660 Prof. Karl Pearson on a General 



y'i y"i U"' y-Uo-) y"-> y" r .*&&&. the corresponding x values in 



Ca, Cfa . . . C V . 



This gives us the type equations : 

 S(c« (y -y) ) = A «S ( c *«) + AjSSCcoc^a ) + ••• + A vS M , 

 S(c0 (y -</)) = A *S ( C « C/3 ) + A /3S (c/) + . . . + A vS( C/3 6v) 5 



S(tv(?/ -?/)) = A aS(cvCa) + A o ^S(cvc / 3) + . .. + A vS(c„ 3 > (iv.) 



We have thus rc equations to find the n unknowns A a, 

 A /3,...A o i/, so soon as the summation terms have been found. 

 But, clearly, 



y P —y = ^v* c a + Apflcp + • • • + V c >- • • ( v 



Multiply by y —y and sum : 



map (t r 0p = A 7 ,aS (c«(y „ - y ) ) + Ap£S (cp(y -y)) + ... 



+± p vS(Cv(y -y)). . (vi.) 



Takings from^=l to p=n } we have ra equations to find 

 the unknowns S(ca (y Q -3/)), S(c /3 (# l -y))...S(c„(y -y)) on the 

 left of equations (iv.) above. 



Now let D = the determinant 



\<x. Aj/3,... Aii/ 

 A 2 «, A 3 /3,... A 2 v 



A„«, A„£,... A„v 



and suppose d pq to be the minor corresponding to the con- 

 stituent of this in the pth row and qth. column. 

 Then 



Ca = p {diafyx ~!J) + d 2a [y 2 -y) + d 3 « (#3 ~#) + ■ ■ • • + &*«(#« -y) \ 



^3= ^ {4p(yi -y) +d2p{y2-y)+d3p(y3-y) + • • -+d n p(yn -y). . (vii 



Hence 



« JJ" 



g 8 [ P s"(<VV)+2S'( 



d pad j) aO" j >(T p' I 1 pp' ) 



} 



. . (viii.; 



the second sum S' embracing all pairs from 1 to rc of unequal 

 p and y/. 



