Treatment of the i Normal Curve ' of Statistics. 171 



formulae in the theory of error, and to apply them to questions which 

 arise in relation to normal distributions and normal correlation. The 

 method is, throughout, elementary, the use of the differential and 

 integral calculus being avoided, though geometrical infinitesimals are 

 to a certain extent employed. 



Geometrical. 



The normal curve is denned by the property that the product of the 

 abscissa and the subtangent is constant ; thus if MP is an ordinate 

 from the base, and PT the tangent, OM . MT = a 2 , being a fixed 

 point in the base. The area bounded by the curve and the base 

 OMT is called a normal figure. The length 2a is the parameter of the 

 figure. The definition of the curve leads at once to its projective 

 properties, and also to the formula for the mean value of x 2n+i or 

 x 2n+2 , where x denotes the distance of an element of area from the 

 central ordinate. 



If the normal curve or normal figure is rotated about its central 

 ordinate, it generates the normal surface or normal solid. It is 

 proved geometrically that any vertical section of this solid (i.e., any 

 section by a plane perpendicular to its base plane) is a normal figure 

 of the same parameter as the central sections; and, therefore, if the 

 sections of the surface by any series of parallel vertical planes are 

 projected on any plane of the series, the curves so obtained are ortho- 

 gonal projections of one another with regard to their common base. 

 The converse propositions are also established geometrically. 



The volume of the solid is obtained in terms of its central ordinate 

 and of the parameter of vertical sections ; and thus it is found that 

 the central ordinate of a normal figure of semi-parameter unity and 

 area unity is 1/ \/ 2tt. 



Let 2 be any closed curve in the base plane. Then it is shown 

 how to construct a curve whose area shall be proportional to the 

 portion of the solid which lies vertically above 2, i.e., to the volume 

 which would be cut out of the solid by a cylinder having 2 for its 

 cross-section. Thus, when 2 is given, this volume can be measured 

 mechanically. 



Statistical. 



Let L and M denote the measures of two co-existent attributes, 

 L x and M x their mean values, a 2 and b 2 the mean squares of the 

 respective deviations from the means, and ab cos D the mean product 

 of the deviations from the means. Then the angle D is called 

 the divergence. The solid of frequency of (L — L^/asin D and 

 (M— Mi) lb sin D, the planes of reference being inclined at an angle D 

 to one another (so that the included angle is 180°— D), is called the 

 correlation-solid. 



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