I 



CuLVERWELL — Soiutioiis ill the Calculus of Variations. 391 



change sign at pleasure, it is quite evident that we can never have any 

 maximum or minimum. For instance, take the integral 



JO 



which, if dx cannot change sign, except when the sign of the square 

 root changes, represents the length of the curve Joining to P, and 

 has a true minimum value corresponding to the straight line. If in 

 it we are at liberty to change the sign of dx arbitrarily, and without 

 changing the sign of the square root, then the integral has no maximum 

 or minimum, but is capable of passing to continually greater or smaller 

 values by suitable variations [i.e. by making the quantity under the 

 square root have a larger or smaller value for the positive or negative 

 values of dx, respectively) ; its value in this case is geometrically 

 represented by the difference of the integrals for dx positive and those 

 for dx negative. 



The same may be easily seen from the result of this Paper. The 

 integral is increased or diminished in passing from OAP to OABP 

 (fig. 3), according as EBx is positive or negative, and therefore if Dx 

 can have either sign, we can always increase or diminish the integral, 

 whether U can or cannot change sign. Thus, an arbitrary change of 

 sign of the independent variable is always excluded, and if a change 

 of sign in the independent variable occurs in consequence of the form 

 of the curve, the sign of E must change with it. 



