Compound Law of Error. 211 



condition that 



zdx dy=l. 



Li 



Transforming back from the principal axes which we have 

 employed*, we find for the general expression 



1 -(mz2— 2Zzy+*y2) 



Z— .. e 2(km-l2) 



2-7T sjkm — l 2. 



By parity of reasoning we obtain as the general form for 

 the law of error relating to any number of variables x 1} x 2 , %& 

 Ac., 





1 ~ K 



,z 1 2+2L, 2 z ] x 2 +2L 13 z 1 z 3 +K 22 :r 2 2 +&c. 







2A 



2 — 



(2tt)2 ^ 





s 



where A is the determinant 









*i 



t>\2 



Z 13 . . . 





^21 



K 2 



'23 • • • 





'31 



'32 



k s . . . 





• 



• 



• 





• 



• 



• 





• 



• 



• 



# 2 =2 jj J . . . %i x i d%\ dx 2 dx 3 .... 



k 2 = 2§§§ ._... (J ^ 2 2 d#l ^ 2 cfa? 3 



Z 12 = 2 j Jj* . . . £ x 1 x 2 dx x dx% dx 3 . . . . = Z 21 , 



the limits of the integrals and extent of the summation being 

 as before ; K x is the first minor of the determinant formed 

 by omitting the row and column containing k x ; L 12 is the 

 first minor formed by omitting the row and column con- 

 taining l 21 , or l 12 ; and so on. 



* The values of the k and m which we have "been employing with 

 reference to principal axes are in terms of our original k } I, m referred to 

 any axes respectively : 



k cos 2 6 — 21 cos 6 sin d-\-m sin 2 6, 



and k sin 2 6 4- 21 cos 6 sin 6 +m cos 2 6 ; 



where tan 26= (k — m)-7-2l, 



See note on p. 208. 



