72 
2. 'Fiie centres of the twelve circles which circumscribe the 
component triangles of the three preceding quadrilaterals, lie, 
two and two, in six points which are in the same circumference 
ABC; and the lines O 0», O^O*, 0j0 7 , are each diameters 
ot this circle, and intersect at its centre. 
3. If perpendiculars be let fall from any angle of a triangle 
upon its internal and external bisectors of the other two angles, 
the four points of intersection, in each case, will range in a 
straight line. 
4. If A m and Aw be two of these perpendiculars, the diagonals 
of the rectangle A m B n bisect each other, and consequently, 
the straight lines in theorem 3 pass through the bisections of 
the sides A B, A C, B C, respectively, thus forming a triangle 
r s t equal to one-fourth of A B C. 
5. If from any point in the directrix of an ellipse there be 
drawn two tangents to the curve, and if these be produced to 
cut the auxiliary circle upon the major axis in X and Y, then 
if X Y be joined, the line X Y will pass through the centre 
O of the ellipse. 
6. Conversely ; if from the extremities of any diameter of 
the auxiliary circle on the major axis there be drawn tangents 
to the ellipse, they will intersect on the directrix. 
The demonstration of some of these theorems is easy, and 
of an elementary character ; that of others depends upon con- 
siderations which have been advanced by the Author in some 
of his papers that have appeared in the Ladies’ and Gentlemen s 
Diary. 
Mr. G. V. Vernon, F.R.A.S., produced Manuscript 
Copies of thirteen years Barometrical Observations made at 
Greenwich, which had been kindly communicated by the 
Astronomer Royal, showing the observations once a day for 
every day instead of the daily means, and presented such 
series to the Section, for which present he received its thanks. 
Resolved, that the Secretary be requested to prepare and 
issue a circular, applying for copies of Logs and Meteorological 
Observations made at sea, and especially on board the various 
mail steamers. 
