THE CUKVE ON ONE OF THE COORDINATE PLANES. 471 



value of xjje will vanish for any direction of e, without even excepting that 

 direction of e which might render xfie parallel 

 to the element dp of the curve. 



If, for example, we put the question to 

 determine the maximum of Tp under the con- 

 dition 



Up = constant, 



then we get the two equations 



SpSp = and YpBp = , 

 which result into 



Bp = yjre = . 



Another supposition to be tried will be to assume that for a point on the 

 limiting curve the direction of t//e will be parallel to dp for any direction of e, so 

 that in that case the possibly not vanishing t//e will be invariable as to e. 



These two suppositions expressed respectively by the condition 



and by the condition 



V(dpy}re) = 0, 



in each case for any direction of e, are not excluding each other, and we will be 

 able to show that the consequences of the second of the conditions involve 

 the consequences of the first of them. 



We may remark that the first of the conditions, namely, 



tye = , for any e, 



will also present itself when we look upon the limiting curve as being the 

 general envelope of all the possible curves which the extremity of p may 

 describe under various limitations. 



We may further remark that the condition 



V . dpifre = , 



namely 



V(dp8 P ) = 0, 



expresses the answer to the problem of finding the curve comprising the area- 

 maxima of all the areas comprised by curves described by the extremity of p. 



In order to conceive how the extremity of p may be caused to describe 

 certain definite curves (both in the case of an envelope and in the case of an 

 area to be considered), we must reflect that the expression of p depends on the 

 three scalar elements of the versor p, and that we may conceive a certain 

 relation between the three to be established, so that only two of them may be 

 considered as independent. But here there arises the theoretical difficulty of 

 our inability of guessing at the true nature of the relations to be established 



