When the points (2%, 2) lie in a 
curve of the form shown (fig. 1), which 
appears to tend asymptotically to the 
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| 
| 
| 
| ' | ordinate-lines «— 7, and «= 2, of the 
| | extreme limits x, and 2, of x, a trial 
| . 
| with 
U 
| z =A “log p —— 1 
It oe an a (1) 
H suggests itself. 
| ° . 
Introducing the median wv, (corre- 
Be cat If ey sponding to z= 0) we find 
En — Um 
[PR aoa 9 : . . . . ° . . (2) 
yy Vo 
e— eu Vind 
zog pe ’) OE kr 
Ly— En — Um 
Now we have to determine the constants z, 2), wn and 2 from 
the given curve. The curve on ordinary ‘squared paper still furnishes 
the value 7. 
From 
dz 4 i il A Mz, S11 bi A „Me = ©) 
Ee ji a = a Rn 
de wv Vy Un e (x — &o)(@n— es Se En (oa Je ee on 
(M= mod =- “loge = = 434295) 
and therefore 
and 
d*z AM(x,— &,)[ 20 — (ot n)| 5) 
== ee {0 
dx? (e—2,,)?(#, — 2)? 
> . oI = . . a << . dz Kk 
we find for the coordinates. (8,8) of the point of inflexion hen 0 
Ave 
& wv, = 3 x Und 4 
S= LRL S= logp=i log. (6) 
a Vn — Lo 
and for the slope & at this point 
en dz AM(an — Xo) AM ‘ 
ie ee = eas 
de ii Ne NEF Ee En dq oO 
In general the position of Ha point of inflexion itself, situated at 
equal distances from both the asymptotes, cannot be fixed exactly. 
On the other hand the position of the inflexion tangent is pretty 
well determinate; its equation runs 
2—C= 0 (a A 
The point of intersection with the axis of « has for abscissa : 
