no. 3.] PSYCHOLOGICAL EXAMINING IN THE UNITED STATES ARMY. 



637 



Solving these equations for x lt x 2 , etc., we obtain the following results: 



a 1 R l2 R t 



(J., 

 ^3 



a t R t 



13 

 ^22 ^23 

 ^-33 **33 



R. 



X l _ d <r n R n3 R n 



#n) 

 R 2 b 

 # 3 c 



R™ 



# n 



°\ #11 ^13 #14 



a 2 it 21 xt 23 ti 2l 



tr 3 R 31 R 33 R 3i 



a A R u R 43 R it 



x 2 _ d a n R ni R a3 R D4 



<7, A 



#, n 



# 2 n 

 #3H 

 #4D 



#n 



Xs = d_ 



<r 1 R n R l2 R u 

 a 2 R 2l R 22 R 2i 

 <r 3 R 31 R 32 R 3t 



<r n i? ni R B2 R Bi 



#m 



#2B 



#sn 



#n 



etc. 



where A = 



a 1 <r 2 a 3 • • • ff n 



"i #11 #12 #13 * * ' #in 



^2 *^21 -^22 -^23 • • * £i 2 \i 



C 3 /l 3I ti 32 •ti-33 ' * * &311 



&n **ni -*^n2 **n3 • • • xlon 



Further reduction leads to : 



x t (<r,+r 12 <r 2 + r 13 (T 3 + r lt o i + ■ ■ ■ + r, D <r p )<? 



where 



SW) + 2S'(r li c i <7 i ) 



(6) 



S' being a double summation for all pairs of values of i and j from 1 to n, except i = j. These 



are very simple results, for if 



X = x 1 +x 2 +x 3 + • • ■ x n 



it is easily shown that 



■ Y _ (T 1 + r 12 (7 2 +r 13 <r 3 +r li <T i + ■ ■ ■ r 1D a a 

 t-xi^- - s 



and that 



Therefore the results in equations (6) are merely the values of x u x 2 , etc., given by the following 

 regressions of x u x 2 , on their sum, for the special case of the sum equal to d. 



£i= r X— 

 <r, Txi ^ 2 



- 1 = r I2 X^r, etc. 



