236 



dx 



= x + (x cos a + y sin a) cos a 

 = (y cos a x sin a) sin a, 



. . N 



z -5-.= (y cos a a; sin a) cos a ; 



therefore, when y cos a as sin a=0 .................. (2), 



-j- and -y- both vanish. 

 aa; ay 



Under these circumstances z becomes = + a. 



Now equation (1) represents a cylinder having its axis 

 parallel to the plane of (x, y). Equation (2) represents a 

 plane which passes through the axis of the cylinder, and 

 which cuts the surface in two parallel straight lines. Along 

 the upper straight line we have z a. All points in this 

 straight line are at the same distance from the plane of (x, y), 

 and at a greater distance than any points not in ttiis straight 

 line. This straight line is in fact a ridge in the surface. 



Another example may be seen in the equation 



This surface is that formed by the revolution of a circle about 

 a tangent line which is the axis of z. The highest point of 

 the circle will by revolution generate a circle, all the points 

 of which are at the same distance from the plane of (x, y\ 

 and at a greater distance than any adjacent points of the 

 surface. 



EXAMPLES. 



T , o a a 



1. Let u = af + xy+y--\ --- h , 



i y 



^ U _ 9 ? au _ 9 ^ 



fa~ +y ~tf> 3$ y - ~tf' } 



3 3 



therefore 2x + y - 2 = 0, 2y + x -^ = ; 

 x y 



therefore (2ce + y) # 2 = a 3 = (2y + x} y* ; 



