538 



SCIENCE PROGRESS 



them, so that the number of forms that the skew (or asym- 

 metrical) expansion may take is indefinitely large. Suppose 

 that P = q and that the series thus becomes the terms of 



/i i\ 10 • /i i\ n 



200 (- + -) > or in general Ml- + -1 . Then we get the series 



which satisfies the conditions of our first approximation 

 (p. 537), that is, the series given by the symmetrical binomial 

 expansion. It would be the series given by drawing ten 

 cards from a pack of ten each of all four suits, and recording 

 the numbers of red cards drawn in (say) 200 trials. Thus : 



Obviously the symmetrical binomial series is only the limit to 

 the asymmetrical series. In chance-series occurring naturally, 

 the former are exceptional, and skew binomials represent 

 chance-series more frequently than do the symmetrical bi- 

 nomials. This point is of some importance. 



The series formed from the symmetrical and skew 

 binomials are, however, discontinuous ones, while the series 

 formed by biological frequency distributions are continuous. 

 We might fit a binomial expansion to an observed frequency 

 distribution, but the former would consist of a series of 

 ordinates only, and would not be a continuous curve, while 

 the frequency distribution, being formed from a series of 

 grouped values, ought to be represented by such a continuous 

 curve. We must therefore obtain the limiting expressions 



for the expansions (p + q) n , first when p + q = — | — , and 



next when p is not equal to q, n being " infinite " in all cases. 

 To find these limiting expressions is not easy, and the proofs 

 usually given are difficult to follow. It is, however, estab- 

 lished that the limit of I — | — \ when n becomes indefinitely 



a;' 



large is the Gaussian curve y = y e ~ ~&% where y is the 

 maximum ordinate, e the exponential limit, and <r the " stan- 

 dard deviation," or " error of mean square." This Gaussian 



