On Jlnalytical Geometry. 



285 



Should this body prove to be a new proximate principle, the name 

 betuUne, suggested by a distinguished scientific gentleman, who has 

 seen some of it, would be particularly appropriate. 



I have 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. I have been unable to detect any of the crystalhne matter 

 in it. 



Art. IX. — On Analytical Geometry; by C. Wilder. 



Let a, B, and C, be points in any surface whatever, connected 

 by the lines AB, AC, and BC or oc, y, and z, the shortest that can 

 be drawn on that surface between those points. 



C 



Then x-\-yyz, y-\-zyx, or 2: > a? — y (a; being greater than y,) or 

 what is the same thing z<Cx -{-y and > ^(a; +y) ^ — 4xy, 



z<^'v{x — yy-i-4xyB.ndyx — y, 

 and which shews by inspection alone, that when A and B are compris= 

 ed between and 1, that we may write z=^'\^{x-{-y)'^ —4Axy, 



z=^/{x-yy-\.ABxy, 

 from these equations, we find 



— z^-{-(x-{-y)^ -z z X y 



xy X y ' y * X * ^ 



^ z"^ —{x—yY z z IX y \ 



xy X y \y ^ X 1 



When therefore the angles of the triangle ABC depend on the relative 



position of the points A, B and C, A and B will be functions of those 



Vol. XX,— No. 2. 37 



