124 
MR. E. P. CULVERWELL ON DISCRIMINATION OF MAXIMA AND 
It may be objected that exceptional cases might occur, in which a z y and A A) 
coincided over S" as well as over S'—S". But it is directed that, of S", 2q is in the 
part common to S and S', and is outside S. The surface &c., is 
therefore one in which discontinuous values for the fluxions of y 1 -f- aA 2 y l: y 2 fl- a A 2 y 2 , 
&c., would appear in the equations (10), Art. 18 ; and therefore that equation cannot 
be satisfied by these values. Hence A 3 is not the same operation as a 3 . 
Observe that the whole point of the proof consists in the fact that the a 3 variations 
are sufficiently continuous, notwithstanding the discontinuity in the highest fluxions. 
To complete the proof, it only remains to show that the surface V', for which S C U = 0, 
satisfies the equations (10), Art. 18, for every point of the region S. Suppose that it did 
not do so for a portion S x of S. Take a compound surface made up of V x over S 2 and V' 
over S — Sj, where V x is the synclastic surface, having at all points of the boundary of 
S x the same values of A y x , A y. 2 , &c., and all but their highest fluxions, as those of the 
function V'. This compound surface gives us an admissible variation, and the integral 
over it is less than that over V'. But, by hypothesis, it is impossible within the region 
S to find a surface giving a smaller value to the integral than that given by V. 
Hence V' must be a synclastic surface, and the proposition is proved. 
Hence a function will be synclastic provided, first, that the condition given in 
Art. 20 is fulfilled for every point in the region of integration ; and second, that 
it is impossible, within that region, to draw another synclastic surface with the same 
limits. It will be easily seen that these conditions are independent of each other, and 
that, if either or both fail, the function becomes anticlastic. 
22. When the highest fluxions of any dependent variable are not all of the same 
order of differentiation, the conditions found in Art. 20, although sufficient, are 
not all necessary. For, as will be proved presently, the values of the highest fluxions 
of the A variations are not all of the same order of magnitude. To ascertain the 
comparative orders of different fluxions, let us consider the equations 
p 1 dAz , 
Ac P - AZp m = J t -^-dx lf 
p 2 dAz , . 
A Sp — A z P = —;— ax o, and so on, 
Jp dx % 
where A z F means the value at the point P, or aq, aq, . . x m ; Az Poi that at the point on 
the boundary which has the same values for all the coordinates except aq, which has 
the value aq' given by <£( x{ aq . . . x m ) = 0, where <f) = 0 is the equation of the 
boundary ; and similarly for the other coordinates. As aq — aq', aq — aq', &c., are 
everywhere of the same order, /3, it might appear that cl A zjdx Y , d Az/t/aq, must each 
be of the order az//3. This, however, is only the case when dAz/dx x , d Az/dx. z , &c., 
do not rapidly change sign. For, while it is clear that dAz/dx must exceed Az in the 
ratio 1//3 at least, it is easily seen that the ratio may be greater if the terms in the 
