116 
MR. B. P. CULYERWBLL ON DISCRIMINATION OF MAXIMA AND 
this quantity must change sign with x, as clx is a constant quantity; this, however, 
is a mistake : it is necessary to know the nature of the curve in order to he sure that 
dx does not change sign when x does.) Hence, according to the first rule, the integral 
will be anticlastic, should the value x — 0 be included in the limits of integration. 
But, if we seek the limits within which the condition in the second proposition is 
fulfilled, we find that, if we start from a point x 0 = — a, a second curve of the same 
species does not intersect the original curve until x = -f- a. It follows that, so far as 
regards the second condition, the integration might extend from x 0 — — a. to aq = + a. 
It may be well to observe that the proof given in Case 1, that when the integration 
is extended beyond the limits stated, i.e., those for which S“U can vanish, the syn- 
clastic property ceases to hold, does not in any way depend on the supposition that 
S 3 U does not vanish. It is shown absolutely, and without exception, that when the 
limit stated is passed, S 2 U can change its sign (§ 16), and the values of S 3 U, S'U, for 
those limits will only enable us to find whether the synclastic property holds at the 
limit up to which it is known to hold, namely, whether it holds up to and including 
the limits found in Prop. 2. Consider the case of a curve, and let ACB and ACB be 
two consecutive curves satisfying the limits and the differential equation 
dY r d 2 Y 2 
“dfo + YH ~ &C - 
then both the first and second variations, SU and S 3 U, vanish in passing from the curve 
ACB to AC'B. But the third variation, in passing from A to B, will not, in general, 
vanish, and may be expressed as a function of the coordinate of A, = f(x), suppose; 
now, as A moves along the curve, f(x) will, in general, vanish at one or more points. 
Hence it follows that, in general, the synclastic property only holds between A and B, 
and does not hold for the limits A and B actually, though there the difference is only 
of the third order; but there may be certain points for which S :: U vanishes and S 4 U 
is of the same sign as Y nn dx, and for these the curve joining A and B gives a truly 
synclastic value to the integral. 
If the synclastic property ceases because Y nn changes sign, it will hold up to (and 
including) the limit, and only cease as yon pass beyond the limit. 
18. It is often convenient to borrow a few terms from geometry when treating of 
functions depending on a number of independent variables x v x. 2 , .... x„. In the 
following paragraphs the word “point” will be used to denote any single set of values 
of the independent variables, while “ region ” will mean a continuous collection of 
