50 Records of the Indian Museum. [Vol. XXIII, 



Thus an asymmetrical frequency curve may be really built up 

 of normal curves having parallel but not necessarily coincident 

 axes and different parameters. The object of the present section 

 is to discuss the possibility of splitting up our asymmetrical fre- 

 quency curve into two component normal curves. 1 



Pearson gave necessary mathematical formulae 4 for this pur- 

 pose in his memoir of 1894. The solution depends on finding the 

 roots of a numerical equation of the ninth degree, and the arith- 

 metical calculations are extremely laborious. Pearson has dis- 

 cussed the application of the theory in several actual cases. 3 



Let /> 2 > M3, /"^ and /* 5 be the moment-coefficients, M the mean 

 and N the total of the given frequency curve. Let m lt m. 2i be the 

 means, <r,, o- 2 , the standard deviations and n u n. 2 the totals of the 

 component curves. 



Then if h is the unit of grouping 



m 1 =M + y, . h and m l — M + y 2 . h 



Also, taking h=i, we have 





°-i* = M2- J/Vyg- iPtfi + Pz 





a % % = h - \v%h\ ~ IPrfz + Pi 





»,-- * 





y\-y-i 





n 2 = + * 





7i- y-i 



Let 



P\='Y\+ 72, p2 = y\-yz and 



Also 



A 4 = 9/ x 2*-3/' 4 > A & = 30/*s^3 - 3/*5 



Then 



_2^-2n s \^P 9 -\ b P i 7 --8^p 2 S 



4^- X 4 />2 + 2/)o S 



Pz = P\>Pi 



ft Hence, so soon as p. z is known, Pi = pjp. 2 can be found, and 

 then y, and y i will be the roots of : — 



Y 1 -p\y + p2 = ° 



The equation for finding p % is one of the ninth degree : — 



24PJ* - 2%k4>£ 4- 36M3W - (24/*8 X 6 " lOlfip,* ~ ( HWK - * Vfcl* 



f (288/x^ - I2X 4 \ 6 ^ 8 - A/)/)/ + (24/x 3 s A 5 - 7/x 8 V)^* + 32/* s *X 4 /> 2 - 24/, :s 6 = 



1 Ibid., p. /_>. " There are reasons, indeed, why the resolution into two is of 

 special importance. A family probably breaks up into two species, rather than 

 three or more, owing to the pressure at a given time of some particular form of 

 natural -election .... Even where the heterogeneity may be three-fold or more, 

 the dissection into two is likely to give us, at any rate, an approximation to the 

 ( hie! groups." 



I he fundamental formulae have been expressed in a slightly modified form 

 in terms of the 0-constants in a recenl paper " On Sexing Osteometric Measure- 

 ments." Biomwtrika Vol. to, 1915, pp. 47c) — 487. 



K. Pearson: "On the Applications of the Theorj ol Chance to Racial 

 Differentiations," Phil. -U<itf. 1901, p. no. 



K. Pearson: "On the Probability that two Independent Distributions of 

 Frequencie are really Samples,,! the Same Population, etc.," Biometrika Vol. 

 io, 1915, p. 123 tt seq. 



