4o8 TRANS. ST. LOUIS ACAD. SCIENCE. 



and (2) becomes 



dp = AdT _ Bdv . (3) 



1°. To find the direction of maximum slope with respect to the 

 V T plane at any point on the surface. For this purpose pass a 

 plane through any point in the surface, and at right angles to the 

 vT plane. Its trace upon the i^T plane is 



T= [:i + av, . . . (4) 



p being indeterminate ; where a is the tangent of the angle 

 which the trace makes with the v axis, or 



^ = -r (5) 



dv 



From (3) and (5) we have 



dp = {Aa _ B) dv . . ' . . (6) 



Calling 6" the slope of any element of the intersection of the 

 plane and the surface, dz being the projection of the element on 

 the V T plane, we have 



— ^^- - "^ . . (7) 



S=^^^^^. . . . (8) 



In determining the direction of maximum slope at any point, 

 it is evident that A and B will be constant, which gives as the 

 required condition, • 



dS A-{-Ba 



= ! = o , 



da {i-\-a)i 



A 

 Substituting the values of A and B, we have 



