112 
MR, E. P. CULYERWELL ON DISCRIMINATION OF MAXIMA AND 
C c 
becomes possible to draw a second curve such that S~U A = 0. For, if S 2 U A could be a 
C' 
positive quantity of the order a 2 , we could alter Sy so that S 3 U A should be positive 
and of the same order, C' being between A and C, infinitely near C. Hence the 
maximum property would have ceased at C', but by supposition it does not cease till 
0 
C, and therefore we have to find C as being the first point for which S 2 U A or 
s 2 u! 1 = o. 
Now S 2 U^ = 0, 1st, because it vanishes independently of the form of Sy, or 
2nd, because a particular form is assigned to Sy, causing it to vanish, or 3rd, because 
x i ~ x o can be divided into separate parts, some of which vanish from the first cause 
and the rest from the others. 
The first case can only occur if S 2 U vanishes identically, i.e., if its coefficients 
vanish, and will not be further discussed. The second case will occur when does 
not vanish between A and the point for which S 3 U A = 0 (this will be evident 
presently). The third case must be investigated on the supposition that the parts 
which vanish independently of the form of Sy are infinitely small (for otherwise 
S 2 U would vanish identically), and for these parts Y )m must vanish, for when the 
limits of integration are infinitely close, and the limiting values of Sy, Sy, . . . Sy (n ~ l 
zero, o ’ U = j Y nn Sy im dx, and this vanishes independently of the form of Sy when 
We have then only two cases to discuss— (a) Y nil does not vanish throughout the 
C 
integration, so that g 2 U a vanishes in consequence of a particular form being assigned 
to Sy throughout the integration, and (b) that in which Y, m does vanish. 
First suppose that Y m does not vanish. Then C is evidently determined at the 
first point to which a second synclastic curve can be drawn, having at each limit 
contact of the (n — 2) th order with ABO A 
* For it is evident that if o 2 U vanishes in passing from ABC to ADC, and if any portion as EF of ADC 
were not itself synclastic, we could obtain an integral greater than that along ADC, and therefore greater 
than that along ABC, by joining’ EF by a synclastic curve EHF, having at E and F contact of the 
(n — 2) th order with ADC ; and therefore the preceding reasoning shows that C would not he the first 
point at which a curve can he drawn so that — 0. Hence we must use the synclastic curve to get 
this limit. 
