XXV XXVI] lutwlm't;,,!, cxliii 



Thus /'. 10 (i3* 25 )-* 787 . l 9ti7.'', 



which is exact. 



Beyond x = 75 the forward dilVrn -MCI- method will still apply, but the number 

 of differences required is excessive, if we start with q 2 or 3. For <y = 4 the eighth 

 difference suffices for 7-figure accuracy; for q = 5 the fifth diH'cn-nce will 8iifli< 

 for like accuracy, and so on ; this supposes working with B x (\, q) instead of 

 I x (\, q). Thus by the time we get to q = 8, there is no trouble. For many statistical 

 purposes four or five figure accuracy is adequate, and accordingly there is leas 

 trouble with forward difference work. 



For such an extreme case as /. w (\, 3'25) the limiting difference that the 

 present table provides is the twelfth. Even if we use this and the forward difference 

 formula we shall be out by slightly more than unity in the fifth decimal place. If we 

 proceed also to the twelfth difference, using B. 90 (, 3*25), we shall be out by less than 

 unity in the sixth decimal place, and assuming the thirteenth difference to be about 

 half the twelfth (as it must be here) we can obtain a value differing from the true 

 value by less than five units in the seventh decimal place. The labour, if straight- 

 forward, is considerable, and some will prefer to obtain the result by expansion 

 methods rather than by using the present table of I x (, q) or B x (^, q) when x 

 approaches unity and q is fractional and small. 



TABLE XXVI. 

 Table to find Modal Ordinates of symmetrical Frequency Curve*. 



n TT 



/Z f2 



cos n ~ 1 <t>d<j>= sin n-1 <j>d$ for n= 1 to 105. 

 j Jo 



Here ?-**<M) = i ^ r<Jn)/r(J(n + 1)) .................. (i), 



where B and F represent the complete Beta and Gamma functions. The value of 

 q n can be found by this relationship from any table of the complete F-function *. 

 But the present table enables q n to be found without a double interpolation for 

 TO < 106. 



The three types of frequency curves with which we have to deal are : 



(a) Symmetrical U-shaped frequency curve 



y=y / ^yv *"<* 



Range : x = a to + a. 



N 



N being the total frequency. 



* For example, Tracts for Computers, Nos. vm and is. Cambridge University Prem. 



