A. RlIlND 
133 
The equation to Type IV. is 
with the redaction formula* 
fln+i = — ^ V-'^l ^ 2(1 tan <^fln] , 
r — n V cos- (j) Tr J 
where tan ^ = i//?- and ?• = 6 A - l)/(2y8o - o/3i - 6). It is clear that when 
7- — n, fiji+i becomes infinite. Now the probable errors of and /3., require us to 
go as far as fj.^. Hence ?■ must be greater than 7 if we are to use this formula, 
and this gives us at once the limiting line 
8/3., -15^1 -36 = 0. 
Of course the difference formula would also fail for fx.^, or /Xj, if r were equal 
to 4, 5 or 6. But all the resulting lines lie outside the above line, which is all we 
need take into account. The failure of the difference formula is easily seen if we 
remember that 
1 di/ _ ((„ + cfiif 
y dx Co + CiX + C2X" 
leads at once for this type to 
j (a„ + (iiu:) ydx = j (Co + c^x + c..:r) dy 
or N (UofXn + tlil^n+l) = - ■N' (c„»//,i_i + Ci (n + 1) yU.,, + C2 (n + 2) 
+ [y (c„x"- 4- CiA'"+i + c.,x'"+")\tZ- 
The difference formula above follows from supposing the term between brackets 
to vanish at the limits. 
But this it will not do unless 
or if a be finite, unless r be > u, or to apply present results r > 7. 
Of course for any real data /j,^ may become large, but it cannot actually 
become infinite. Fairly good fits — owing to the agreement of the first four 
moments — may be found even near the line 8/3-2 — — 36 = 0, but if we want to 
get the probable errors of /3i and fi., in this neighbourhood, it is best to calculate 
the higher moments and /S^, jS^, /3s and /S^ from the actual data. Outside this line 
* See Biometrika, Vol. 11., p. 281. 
