72 



2. The 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 Ofio, OoOg, O5O7, are each diameters 

 of 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 A w be two of these perpendiculars, the diagonals 

 of the rectangle Am B n bisect each other, and consequently, 

 the ^.raight lines in theorem 3 pass through the bisections of 

 tliQi sides A B, A C, B C, respectively, thus forming a triangle 

 *tif^ 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 auxihary 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 thQ 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. 



