538 
at vi +p 
a 
5 XK De —- 
a—o° yw +1 
i 
(11) 
Now 
z= du + const = 4'log v +- const = AM ¢log v + const, 
dz dz dv' Y v ALB! 
— = —.— =AaM| — — So) = = ——__ , (12) 
du dv’ du vB pl} MM (v'B(yr' +1) 
lz 
den 
dz aM du) dv _ AL — Bye 
and 
yo +8 
den Ade ER M (u' +B(yv 41 
The factor 
(13) 
GT a(a + t—6) 
1—py=14 ——— —) 
(a +4-tTy(a—o) ~ (a+r)(a—o 
is positive, provided that 6 and r are sufficiently small. 
Also the quantity v= 10% is always positive, as long as w’ is real. 
in Se AE ae ye nj 
The quantity v is, —-— being supposed positive, also positive, 
a--6 
without restriction when 83 and y are both positive, provided that 
. . . ! | 
v’ >— B when @ is negative and provided that v' < —— when y 
y 
is negative. 
The domain of reality for «, and hence also for z, is limited by 
1 
the values — 8 and — — for v', or by the values u’ — loy (—&) and 
a 
1 
u’ = log(——) for u’. These limits really exist if B< 0 andy <0 
+ 
resp., or <0 and t >O resp. 
When 8 is a small negative quantity, the value u'—=log(-—p)=—B 
is negative and rather large. 
by af 
When 7 is a small negative quantity, — — is a large positive quan- 
é 
1 ; ee 
tity and the value wu’ = log (— Jt C is positive and rather large. 
“i & 
As for u’ == B we have »>=0, or ú == — oo, hence 2 Sa 
the ordinate line u’ = — B is a vertical asymptote lying to the left 
at a rather great distance from the centre of the domain ; and, as 
for wu’ =H C ‘we havé v=o, or u=-+o, hence z= + o, also 
the ordinate line wu’ + C is a vertical asymptote lying to the right 
at a rather great distance from the centre of the domain. 
So, when @< 0 or 5 <0, the real domain is limited by a vertical 
