1895.] Mathematical Contributions to Theory of Evolution. 257 



Calculated as above the composition of the undiluted black-damp 

 was thus : 



Nitrogen 85'30 



Carbonic acid 14*70 



This result differs little from that given by the other samples. 



II. "Mathematical Contributions to the Theory of Evolution. 

 II. Skew Variation in Homogeneous Material." By KARL ; 

 PEARSON, University College, London. Communicated by 

 Professor HENRICI, F.R.S. Received December 19, 1894. 



(Abstract.) 



PART I. Theoretical. 



In the deduction of the normal curve of frequency from the sym- 

 metrical point binomial, three conditions are usually assumed to be 

 true : 



(a.) The chances of any " contributory cause " giving its unit of 

 deviation in excess or in defect are presumed to be equal. 



(6.) The number of " contributory causes " are supposed to be 

 indefinitely great. 



(c.) The "contributory causes" are all supposed to be indepen- 

 dent. 



(c) amounts to the assumption of a binomial form (p + q) H j (a) to 

 the equality of p and g, (c) to the indefinitely great value of n. 



It is shown in the paper that there is an important geometrical 

 relation between the normal frequency curve 



and the symmetrical point binomial, OJ + i) w , which is true inde- 

 pendently of the magnitude of n. Thus the condition (b) is not 

 necessary to the very close fitting of symmetrical point binomials to 

 normal curves for even very small values of n, such, for example, as 

 8 or 10. This has been long recognised in statistical practice if its 

 source has not been noted. 



We can remove the condition (a) from our d- priori limitations by 

 finding a curve which is related to the skew binomial (p + q) n in pre- 

 cisely the same manner as the normal curve is related to the sym- 

 metrical binomial (^ + ^) w . The equation to this curve is 



h*)V". 



a 



