108 
MR. E. P. CULVERWELL OR DISCRIMINATION OF MAXIMA AND 
synclastic for wider limits. For it does not seem by any means evident that the 
function cannot be synclastic unless the transformation is valid. It might still be 
possible to reduce it to a sum of several squares, for instance, after it had ceased to be 
possible to reduce it to a single square term. At first sight this would appear not 
improbable, for it would mean that it was still possible to determine the first few 
coefficients, z x , z 2 , ... . z r , &c., so that none of them vanished, although it was impos¬ 
sible to determine z r+l so that it did not vanish ; then the integral would be reduced 
to the sum of ( n — r ) squares, and the conditions would be that all their coefficients 
were positive. 
These difficulties seemed so great that it appeared better to attempt the problem 
by an altogether different method, namely, that of supposing that the synclastic 
property does hold for a given length of the curve and then ascertaining where the 
property ceases to hold. 
It is evident that if this could be done it would be sufficient to find the synclastic 
condition for an infinitely small range of integration, and this suggested the method 
now to be given. As already stated, the discussion of the further condition for 
synclasticism will be postponed to Art. 21. 
Part II. 
The General Method. 
13. A full account will now be given of the general investigation as applied to the 
case of two variables, and a somewhat shorter discussion of the general case will be 
found in Arts. 20, 21. 
Consider the conditions under which the equation 
\j{ x > y + %y, y + Sy, . . . y u) + Sy {n) ) dx = j V f{x, y, y, . . . y M ) dx 
+ L. (rfy ^ +. f + ' ' ' + ^ 8^/< " , ) dx 
+ i L, (flV + 2 £j7y h J &, J + % + • • • + dx 
+ A L, (|f ^ + 3 Wiy V %+••• + W) dx + &C - 
is valid. It is obtained by writing y + Sy, y -f Sy . . . for y,y . . . in f (x, y, y . . . y w ), 
expanding by Taylor’s Theorem and then integrating. Taylor’s Theorem requires 
that numerical values of the quantities Sy, Sy, &c., shall not exceed certain limits, and 
that the values of x , y, y, &c., shall not be such as to make the coefficients in the 
expansion infinite. Hence, if f(x, y, y . . . y M ) satisfy the latter condition for every 
value of x included in the range of integration, and if we take Sy, Sy, &c., small 
