162 tr. S. COAST AND GEODETIC SURVEY. 



We have also among r, u, and the alteration u — u^ of the 

 angle u the relation 



2r sin (u — u^) = (c — d) sin 2u, 



which expresses that, in the triangle ORM' , the sines of 

 two of the angles are to each other as the sides opposite. 



The maximum and the minimum of r correspond to the 

 principal tangents and are, respectively, c and d. 



Let us call r and r^ the ratios of lengths in two directions 

 at right angles to each other and let ^ be the alteration that 

 the right angle formed by these two directions is subjected 

 to. From the well-known properties of conjugate diam- 

 eters in the ellipse we have 



r^ + r^i = c^ + d"^ 



rr^ cos \f/ = cd 



or, in terms of the scales along the parallels and meridians, 

 the semiaxes are given by the equations 



c' + d' = 'k\ + lc\ 



cd = 'kjnkp cos x//. 



For all angles not changed by the projection the product 

 of the ratios of lengths along their sides is the same. 

 In fact, let OA (fig. 45) and OB be the two principal 

 tangents; let MON be any angle whatever; and let 

 M'ON' be its projection. Let us denote by r' and r" 

 the ratios of lengths along OM and ON and by u and u' 

 the angles AOM and AOM' , 

 Then 



r'' sin AAON'=-d sin Z. AON', 



but we know that, when the alteration MON— WOW 

 is zero,' the angle AON is the complement of u' and the 

 angle AON' is the complement of u\ so that the second 

 equation gives 



r" cos u = d cos u' . 



By multiplying these equations member by member we 

 obtain 



t' r"=^cd, 



