KAiiL Pearson 
9 
Laplace's method gives the " normal curve " as an approximation to the original 
series, when we make very limiting assumptii)ns as to h, ;u, P. It led to his 
suggestion that |e~^' "rf.r should be tabulated, and eventually to the CdUiputation 
of the so-called '■ probability integral." 
(5) It will be seen that Laplace's method is only a rough method of approxima- 
tion to our original series for C,.. That series is only a special case of the general 
hypergeometrical series, and we have at once thrust upon us the problem : Can no 
l.r:"- 
better curve be found than the Gauss-Laplacian y = rjne to represent the sum 
of any number of terms of the hypergeometrical series ? The answer is that curves 
infinitely better which demand no restriction on the values of vi, n, P can be 
determined, and these curves can with just as much validity be called probability 
curves as the above normal curve ; and the integrals of their areas up to any gjven 
value have equal claim to be " probability integrals." 
Let us return to Laplace's starting-point on p. 6. We have at once for 
the ratio : 
C,+i _ {p + r+\)s (p + r + 1) (711 - r) 
Hence 
Or {r+l){q + s) {r + l)(q + m-r) 
Cr+i — Oi._ p{m + 1) —n — rn 
C,+, + G, m {p-^2) + q + r[2{ra-l)-p + q] - 2/ ' 
Divide by /; and put pjn = P, qjn = Q as before, and we have 
\ n/ \ n J n 
Now suppose Co, C'l ... C',., ^,.+1 ... to be plotted as a histogram, i.e. as a series of 
rectangles of base c and heights C'„/c, Cj/c ... 6V/c, C,.+)/c' .... Let the tops of the 
midordinates of these rectangles be joined, so as to form a polygonal figure 
Po, Pi, P.2 ... P, , P,+i ■■■ ■ If we take the origin at the ordinate GJc we shall have 
for abscissae of the angles X,. — rc, Xi-^-^ = rirc . . . and for ordinates y,. — Gr/c, 
y,.+i = Gf+ijc. Thus the abscissa and the ordinate of the midpoint of a side of the 
polygon will be , = ( + and v/^ ,^,1= 2 (CV+i + G,.)lc. Accordi ngly we have, if 
A.« = c : 
2 c\P{m + l)-\^-X 
r + J 
Now let us ti-ansfer the origin to the mode, i.e. take 
- .r,. + = c IP ( m + 1 ) - ^1 - + i. 
