146] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 135
but we showed, in the article referred to, that A'Sl varies inversely as the
acquired twist velocity, whence the theorem is proved.
This is, in one respect, the simplest construction, for it only involves the
chord AA' and the homographic axis.
The chord AA' must envelop a conic having double contact with the
circle (Fig. 28), for this is a general property of the chord uniting two corre-
sponding points, A and A', of two homographic systems. Let I be the
point of contact of the chord and conic (Fig. 28). Then AA' is divided
harmonically in I and T; for, if XY be projected to infinity, the two conics
become concentric circles, and the tangent to one meets it at the middle
point of the chord in the other; the ratio is therefore harmonic, and must
be so in every projection; whence,
AI = AT
A'l A'T’
but the last varies as the square of the twist velocity acquired, and hence we
see that—