8 RICE ART. B 



In this case all the d(l)(x)/dxr are unity and equations (7) take 

 the form 



<f>{x) = 0, 

 dxi dxo ' ' ' ' dxn 



(8) 



In order to distinguish between the sets of values which yield 

 a maximum and those which yield a minimum, we must con- 

 sider the terms in the expansion of f{q +- bq) — f(q) which in- 

 volve 6 and higher powers of d. Thus we now write 



f(q + dq) - f{q) = |^ 



+ higher powers of d, (9) 



where ttrs is the value of the second differential coefficient 

 b^f(x)/dxT dxs when a set of values (qr) obtained from the equa- 

 tions (7) are substituted for the variables (xr). Now if this set 

 of values yields a minimum, then the right-hand side of (9) must 

 be positive for any possible values of ^r- If we now assume that 

 the term in 6"^ preponderates in value over the remaining terms 

 in 6^, 6*, etc. (which will be the case if the differential coefficients 

 satisfy the usual conditions) then the condition for a minimum 

 value is that the quadratic expression in {^r) 



an ^1^ + ^22 ^2^ + 2 ai2 ^1 ^2 + 



should be positive in value for any set of values of (^r) which 

 satisfy the condition imposed. Actually the conditions which 

 make the quadratic expression positive for any values of (^r) 

 unrestricted by any condition have been worked out by the mathe- 

 matician; so these conditions will be sufficient for the criterion 

 of minimum in our problem, though they may not be absolutely 

 necessary for our restricted values. The conditions can be 

 stated as follows. Consider the determinant of the n*^ order 



