159 
To these equations we add the transformation formula for wy,s, 
which may be derived from (28) *) 
Uh at (CR re ih, Megs. Mes rk ot. Aawasdes) 
§ 9. We shall now ‘consider the 6 quantities (27) which we shall 
especially call “the quantities y”’ and the corresponding quantities 
abe Vibes Teens. Ws 
According to (80) these latter are homogeneous and linear functions 
of the former and as (because of (5)) the coefficient of 4 in Was is 
equal to «the coefficient of ys, in ya, there exists a homogeneous 
quadratic function L of y w,,, Which, when differentiated with 
« 
41th ee 
respect to these quantities, gives wp ..w,,. Therefore 
Aes 
ut 34 
ibe ED At (ad a SA Pay wee a Wey ee ( ) 
and | 
LS EDA We AREN rd ON (35) 
If, as in (34), we have to consider derivatives of L, this quantity 
will be regarded as a quadratic function of the quantities y. 
The quantity L can now play the same part as the quantity that 
is represented by the same letter in $$ 4—6. Again LdS is invariant 
when the coordinates are changed. *) 
we have 
V —g> (cd) Yac vod Wea = > (cdef’) vac Yoda JecG fii Wer == = (df) Ybd 9 fdWaf—=Wab. 
' The last two steps of this transformation, which rest on (a) and (5), will need 
no further explanation. In a similar way we may proceed (comp. the following 
notes) in many other cases, using also the relations © (ad) pha7ha=1 and 
© (a) Pha zea = O (the latter for b= c), which are similar to (z) and (6). 
1) If we start from the equation for s’a) that corresponds to (29) and if we use 
(7) and (28), attending to VW —g’ =|p'V —g, we find - 
Wah — = = (cd) dca gdb Li fo 
sad. 
== Per = (cde fhij k) pec pfa Phd Pib X je Tkd Jef JhiWjk- 
ee 
This may be transformed in two steps (comp. the preceeding note) to 
—— & (ef hi) pfa Pib Jef Ihi Veh: 
VY —g 
In this way we may proceed further, after first expressing Jch as a function 
of dim by means of (32). 
*) Instead of (35) we may write L = 42 (ab) bas vas and now we have accord- 
ing to (28) and (33) 
