272 G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 
In virtue of (56), or better, of its conjugate 
K(pbq) + dp.g=9, 
we have 
K(peg) x ppg =— <p.g.p. bq, 
and thus, owing to gp = 1, 
S*(ppq) — V*(peg) =— peg. 
Therefore 
2|aBy| <daba, 
namely, 
(65) P—Q=—44ppq + 42p4,. 
Comparing this to the expression (34), we have 
(66) rTM =—4appq, 
as (67) 5 Qo", =— 42 p79... 
Of course the second members are scalars like the first, and we have : 
(68) Sall =—4appq — 43pi¢i. 
Thus far we can form the expressions independently of the conditions of 
integrability. 
Introducing now these conditions, we replace a, by the expressions (51), 
which give 
, eee = 
PA =— PYa = 5 V(4<1v) 
, , it AD . 
(69) PY =— Ph = 5 V(B<1v) 
, , if! é 
Pq =— py. = 5 Vivir). 
These expressions take place also, in all rigour, when 4@, #, ¢, a, 8, y, are replaced 
respectively by 2, y, 2, 2, J, ete. 
We have thus, taking piq x pq; = p.9.- 

> | LYZ| PGer =— =2 igh) Vin), 
which by (E) § 1, becomes 
mF : (<Iv)’, 
Thus: 
TaD 11 1 
(70) DV = 3 ( <u)’ ea ié zo, : 
Then we draw from first and third members of (69), 

