ERROR TO CASES OP NORMAL DISTRIBUTION AND CORRELATION. 105 
called the median of the curve, X'OX the base, OH the centred ordmate, and A'A the 
parameter. 
The curve is obviously symmetrical about the median, and asymptotic to the base 
in both directions. 
The area bounded b}^ the curve and the base will be called a normed figure. 
§ 2. Formation of Family of Curves by Projection. —Let a new curve be formed by 
orthogonal projection of a normal curve with regard to the base in any ratio. Let 
MP and NQ be ordinates to the original curve, and MP' and NQ' the corresponding 
ordinates to the new curve (tig. 1). Then MP : MPA: NQ : NQ'. Hence PQ and 
P'Q' will intersect on the base. Let N move up to and coincide with M. Then PQ 
and P'Q' become the tangents at P and P' to the two curves, and therefore these 
tangents meet the base in the same point T. Hence for the second curve we have 
also OM.MT = OA-, and therefore this is also a normal curve of parameter A'A. 
JSimilarly, if the curve is projected with regard to OZ in the ratio a : b, the new 
curve will be a normal curve of parameter 2b, having the same median. 
Fig. 1. 
§ 3. Limitation to Curves so obtedned.—Thws,, by projection of a single normal curve 
with respect to the base and the median, we can get an indefinite number of normal 
curves of different parameters and different central ordinates. Conversely, if S and 
S' are two normal curves placed so as to have the same base and the same median, 
either can be got from the other by projection. Let the parameters be 2a and 2b 
respectively. Project S into a curve S" of parameter 2b, and let S denote the family 
of projections of S" with regard to the base. Then the tangent at each point of S' 
coincides with the tangent to the particular curve of S which passes through this 
point. Hence S' is one of the curves or else is the envelope of these curves. But 
the curves have no envelope at a finite distance. Hence S' is a projection of S". 
§ 4. Standard Normed Curve .—It is, therefore, convenient to take a standard 
normal curve, and to consider all other normal curves as obtained from it by projection. 
VOL. CXCII.—A. 
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