(4) 



182 . Dr. H. Jeffreys on 



So far the systems of Orthogonal curves are unspecified; 

 the contour-lines, or lines of strike, will be taken to be 

 X = constant, and the dip-lines will be //,= constant. Thus 

 u = V, v — 0. Now if ds denote an element of length in any 

 direction on the surface, arid rectangular coordinate axes are 

 chosen, the axis of z being upwards, the contour-lines are- 

 specified by the condition that z = constant along them. If 

 the surface has the equation z—f(x, y), then along a 



contour-line ~ =0, and therefore ^- 4-^^=0. 

 as o% oy dx 



Thus A can be taken equal to z, and as the contours are- 

 parallel to the plane of (#, ?/), it follows that their projections 

 on the plane of (#, y) will also satisfy the differential 

 equation 



dx ~dxj ~dy * 



Now the projections of the dip-lines must be perpendicular 

 to those of the contour-lines, and therefore along them we 

 must have 



dy = ~df j-df^ 

 dx 'dy/'dx' 



The solution of this equation can be put in the form 



Known function of x and y = arbitrary constant, . (5) 



and this function can then be taken to be fi. 



The element of length on a dip-line measured in the 

 direction of the flow is cosecacfe, and therefore 7^= —sin a. 

 h 2 is as yet undetermined. The equation of continuity is 



•&2K- <*> 



The above treatment is independent of the form of the 

 surface considered, and shows that if the value of Vf is known 

 along any contour, it can be found along any other by an 

 integration along the dip-lines. Thus a general solution can 

 always be found. 



At this stage the law followed by the friction may be 

 investigated. 



As long as V is a function of f, Vf must be of order 

 f A cosec a dz, or Al, where I is of the order of the horizontal 

 dimensions of the area. The condition that the friction 

 may be proportional to the square of the velocity is the 

 Osborne Reynolds criterion, that Vf shall be greater than 



sm 



