( 216 ) 
The deduction of /(a*) chosen by us loses its validity as soon as 
we have 
2v = V(a— d)* + 4£y = 0. 
For then X becomes =0 and p = q- In this case however c = 0, 
so that the differential equation (14) now runs: 
/V) ,2/» m 2 g'\x) * * ’ ‘ 1 ] 
The integral of this is 
< PM±± .(2#) 
JW*)+1 
where, to satisfy (^4) and ( D) we must put 
/(*) = *'; 
f .. W 
The expression for (p n (sc) we find most easily by considering that 
i\n) = nu r ‘—' and Q(n) = 2 u n . We thus arrive at 
'n{*)=9- 
~j(«+d) + n(a-6))g(x) + 2»^~| 
_2»y^) + ((«+d)-n(«-tf)}J 
The case v = 0 forms the so-called 'parabolic case*) of the linear 
substitution, because the invariant points g(y ) = g(x) coincide. 
Also of the standardfonn (D) we shall defer the applications 
until later. 
§ 4. Beforehand we shall try to heighten the order of the differen¬ 
tial equation by means of whose integration the standard form originat 
By differentiation of (12) we find with the aid of (3) 
2 W - Kr. -1IV) - (X, -1 x,»)| + nr. - ar.r. + er,‘) - 
- (X, - 6X,X, + 6AV)j = — jfa, - - 
- (f, - 4 i,|, + Sf,')J. 
As fartheron holds: o' Y, -j- Qr lt = q'X, -f eii > we can a * s0 
q ‘ j(r, - 4 Y, Y, + 3r,*) - (X, - 4X.X.+ 3X,')! + 2 ?’(( r ■»— 2 " 
- (X,—| X,»)i,} = _ (to, - 4 M , + 9 V ,>) - (§, - + 3 S.’))-< 22) 
1) Gf. Klein—Fricke Vorles. u. d. Theorie der Ell. Modulfunktionen, Teubner, 
1890, p. 164. 
