100 
MR. E. P. CULVERWELL ON DISCRIMINATION OF MAXIMA AND 
when Sy is a suitably chosen function of x, and the opposite sign when another 
expression is taken for Sy. In the former case the integral must be either a maximum 
or a minimum; a maximum if the' sign of S 3 U is Constantly negative, and a minimum 
if it is positive. In the latter case the integral is neither a maximum nor a minimum; 
its only characteristic is that the first variation vanishes. It will save much useless 
verbiage if we use a single word to express this latter class of integral, and the word 
“ anticlastic,” borrowed from geometry, seems a suitable one. It has also the advan¬ 
tage of suggesting another term, “ synclastie,” for those functions which give either a 
maximum or a minimum. With this explanation the problem before us is to ascertain 
whether the integral U is synclastie or anticlastic. If it be the former, a glance will 
enable us to determine which the result is, a maximum or a minimum. 
4. We might reduce, by a real ]inear algebraic transformation, the part of S 2 U under 
the integral sign to the sum of n squares, and say that, if their coefficients be positive 
all through the integration, then, whatever be the limiting values of the arbitrary 
variations, the second variation must be essentially positive (at least unless dx 
changes sign during the integration); but the converse of this is not true, for it does 
not follow that, if the coefficients of the squares have different signs, it is possible to 
make the second variation change sign. If Sy, Sy . . . . Sy {n) were all independent, 
this converse would be true ; hence one element in the problem before us is to introduce, 
in a suitable form, the interdependence of the quantities Sy, Sy ... . Sy (n \ Again, 
if the limits are not all arbitrary ; there will be further limitations to the range of 
values taken by Sy and its fluxions ; and the second element in the problem is to find 
how these limitations to Sy and its fluxions affect the sign of the second variation. 
5. Jacobi’s method appears to me open to the serious objection that it is necessary 
to its validity that the first (2 n —1) fluxion of Sy should be continuous; so that the 
discussion only proves that a curve AB fulfilling the synclastie condition gives a 
better result than any infinitely near curve fulfilling the same limiting conditions, and 
which is continuous to the 2 n th fluxion of y. And it would not show that it was not 
possible to find other broken curves fulfilling the same limiting conditions, and giving, 
at our pleasure, a value to the integral either greater or less than that given by the 
curve AB. Thus, for instance, in the case of least action, it would not show that the 
action in the free trajectory was less than in any constrained path (which it is, in fact), 
but only that it was less than that in any path for which the tangent had but one 
position at every point. To show for the general case that Jacobi’s proof assumes 
