for a System of Correlated Variables. 15 



Then 



0, 



d Jl = -{[.. J0 V . .cW n e- Q ° sin (r l e 1 + . . + rA) -£- ("A + ■ ■ • + 'A) 

 <* M i JJ x 



. .^j. . xWne'^ 9 sin (r^ + . . . + r n n ) (a^i + M 2 - • • &c), 



-f 



or 



J5 = -ff de v .Jd n e'^ S in (^+...+fA)g .... (21) 



Again integrating e~ q ^ cos (>'A...+t'A) for X by parts, 

 remembering that Q fl bding positive the integrated term 

 vanishes at both limits when l is infinite, we have 



i 



dd 1 e- Q ocos( : r l d 1 +-- + r n0n) 



= - r de Y e- q 9 sin (/-A + . . . + rj n 





(22) 



Integrate both sides of this for 0, ..0 n . That gives by (21) 



and (22) 



1 rfU dV TT 



U= j— or -^ = -nU. 



t'i dii x att 1 



Similarly d jj 



-j — = —r»U, <Vc, 

 du. 2 



and therefore, since 



dr x , dr 2 „ 

 u n r«2j— + &c.=ni 



\] = Ae~' (r ' Ul+r2U2+ '" +r ' lU ' l) = i ^e ~*~ r ". . • (23) 

 where A is constant. 



Let Q =itru, ] 



or Q tt =ia 1 w 1 2 + &i 2 wiM 2 +...-fia2M2 2 +&c.,V. . (21) 



Q r = far{ 2 + frgiva + . . . + W*2 2 + &C. J 



We have thus three equivalent forms of the index Q. 

 Now Q M is the same function of ^...^ that Q e is of 0i...0„. 

 This is Todhunter's result modified only by a change in 

 notation, he using a and 26 where I have used ia and b. The 

 proposition stated in art. 11 is thus proved. 



25. It is convenient at this point to consider the relations 

 between the coefficients a, b and a, fi, and between them and 



