I.—On toe Law or Error in Tarcet-Suootinc. By E. UL. 
Der Forest, Watertown, Conn. 
THE complete expression for the symmetrical law of error in the 
position of points in a plane is 
pel A,h,dady _ (hy2a? + ho? y’), (1) 
70 
where 2 denotes the probability that an error committed will fall 
within any small rectangle dxdy whose codrdinates, at its middle 
point, are « and y. The axes should be taken to coincide with the 
free axes of the group of shot-marks, when these last are regarded 
as the masses of material points all equal to each other. The origin 
is at their centre of gravity, and is the point for which the proba- 
bility zis a maximum. (Compare my article in Zhe Analyst, Des 
Moines, Lowa, vol. viii, p. 73.) Though dz and dy are in strictness 
infinitesimals, the formula is evidently approximately true when 
they are regarded as any small finite distances. Points for which z 
is a given quantity will lie in an ellipse, and all such ellipses are 
similar and concentric as long as the constants h, and h, remain the 
same. ‘These are determined by the relations 
1 I 
aa Ws aa Orr @) 
where p, and p, are the quadratic mean errors in the x and y direc- 
tions. If the probability of deviation from the maximum is the 
same in all directions, then 
=P, + Py =2p,'=2p,. (3) 
is the squared q. m. error measured directly from the origin, and 
1 
hh zh.=— 4 
eect (4) 
is the constant to be introduced in (1). Denoting «*+-y’* by 7”, (1) 
is reduced to 
a pnd 
pao eee (5) 
where the ellipses of equal probability have become circles, and the 
axes may be taken in any convenient direction. As this formula is 
TRANS. Conn. AcapD., VoL. VII. 1 SEPT., 1885, 
