The Normal of Curvature. 231 
Art. XXV.—The Normal of Curvature; by Grorce CiinTos 
Wurrttock, Prof. of Chem. and Nat. Hist. in Genesee College. 
Tue complicated method of the osculatory circle and its met- 
aphysical difficulties, has conducted me to the following solntion 
f the problem of curvature, founded on the natural and simple 
relations of intersecting normals to the are embraced by them. 
he equation of the normal to a curve o 
at the point (y, x), is 
1 dy 
Ea hs od Mik [y=3 | 
or ( —Y¥,)=2,—-2, 
Y,, &, being the co-ordinates of any 
point in the normal; so ; 
(ythy (ytk—y,)=.—(t+h), 
is the equation of the normal intersecting the curve in the point 
(yt+k,r+h). Or, for ¢, the point in which the normals inter- 
sect, distinguishing the codrdinates of this point by y., 7,, the 
equations of the normals become 
y’( ~¥s) — Be F, 
ytky(y—yo) Hy thyk=a,—2—h} 
. [((ytky yy - yc) Hy th k= —h, 
Ly a k 
and YEO AY (y-yn)4(y thy G=—13 
from which, reducing h to zero and observing that [*] =y' and 
pee 7 = pee =(9'x)'=9"z, [y=9r], 
we have y"(y— Yo) ty’? +1=90, 
ay? 
sar JER 
pa gee tye 
or Yo-Y and by substitution, 
Therefore, putting 9, which we denominate the “ Normal of 
Curvature,” for the length of the. normal embraced between the 
point « and the curve when h=0, there results, 
(y?2+1)* 
_ 
S=V (yo—y)? +(@- 25)? = : 
It is evident that the Normal of Curvature applies much more 
directly and simply to the problem of central forces than does 
the Radius of Curvature. fo 
Genesee College, May 7th, 1853. 
