18 ALF GULDBERG. M.-N. Kl. 
by the eliminations of these quantities, obtain certain relations connecting 
the differential coefficients of z with A, B, C, ...G, A. But the 
peculiar form of the functions in the first member of the above system 
enables us to effect this elimination so as to lead to four final equations 
independent of v and u. 
Thus multiplying (e) by = (/) by - and (g) by 2 “ and adding, 
we find, on rejecting the common factor u, that 
eu ou ou 
: BEE: eu\ ou 9u\ ou 
Again, multiplying (a) by lols —, (2) b de) 3" (g) by 
= (=) (5) and (4) by — 5 me, adding, and again rejecting the common 
factor z, we have 
ou\ ou u\ du au\ (du du ou 
ar) 
2 
Further, multiplying (a) by = = (6) by — = oe (d) by 4 
or dt 
(/) by — no 2 and (g) by — lee 
om 
+ BE =| adding, and rejec- 
ting the common factor u, we find that 
du du Ou Ou ou |? au, ou ou\ ou 
aper TIQUE E ay) a» az) 
ee. 
dy } os 
Lastly, multiplying (a) by - | (ce) by — = = ‚(d) by = 4 (e) 
b MV ad (ed by = Li ed 
Y (3x) 37 81 9Y ay) å 
the common factor u, we have 
+ (::) « adding, and rejecting 
