654 BELL SYSTEM TECHNICAL JOURNAL 



If p{x) is varied at a particular argument Xi = Si , the variation in r(x) is 



8r{x) = q(xi — Sr) 

 and 



bU = — I qixi — Si) log rixi) dxi — X log p{s^) = 



and similarly when q is varied. Hence the conditions for a minimum are 

 j q{xi - 5i) log r{x^ = -X log p(si) 



j p{xi - Si) log r{xi) = -fji log q{si). 



If we multiply the first by p{si) and the second by q{si) and integrate with 

 respect to s we obtain 



Hs = — X Hi 



H, = -n H, 



or solving for X and )u and replacing in the equations 



Hi I q{xi — Si) log r(xi) dxi = —H3 log pisi) 

 H2 j p{xi — 5i) log r{x^ dxi = —H^ log p{si). 



^(^^■) = /9 '\n/2 ^^P ~ h^AijXiXj 



Now suppose />(a[:i) and q{xi) are normal 



(2x) 

 ^(^t) = L^W2 ^^P ~ h^BijXiXj. 



Then r(:ri) will also be normal with quadratic form dj. If the inverses of 

 these forms are ij^y, bij, Cij then 



We wish to show that these functions satisfy the minimizing conditions if 

 and only if a,y = Kbij and thus give the minimum H^ under the constraints. 

 First we have 



fi 1 

 log r(x») = - log — I Cij I - \i:CijXiXi 



/n 1 



q{xi - s^) log r{xi) = - log — [ dj \ - ^^djSiSj - \i:djhij . 



