EEDUCTION OF TKIANGULATION. 69 



There is another class of conditions, known as side equations, which 

 can be best explained by reference to a figure. In the example, diagram, 

 suppose the figure 0, 1, 2, 3 to represent the projection of a pyramid, 

 of which 1, 2, 3 is the base and the apex. A geometric condition of such 

 figure is that the sums of the logarithmic sines of the 

 angles about the base, taken in one direction, must 

 equal the similar sums taken in the other direction, 

 i. e., the product of the sines must be equal. In the 

 present case, log. sin 0, 1,2 + log. sin 0, 2, 3 + log. 

 sin 1, 3; should equal log. sin 1, 2, + log. sin 2, 3, + log. sin 0, 1, 3. 



The number of side equations which can be formed in any figure is 

 equal to the number of lines in the figure, plug 3, minus twice the number 

 of stations in it or I + 3 — 2 n. In a quadrilateral, 6 + 3 — 8 = 1. 



The numerical term in each angle equation is the difference between 

 the sum of the observed angles on the one hand and 180° + the spherical 

 excess on the other. This is positive when the sum of the observed angles 

 is the greater, and vice versa. The coefficients of the unknown corrections 

 arc in each case unity, unless weights are assigned. 



The numerical term in each side equation is the difference between 

 the sums of the logarithmic sines, taken in the two directions. The coeffi- 

 cients of the unknown corrections are the differences for one second, in the 

 logarithmic sines of the angles. 



The method of making up and solving these equations and applying 

 the corrections to the angles can best be shown by means of an example. 

 That here given is the simplest case involving both angle and side equa- 

 tions, namely, the case of a quadrilateral. The method of forming correla- 

 tives and normal equations, and their solution, is similar to that employed 

 in station adjustment, and therefore the details are omitted. 



In the equations of conditions and correlatives, the angles are desig- 

 nated by directions, to which the corrections are finally applied. Thus 

 the angle of 302 is designated as — 3/0 + 2/0, the sign — being given to 

 the left-hand and the sign + to the right-hand direction. 



