120 
MR, B. P. CULVERWELL ON DISCRIMINATION OF MAXIMA AND 
in the value of y, such as would occur in a path AC . . . DB, but there may be 
sudden changes in y, the inclination of the tangent (as in AEFB). 
20. To facilitate the discussion, a fluxion will be said to be one of the “ highest 
fluxions ” when no fluxions of that fluxion appear in the function whose integral is to 
be made synclastic. Thus, d~yjdx x dx. 2 may be one of the “ highest ” fluxions, although 
ddy/d.x 1 is not; for there may be no terms which can be written D. d 2 y/dx l dx 2 , where 
D represents any combination of d/dx, djdx 2 , &c., while there might be a term 
d/dx o, ddyjdxf 
The limits being supposed fixed, the following proposition can be easily proved : 
—If the highest fluxions of the variable y x which occur in U be all of the same order 
n x , those of y 2 of the same order n 2 , and so on, then the conditions that U shall be 
synclastic when the integral is only extended over a small region R are the same as 
those that the quantity 
(g X* + % Y= + &c. + 2 JSL XY + &c.) <fe, dx,... dx, 
shall be incapable of a change of sign, a, b, &c., representing the highest fluxions, and 
X, Y, &c., being any arbitrary quantities. 
The conditions for the case where the highest fluxions of any dependent variable 
are not all of the same order will be discussed afterwards. When it is said that the 
region R of integration is small it is meant that the greatest ranges of value of the 
coordinates x x . . . x m are small. Thus, if the region is given by 
x i + x i +•'..+ xf — v 3 < 0, 
then r must be small; of the order /3, suppose. This being so, it is easy to show that 
we may neglect the variations of all but the highest fluxions of the dependent variables 
when finding the sign of S 2 U. For the change in A £ in passing from a point P 0 on 
the boundary of R to any point P within it is 
rP f p [ dAz dAz \ 
PA - PA. = J y A* = j P y- dx, + ~ dx, + Ac); 
and, since the total range of dx x , . . . dx m , in the integration is of order (3, the order of 
the integral will be that of the quantities (3 dAz/dx x , (3 dAz/dx 2 , &c., or at least it cannot 
be greater. Again, the limits being fixed, (At) Po , the limiting value of Az, must be zero 
unless 2 is one of the “ highest fluxions.” Hence it follows that all the other fluxions 
are small (of the order (3 at least) compared to the highest; and that they can be 
omitted from the integral when all we wish to determine is its sign. It is, however, 
necessary to show that the entire integral does not vanish, for, as the values of all but 
the highest fluxions, and therefore of the highest but one, are given for all points of 
the boundary, it is evident that 
