MATHEMATICAL NOTES*. 



1. IF a plane passes through any point of a surface, and 

 makes any function of the intercepts it cuts off from the axes, 

 a maximum or a minimum when it touches the surface, this 

 maximum or minimum value is constant for all points of the 

 surface ; and, conversely, if for every point of a surface, a given 

 function of the intercepts of the tangent plane is constant, this 

 function is, with reference to any single point of the surface, 

 a maximum or minimum for the tangent plane. 



This appears at once from the following considerations: 

 Xj y, z being a point in the surface, 05 , y , Z Q , the three inter- 

 cepts, < (j , y , z ) the given function, if we seek to determine 

 the surface so that <f> shall be a maximum or minimum, we have 

 the equations 



< IK? ......... < 



fjb being a factor. From these equations we get 



and therefore the differential equation of the surface is 

 dx 



_ r ,x. 



" 



* Cambridge Mathematical Journal, No, XIX. Vol. rv. p. 47, November, 1843. 



