Ui 



Putting' 



S. 





— "' '/"/ — (2) 



n nöj.Og 



wliere ii. denotes the number of equations, o the standard deviation 



and /),f, tiie eorrelation-coetïicient (cc.) between .Hj, and j'^, the ii — 1 



equations deduced from (1) can be substituted bv the equivalent set 



of equations : 



»•,, — rt,r,3 + (Ï, + ,«^rj, + . . . a„?'.i„ I 



By the quantities d thus calculated, the quantities /; l)econic 



0, 0, a,i 



Obviousl}- the equation (J) holds good only to a limited degree 

 because the data are necessarily incomplete; a measure of the com- 

 pleteness is obtained by putting 



from which 



or, by substitution of tiie values f2) : 



K,' = a/ -f a,' -f n,^ . . «„» 



+ 2a„a,r,, -(- 2<i,a^r.^, . . . 2a^a„rj„ 

 +-2",rt,7',4 + ^atd^r,^ . . . 2<(30„»'3„ 



-|- 2a,i_ia,ir„_i . „ 



The quantity A* represents the general c.c. of e(|uati()ns (I i and 

 the i)robable error of one determination of ,c, becomes 



w = ««1 \/l-R' a = 0.67449 



The partial c.c, defined as the c.c between ,(), and .c, when all 

 other values .*; are zero, is calculated by solving also the equations 



and is given by the expression 



9/'7 = ^^1"! ^'11' ■ 



the sign of o being that of the quantities fi. 



2V 



