96 
MR. E. P. CULVERWELL ON DISCRIMINATION OF MAXIMA AND 
by the illustrious Jacobi, who treats only of functions containing one dependent and 
one independent variable. Jacobi says (Todhunter, Art. 219, p. 243) : “ I have 
succeeded in supplying a great deficiency in the Calculus of Variations. In problems 
on maxima and minima which depend on this calculus no general rule is known for 
deciding whether a solution really gives a maximum or a minimum, or neither. It 
has, indeed, been shown that the question amounts to determining whether the 
integrals of a certain system of differential equations remain finite throughout the 
limits of the integral which is to have a maximum or a minimum value. But the 
integral's of these differential equations were not known, nor had any other method 
been discovered for ascertaining whether they remain finite throughout the required 
interval. I have, however, discovered that these integrals can be immediately 
obtained when we have integrated the differential equations which must be satisfied 
in order that the first variation may vanish.” 
Jacobi then proceeds to state the result of his transformation for the cases where 
the function to be integrated contains x, y, dy/dx, and x, y, dyjdx, d : y[dx~, and in 
this solution the analysis appears free from all objection, though, where he proceeds to 
consider the general case, the investigation does not appear to be quite satisfactory in 
form, inasmuch as higher and higher differential coefficients of hy are successively 
introduced into the discussion (see Art. 5). Jacobi’s analysis is much more com¬ 
plicated than Brunacci’s, its advantage being that the coefficients used in the trans¬ 
formation could be easily determined; hence it supplied the means of ascertaining 
whether they became infinite or not. 
dz = 0 all along tliis curve, and it may, therefore, be added to the integral (2) without altering its 
value. Now, 
f W (a dx + p dij) = 0 = [J (/3 & 2 ) + |- (a& 2 )) dx dy 
~ j"j 11 ^ ^ ^ ^' 
Adding this quantity to the integral, we get for the quantity under the integral sign in (2) the 
expression 
^A -}• L ^ ^ (B "t /3) £2 dp -f- 2 (C + a) Sz cq + P Sjj“ 2R t>p - f- Q 5<gj- dx dy. 
This expression cannot change sign if PQ — R 2 is positive, and 
(PQ-R 2 )(p(a + ^ + ^)—(B + /3) 2 ) —(P(C + «)-(B P /3)R) 2 >0. . . . (3) 
\ dy dx/ 
We can determine a and (3 so that (3) shall he true, and hence, if PQ — II 2 be positive, the second 
variation will be invariable in sign. 
The objection to this method is that there is no means of ascertaining whether a and /3 remain finite 
or not. If the region of integration be small, they can always be determined so as to satisfy (3). But 
in general they become infinite when the integration extends over a large axea. Thus, to complete the 
solution, it is necessary to have some means of finding within what range of integration the criteria are 
sufficient. It was because Brunacci’s method did not easily lend itself to the discussion of this problem 
that Jacobi devised his far moie intricate method. 
