GEOMETRIC PROBLEMS RELATING TO CURVES, ETC. 33]. 
ique circumstances, and witha minuteness of evidence relative to the 
social condition and the vital characteristics originally pertaining to 
her whose sepulture was involved in the ravages of the Crimean war, 
which led to its acquisition: the facts recorded in this paper, may 
possess some slight value as a contribution to data now accumu- 
lating from the labours of many independent workers, and destined 
ultimately to establish physical ethnology on a sure and well-deter- 
mined basis. 
GEOMETRIC PROBLEMS RELATING TO CURVES HAVING 
DOUBLE CONTACT. 
BY J. W. MARTIN, LU.D., TORONTO. 
Read before the Canadian Institute, 10th March, 1860. 
Given a circle,and a point o inside it; if a line passing through 0 
and cutting the circle in the points a and 6 be divided externally in m, 
so that OE SLD ep segn.ents of fixed chord passing through o 
am X bo ¢' 
then tangent to circle from m will be to perpendicular from m on rt the 
polar of o as secant of angle which cc’ makes with diameter of circle 
passing through o to unity. 
If ac bc’ be produced, they will meet at p, a point on rt; and if 
from p we draw a line parallel to ec’ it must pass through m, the 
anharmonic ratio of the pencil p. a0 6m being as co:c'o, and as the 
angle bpm= be o=bap (pm)? =am xX bm=square of tangent to circle 
from m, locus of m..is s—e?a2=0, s=o being equation of circle, and 
a=o that of the line 7¢. In like manner, if p be joined with 
/ 
middle point of cc’ joining line meets ad in m’. So that eas 2 bm age 
am'xX bo c’o 
and locus of m’ is s+e'%a?= 0, e being = to ce’ divided by sum of 
perpendiculars on rt from c and c’. The conics s—e?a?=o0, s—e/?a? 
=9, are polar reciprocals. The lines coc’, fo f’, each of which makes 
with diameter of circle passing through 0, an angle whose secant=e 
are parallel to the asymptotes of the conic s—e?a?=o, and _ polars of 
the points where the asymptotes cut (rt), while the line joining their 
