B 



MATHEMATICAL NOTE 



JAMES RICE 



1. The Method of Variations Used for Determining the Condi- 

 tions under Which a Function of Several Variables Has a Maximum 

 or Minimum Value. In the discussion of the conditions for 

 equihbrium of a system and of the criteria of stabihty of a state 

 of equihbrium, the following mathematical problem is presented : 



To determine the values of the variables Xi, Xi, . . . . Xn for 

 which a given function of these variables, f(xi, X2, .... Xn) has a 

 maximum or minimum value, the variables themselves being 

 subject to a condition such as 



<i>iXly X2, Xn) = 0, 



where (f> is another given functional form. 



Considering a definite set of values for the variables, say Xi = qi, 

 X2 = q2, . . . . Xn = Qn wc compare the value of the function 

 for this set with the value for any neighbouring set, such as 



Xi = qi -\- Sqi, a-2 = ?2 + Sq^, x„ = g„ + 5g„, where 5gi, 5^2, 



.... 5g„ are infinitesimal quantities. These infinitesimal quan- 

 tities are not completely arbitrary in their ratios to one another; 

 for we have to choose them to satisfy the conditions 



<t>(qi, qz, qn) = 0, 



<f>{qi + 8qi, q2 + 5^2, qn + 8qn) = 0. 



It is convenient to write for 8qi, 8q2, . . . . 5g„ the symbols ^^i, 

 ^^2, . . . . d^n where 6 is an infinitesimal positive magnitude whose 

 value can be reduced without limit and ^i, ^2, .... ^n are finite 

 quantities. The difference between the value of the function / 

 for the set of values (xr = qr) and the value for the set 



(Xr = ?r + 8qr) IS * 



fiqi + bqi, qi + bq2, ?„ + 8q^ - f(qi, q2, qn). 



* The enclosing bracket in (xr = (/r) or (g,) indicates that we mean 

 Xi = qi, Xi = q2, . . . . Xn = Qn, orqi, qz, . . . qn. 



