285- 
Should this icrls prove to be a new proximate principle, the name 
-— betuline, suggested by a distinguished scientific gentleman, who has 
e of it, would be particularly appropriate. 
_ Thave obtained considerable quantities of resin from the thin cuti- 
cle of the black birch, (Betula nigra). In its general properties it 
resembles that obtained from the. white birch. It is, however, of a 
darker color, and gives off, when burning, a slightly empyreumatic 
_ odor. Ihave been unable to detect any of the crystalline matter 
in it. . , ; 
Arr. IX.—On Analytical Geometry 5 ‘by 
“Ley A, By ‘and C, be points in any surface 7 ever, connected 
by the lines AB, AC, and BC or a, y, and z, the shortest that can 
‘drawn on that surface between those points. 
Cc 
7 Then 2+-y>z, yt+z>a, or z>x—y (x being greater than y,) or 
What is the same thing z<#+y and > >V (a+y) — 4ary, 
— gc WV (a—y)? +4ry and >a—y, 
and Which shews by inspection alone, that when A and B are compris- 
ed between 0 and 1, that we may write z=V (vty)? —4Azy, 
z=v (x—y)? +4Bry, 
~ fom these equations, we find 
—z? z= ii, Pak 
cape - mesa ee er Z 42, 
—(r#—y)? 22 F 4 
es fate ies 2 (244-5) 
When therefore the angles of the triangle ABC depend on the relative 
Position of the points A, Band C, A and B will be functions of those 
Vol, XX, —No. 2. $7 
