160 



U. S. COAST AND GEODETIC SURVEY. 



of the alteration is not changed. Thus of two angles 

 which are found to be changed by equal quantities each 

 is the complement of the projection of the other. 



If we wish to calculate directly the alteration which any 

 given angle u is subject to, we should make use of one of 

 the two formulas 



(c-d) ism u 

 tan(u-u) = ^^^^^^.^ 



/ ,x (c—d) sin 2u 



tan (u— u )= , J . / JN TT-f 



c+d+{c—d) cos 2u 



which follow immediately from the previous formulas by 

 easy analytical reductions. 



Fig. 45.— Angular change in projection, first case. 



Let us now consider an angle MON in figures 45 and 46, 

 which has for sides neither one nor the other of the prin- 

 cipal tangents OA and OB. We can suppose the two 

 directions OM and ON to the right of OB and the one of 

 them OM above OA. According as the other ON will be 

 above OA (fig. 45) or below OA (fig. 46), we should calcu- 

 late the corresponding angle M^ON' bv taking the differ- 

 ence or the sum of the angles AOM and AON\ which 

 would be given by the formula stated above. The alter- 

 ation MON—M[ON' would also in the first case be the 

 difference, and in the second case would be the sum of 

 the alterations of the angles AOM and AON. When the 

 angle AON (fig. 45) is equal to the angle BOM\ we know 

 that its alteration is the same as that of the angle AOM, 

 so that the angle MON will then be reproduced in its true 



