184 
CALCULUS 
6. Continuation. Several Auxiliary Equations. The method of 
Lagrange’s multipliers applies to the general case that the variables 
are connected by an arbitrary number of auxiliary equations. For 
example, let 
(1) u = F(x, y, z, t), 
(2) 4>(a>, y, z, t) = 0, ¥(a>, y, z, t) = 0. 
If equations (2) can be solved for z and t, then, on substituting 
these values in (1), u becomes a function of x and y. Equations (1) 
of § 3 now take on the form 
(3) + + iA=0, F 2 + f£+f3=0. 
ox cx cy cy 
The derivatives dz/dx, etc. are determined by the equations: 
(4) 4q -f 4> 3 -—I- 4q — = 0, 4> 2 + 4*3 —h 4> 4 — = 0 ; 
cx cx cy cy 
(5) qq + *3^ + * 4 |* = 0, 'k 2 + * 3 ^ + * 4 f* = 0. 
cx cx cy cy 
Thus we find the conditions 
F l 
^3 
f 4 
Fo 
F s 
F\ 
4>i 
*4 
= 0, 
<J> 2 
4*3 
4> 4 
*3 
*4 
4q 
*3 
*4 
The four equations (2) and (6) determine the four unknowns 
x, y, z, t, and for this system of values equations (1) of § 3 hold. 
Lagrange’s method consists in forming the function 
u = F + A4> + ycV, 
where A and ^ are constants, to which shall later be assigned suitable 
values, and where u is considered as a function of the four indepen 
dent variables, (x, y, z, t). It is to this function that condition (1) of 
§ 3 is now applied, and thereby result the equations: 
(0 + ^4q + /¿'Iq = 0, F 2 + A4> 2 + ¿I'k.j = 0, F 3 4- A4> 3 + /x.'Fj = 0, 
F 4 -f- A < t > 4 + /A'k.} = 0. 
From the last two of these A and ¡x are to be determined, and these 
values are then substituted in the first two. The two equations thus 
obtained are precisely the equations (6). 
The extension of the method to a function of n variables, 
u = F(x L , x n ),