THEORY or POLYCONIC PROJECTIONS. 



161 



magnitude by the angle M'ON\ Thus to every given 

 direction we can join another, and only one other, such 

 that their angle is preserved in the projection. However, 

 the second direction will coincide with the first when it 

 makes with OA the angle which we have denoted by TJ. 



The angle the most altered is that which this direction 

 forms with the point symmetric to it with respect to OA^ 

 it is represented upon the projection by its supplement. 

 The maximum alteration thus produced is equal to 2co. 



Fig. 46.— Angular change in projection, second case. 



This can never be found applicable to two directions that 

 are perpendicular to each other. 



The length OM in figure 44 having been taken as unitv, 

 the ratio of lengths in the direction OM is measured by 

 OW. Let us denote by r this ratio; we can calculate it 

 by means of one of the formulas 



or 



r cos u =c cos u 

 r* sin ^fc' = (? sin u 



r2 = c2 cos^u + d'^smH. 



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