104 
■\rR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
once from (3); for if O is the foot of the central ordinate, and if MP is any other 
ordinate, and the tangent at P meets OM in T, then sub-tangent MT = — 2 dxjdz. 
(5.) Statistical Equation, = {k l)X 2 Xi., where X;,. denotes the mean I'th power 
of the deviation from the mean in a distribution whose curve of frequency is a normal 
curve ; h being any positive integer. This relation follows from (3). Since, by the 
definition, Xj = 0, it gives X/, in terms of X., for all positive integral values of h; and 
it may therefore be regarded as the equation to the curve, the position of the central 
ordinate being arbitrary. 
Of these different equations the first is in some respects the most important, as it 
is the direct expression of the relation on which the special property of normal dis¬ 
tributions depends; the property, that is to say, that if the measures of a number of 
independent attributes are normally distributed, any linear function of these measures 
is also normally distributed. The second equation is, of course, essential for any 
numerical calculations. The last two, however, have certain conveniences when an 
elementa,ry investigation is desired. I have therefore adopted the geometrical defini¬ 
tion of the curve, and have deduced the statistical equation ; and then have used 
either or both of these as occasion might require. 
The memoir is divided into four parts. Part I. deals with elementary theorems ; 
most of these are well known, but it is convenient to have them collected, and 
established by comparatively simple methods."^ Part II. contains the investigation 
of the principal formulse in the theory of error as applied to numerical statistics. 
In Part III. these formulse are applied to cases of normal distribution. Part IV. 
deals with normal correlation, and is subdivided into two portions. The first con¬ 
sists of a discussion of the more important phenomena which occur when two 
attributes are normally correlated ; while the second contains the applications of the 
theoiy of error. Some of the formulae given in Parts III. and IV. have already been 
obtained by Professor Karl Pearson, but by a different method. 
Part I.—General Properties of the Normal Citrve and of Normal 
Distributions. 
The Normal Curve. 
§ I. Definition of Normal Curve ,— Let O be a fixed point in a straight line X'OX, 
and let a point P move so that, if MP is the ordinate to P from X'GX, and PT the 
tangent at P, intersecting X'OX in T, the rectangle OM.MT is constant and = cd. 
Then the path of P is a normal curve. 
Let OZ be drawn at right angles to X'OX, intersecting the curve in H, and let 
points A' and A be taken in X'OX, such that A'O = OA = a. Then OZ will be 
* It will be seen that some of the proofs are only expressions, in geometrical form, of familiar 
methods of differentiation or integration. 
