102 
MR. E. P. CULVERWELL OX DISCRIMINATION OF MAXIMA AND 
Multiply (6) by By 2 /z. dx, and subtract it from the part under the integral sign in 
equation (5): then we have for the part under the integral sign 
and, writing out the complete variation, we get 
the final expression; from this it is evident that, if Y n dx changed sign in the course 
of integration, the function could not be synclastic. 
6. If the variation By were such that By suddenly changed from one finite value to 
another, By must be infinite at that point, and the integration would not be permis¬ 
sible, or, at least, it would require a justification which, so far as I am aware, has 
never been considered necessary. 
Again, it will be observed that the limiting values of By are introduced in the 
process (though for this case it is easily seen that in the final result the terms in 
which they appear are identically zero). 
What has been shown in this case is true in general : the first 2 n fluxions of By are 
brought in under the integral sign, and of these the last n fluxions . . . y are 
got rid of by integration. Again, the limiting values of the first 2 n — 1 fluxions are 
introduced outside the integral, and, of these, the n fluxions y (n> . . . y ( ~ n ~ l) disappear 
through their coefficients being identically zero ; but the direct proof of this in the 
general case would, so far as I can see, require transformations of even greater length 
than those usually applied to the part under the integral sign. 
7. It was with the object of ascertaining whether it was necessary, in the general 
case, to assume the continuity of the value of By and its 2 n — 1 fluxions that I first 
considered the problem ; and I communicated to the British Association at Montreal 
an account of a method which was free from this objection, and had the additional 
advantage of a simplicity which enabled it to be easily extended to any number of 
dependent variables; and it is, in its main principles, applicable to the most general 
case. 
8. The application of this method to the preceding case is as follows. Taking the 
second variation in the form on p. 99, equation (4), in expanding V.U, we have 
