92 Prof. F. Y. Edgeworth on the 



correction of, or addendum to, the above expression an ex- 

 pression which is described as the third differential with 

 respect to / of the value of P above written (as a first 

 approximation) multiplied by a constant l L which 



Performing this operation* and putting (as before) each of 

 the &'s = 0, and therefore Z=0, 



and (as before) substituting x for c, Ax for 2rj, c 2 for 4:k\ we 

 find the addendum (to the formula for the symmetrical 

 probability-curve), in our notation, 



2/ -^Vx 2 x z l A 



Adding the correction to the first approximation, we have our 

 formula for the asymmetrical probability-curvef. 



II. A second proof is derivable from the reasoning 

 by which Mr. E. L. De Forest in the ' Analyst' (Iowa) 

 (vol. ix. p. Ib3) obtains for the asymmetrical probability- 

 curve a certain formula which has been independently 

 discovered by Prof. Karl Pearson J . The expansion of this 

 formula in ascending powers of x will be found to coincide 

 with the expansion of our formula in ascending powers of x ; 



■provided that the second and higher powers of ~4 may be 



neglected — a condition which is employed by Mr. De Forest 

 in his proof §, and may readily be established || . 



* This and subsequent operations are performed more fully in the MS. 

 deposited in the Archives of the Royal Society. 



t When writing this paragraph I had not adverted to the similar work 

 in Galloway's treatise on Probability (forming the article on that subject 

 in the seventh and eighth editions of the Encyclopaedia JBritatinica), 

 art. 136. 



% Philosophical Transactions, 1894. Proceedings of the Royal Societv, 

 1893, p. 331. Cf. ' Nature,' 1895, p. 317. 



§ E. g. loc. cit. p. 138, regard being had to the definition of his symbols. 



|| For example, let the elemental frequency-locus consist of two points, 

 at a distance b, assumed with respective frequencies p and q. Then the 

 mean square of error for a single element is pqb 2 , and the sum of these 

 means for all the elements, our k, is npqb 2 . Also the mean cube of error 

 for an element is ±pq{p-qW_(cf. Phil. Mag. vol. xxi. (1886) p. 320) j 

 whence j-7-k^—(p — q)-r- Vn Vpq. Which is small when VSi is large 

 relatively to (p — q)-7-*/pq, a quantity which vanishes when the element 

 is symmetrical, and is finite for all but infinite degrees of asymmetry. 

 By parity it will be found thatj-^-^ i n general — - a finite quantity -r Vm; 

 so that it becomes small when n is sufficiently large. 



