Probable in a Correlated System of Variables. 101 



determined when the first n are known, and in using formula 

 (ii.) we treat only of n variables. Now the standard de- 

 viation for the random variation of e v is 



■=\A( ll! £)3' • • • ™ 



and if r pi be the correlation of random error e p and e q , 



O-pO-qVpq 



lp" l q 



(viii.) 



Now let us write =^- = sin 2 /3<?, where /3? is an auxiliary 



angle easily found. Then we have 



a-q— \/N sin/3, cos @ q , 

 — tan j3 q tan/Sp. 



P9 



(IX.) 



(x.) 



We have from the value of R in § 1 



R = I 1 — tan /3 2 tan fi l — tan /3 3 tan /3i . . . — tan /9 m tan /3) 



! — tan /3i tan yS 2 1 — tan /3 3 tan /3 3 . . . — tan /3, t tan /3 2 



— tan /^ tan /3 3 — tan /3 2 tan /3 3 1 ... — tan fin tan /3 3 



- tan /3j tan /3w — tan /3 2 tan /3 n — tan /3 3 tan /3« 



= (-l)» tan 2 ^ tan 2 /3 2 tan 2 £ 3 . . . tan 2 /3„ x 



- cot 2 /3i 1 1 1 



1 -cot 2 /3 2 1 1 



1 1 - cot 2 & 1 



111 -cot 2 /3„ 



= tan 2 /3i tan 2 /3 2 tan 2 /3 3 . . . tan 2 /3„ X J, say. 

 Similarly, 



R n = (- l)«-i tan 2 yS 2 tan 2 j3 3 tan 2 /3„ x J„, 



R 12 = ( - l)"- 1 tan # tan /3 2 tan 2 /3 3 . . . tan 2 /3„ x J 12 . 

 Phil. Mag. S. 5. Vol. 50. No. 302. July 1900. M 



