12 KARL PEARSON 



multiply by ^ M and integrate all over the plane, 



fee C 00 



2-rr F (r 2 ) x „rdr = b„2ir \ at^^rdr = b, s {^} 2 . 

 Jo Jo 



Hence b u = -j^r % \F(r*)x.rdr (xxii). 



115 J o 



Now ^ M consists of an algebraic series in ( - ) . Thus the discovery of the 



value of the integral I F (r 2 ) x^rdr depends solely on the determination of the 



Jo 



odd moments of F (r 2 ) between and oo . We conclude therefore that an 

 expansion in w-functions involves merely the determination of moments, such as 

 every statistician has been accustomed for years to calculate. This is not the 

 proper occasion to deal fully with the properties of the co-functions, nor to 

 generalise them for odd powers of r, and to consider the convergency of 

 expansions in terms of co-functions. They will be discussed on another occasion, 

 but the present writer believes that they will be found of not inconsiderable 

 service, not only in statistical problems, but in certain physical problems where 

 intensity round an axis diminishes with the distance. 



(5) Two further problems are of service for the theory of dispersion. 

 Suppose 



F^)=S^{K^), 



s=0 



Integrate over the plane and remember that Xo = l> 



f co s = oo /"oo 



2tt I F(r' 2 )rdr = S 2-rr & M <o M Y re£r 

 Jo s = o Jo 



= b (xxiii). 



Thus the first coefficient is merely the total volume of the surface z = F(r tl ) } 

 taken over the plane. 



Next consider the second moment 



f<*> s = oo I" oo 



2tt r*F(ir*) rdr — S 2tt \ b^ . Uj, . r'\ rdr. 

 Jo ' s =o Jo 



Every term of the summation vanishes except for s = and s=l, and the left- 

 hand side is the second moment of the function about the axis perpendicular 

 to the plane through the centre = volume x (swing-radius) 2 — b x K\ say. Thus : 



M^ = i r 6i e-*^dr + l f\e-^(l-^) r»Jr 

 o" Jo o-Jo \ 2 ar 2 / 



= 26 cr 2 + b 2 (2 - 4) o- 2 = 2 (6 - b 2 ) a 3 , 

 or b t = b t {l—j i E i /o'} ( xx i v ). 



