ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 121 
To find this curve, take a second series of points Nq, N,. . . . N,,+i, also at equal 
distances ijs/n, and in such a position with regard to the former series that 
= a|^/n, NpMp =l3lx/n ; 
and at the points Nj, Ng, . . . N„ erect ordinates NiQi, N 2 Q 2 , . . . N^Q,^ (fig. 8) equal 
to the coefficients in the expansion of \/ n{/3x -j- Thus 
Fig. 8. 
These ordinates lie in the successive intervals between the ordinates MyPo, MjPi, 
, . . M,jP,,i ; and it is easily shown that (except where it is the maximum 
ordinate) is intermediate in magnitude between M^,_iP^,_i and M^,Pj,. Also we have 
+ ^-N^+iQ,,+i = V (C;:l + C;r^) = <v/7n= M^,P„. 
But : /3 : a ; and therefore P^, lies in It follows that, in 
the limit, Q^Q^+i becomes the tangent at P^. 
Let meet X'X In T^. Then 
n . 13'^-^-^ {/3C^Z\ - aC;r‘] 
— 2)) «}. 
Hence if v/e choose the point O so that 
\/ 7 i.OMj, = — na 2 > = — {71 — p) a, 
we have 
* When n is not infinite, the relation OM,, .M^T^, = a/I shows that, if 2 denote any one of the family 
of normal curves of parameter ccji having their median at 0, the sides of the polygon N 0 Q 1 Q 2 • • 
VOL. CXCIl.-A. 
- N^+iQp+i 
R 
