264 G. PLARR ON THE ELIMINATION OF a, 6, y, ETC. 
(38) Baa(Sda)] =, SaN—P. 
The expression in the first member gives an example of how far one might 
dare in drawing conclusions from the identifying of a, 8, y, with a, B, y. Namely: 
If we put in it Sa8 = 0, S@a = 0, etc., Saa = SBB = Syy = —1, and con- 
sider the remaining terms, they will be expressed correctly by — S<©, where 
O6=a,+ 6+ y; not alone for its value, but also for «9. If we take for 
a,, Bs, y, their Es(uresenTig (21), we get 
_ it: 
0 = —5 V(aw, + Ba, + ya.) » 
and now, for differentiations, one must not be tempted into equating the second 
member to 5 VI, which value would, according to (18), take place if aw,, B 
etc., were dw,, Betc. Accordingly if —S<© was becoming z S<VII, then 
equation (38) would not be satisfied, because of the term — P in the second 
member, which shows that the operation would lead into error. 
It would seem that between P and Q, in other words, between IT and 2a;’, 
there exists no relation expressible in simple terms, such as a mere numerical 
relation a priori, that is, independently of the conditions of integrability. It 
is through those conditions that we shall find the ratio between the valuee of 
P and Q to be satisfying to P + 2Q = 0. 
§ 3. The Elimination. 
The conditions of integrability of the proposed scalars are expressed, either 
under the form: 
(39) Vaie=0, Vas 07, Vay = 0; 
or under the form 
(40) Vpua = 0, ete. 
Both sets of equations are comprised in 
(41) (dwa — Pua) = 0, (dub — pus) =0 
(dlwy — Duy) = 0. 
Developing and putting 
(42) = ead 
we get the equations: 
dv.a—abv+ da— Ppa =0 
(43) dv.8 —Spv+ a6—pe=0 
Iv.y ay but d4y— by =O. 


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