-—IV=0. 
Introducing the expressions (23) of I, etc., in function of II, the result will be 
4<qv—2SII — VII = 0 
—4<qv—2SII + VII=0. 
The two consequences to be drawn from these equations are therefore: 
sit =6 
(45) 
VII = 4 <u. 
In operating by S<j on the second of these equations, we get for the first 
member 
Sai, 
because, whatever be the value of SII, the result qSII will be a vector, and 
therefore S[ <(SII)] = 0. 
Thus 
S<aVII = Salil. 
For the second member we get 4<17v, which is a scalar by itself. 
The result of the operation S <j is therefore the condition: 
(46) SalI = 4470. 
The second equation (45) and the conditions under their form (44), will 
enable us to express S<jII in function of <v alone. For this we take the 
expression (36) formed a priori for SII, which for SII = 0 becomes 
Ssaoll=P+Q. 
