\L\II XLVIII] Inlrwlm-tinn OOXXfil 



Numerically ,// = ,Z?j = 0'3989,4228 .<xi) 



r-.r all values of e, and Table XI, VII gives the values of 2 /y , ,//, ,//,, ,//,, ,#,, ,#,, 



iBl, 1#3, 8#3, 3#3, l/*4, 2/^4, 3/>4, A, 2/4 J /'l, l6 f , , 36',, ,C, , /,' tl //, ,6', , ,C, <V 



id) 1^4 for values nl t from 2 to 10. 



Table XLVIII gives the values of -L, i, -L f ^, _J_ p ^ for intcgf . r va | uef , 

 of n from 50 to 200. This is convenient if n be an integer, but when, as i .ft. M 

 the case, n is fractional it is probably shorter to find -^, -, -^ and -| directly, 



Tl Tl 71 ft 



and the other fractional powers by multiplication. 



Certain precautions must be observed in using formula (viii). Namely : 

 (a) As given, it is for a value of x>x, the mode. 



(6) It assumes 'that I is greater than m ; if I be less than ra, then the sign 

 before the (7-series must be changed to plus. 



(c) If, on the other hand, x< x, and / >m, we obtain our value by working 

 from the other end of the curve. In this case, for I > m, we have : 



and our formula becomes 



= L l+l M m+1 (1 - x 



-- x\ 

 n I 



f 1 

 X [u 



u* + e (w 2 + e)(w a +2e) 



2 + 6) (M 2 + 2e) " (u* + e) (u 2 + 2e) (u* + 3e) + GtC ' j J (xn)> 



where the (7-series must be reversed in sign if I be < m. 



In order that this formula may be effective e must be fairly large, i.e. / and m 

 must differ widely, or, if they be nearly equal, and therefore e small, for example 

 2, 3 or 4, then u must be fairly considerable to give ample convergency in the B 

 and C series, say at least 3 to 4. But looking to the value of u in (vii), this means 



llm 

 x-;c>ux^-i, 



Abscissa-Mode / Im _ (n + 2)*(n + 3) 



Standard Deviation >U V (l+l)(m + l) ~ w 3 



by (vi). Or, if I and m are considerable, the radical approaches unity, and con- 

 sequently the distance from the mode of the point to which the area is taken must 

 be 3 to 4 times the standard deviation. 



For example, if / = 20, m = 80, then by (vi) 



a- = -03984. 



