114 
MR. E. P. CULVERWELL ON DISCRIMINATION OE MAXIMA AND 
(which is legitimate, as hy n is nowhere infinite). Now take two points P and Q on 
ADC and CG respectively, and join them by a synclastic curve PRQ having contact 
of the (n — 2) th order with ADC and CG, P and Q being sufficiently close for the 
synclastic property to hold between them. Then, since I (PRQ) is a maximum, 
I (PRQ) > I (PCQ); 
add 
I (ADP) + I (QG) = I (ADP) + I (QG), 
and get 
I (APRQG) > I (ADCG), and therefore > I (ABCG), 
showing that the integral taken along ABCG is not a maximum, and evidently it is 
not a minimum, for, if it were, every part of it would have to be a minimum, and the 
part from A to B, for instance, is not a minimum, but a maximum.* 
Case 2.—In this case, if Y nn changes sign as well as vanishes, the integral becomes 
anticlastic when the limit is beyond T (the point where Y nn = 0). This is obvious, as 
the part immediately beyond T then gives a minimum value to the integral whose lower 
limit is T, while from A to T it gives a maximum value. If Y nn does not change sign, 
it is easy to see that the maximum property does hold beyond C, with this nominal 
exception. A curve ABGCAA'G can be found the integral along which is equal 
to that along ABCG, C' and C'" being infinitely near C, and C'C"C'" being any curve 
having contact of the (;? — l) th order with ABCG at C' and C'". It is not difficult to 
see this by reasoning similar to the above, but it is shorter to observe that if we alter 
very slightly the value of Y nn , so as just to make it preserve its sign without vanishing, 
we alter the value of the integral very slightly, and, therefore, &c. 
Case 3.—Very slight consideration shows that the synclastic property ceases at T. 
17. Before passing to the general case, it will not be amiss to add a few explanations. 
The argument is not that it is possible to pass from the original curve to any 
other infinitely near curve by repeating again and again for each part of the curve a 
c c c 
* It is sometimes considered sufficient to say that, as MJ A = 0, and £ 2 U A = 0, -while £ S U A changes sign 
with by, the maximum property must cease at A. In Todhukter’s “ Adams’ Prize Essay on the Calculus 
of Variations ” this reasoning is employed (Art. 24, p. 25). But it is invalid for two reasons. 1st, £ 3 U 
and all higher variations may vanish, as in the case of great circles on a sphere; and 2nd, it shows only 
that a curve can be got giving an integral greater than that corresponding to ABCGf by terms of the 
third, not of the second, order. Now, as the second and third variations are quite independent of each 
other, there is nothing whatever to show that when a value is given to by other than that which makes 
S 2 U = 0 this variation £ 2 U can be made to change sign. It is absolutely necessai’y to show this, for 
otherwise the integral from A to G would really be a true maximum, though of a very curious nature. 
—See the remarks at the end of Art. 17. 
