THE MEASUREMENT OF VARIATION. 31 



mathematical expression. We saw a few pages back 

 that an expansion of the binomial (^ + ^) for 20 or 

 more times gave a series of values which differed very 

 slightly in their frequencies from that required by the 

 Law of Error. Supposing, now, a binomial in which the 

 two terms are unequal is expanded, then obviously an 

 asymmetrical series of values is obtained. For in- 

 stance, instead of (-J + J) let (f + i) or ( +, ) be 

 expanded, and series are obtained of which the diagram- 

 matic representations are given in the two curves to 

 the left of the accompanying figure. The symmetrical 

 curve represents the expansion of (i + i), the expres- 

 sion being in each case expanded ten times. The areas 

 enclosed between each of these curves and the base line, 

 or the so-called polygons of variation, are obviously of 

 exactly equal extent, in that the sum of the two terms 

 expanded is in each case equal to unity. 



It follows, therefore, that these asymmetrical series 

 can be represented by the expansion of the expression 

 (P + (?)"* Supposing that n is infinitely large, then 

 curves representing the expansion would stretch out to 

 an unlimited extent in each direction, and though con- 

 stantly approaching nearer and nearer to the abscissa, 

 would never touch it. Supposing n is some finite num- 

 ber, as 20 or 40, then obviously the series is finite also, 

 and its curve is limited in extent. If the two terms of 

 the binomial are unequal, then the curve approaches the 



* The algebraical expansion of this expression is: 

 (p + q) n = 



-{ngn - 



