THEORY OF ERRORS. XCVU 



For brevity, put 



g = i8o° -j- spherical excess, h = 



Then 



' + '+- 



Q = h {g — X — y — z) = he, 



^x = Ug— x—y — z), 

 P 



x^ \x=^ Ug — X — y — z) -\;- X, 



p 



with similar expressions for the other two angles. 



Now by the formula on p. xcv the square of the mean error of an observed 

 angle of weight unity is (since there is but one condition to which Aa:, Aj, A2 are 

 subject), 



p{p.xj + qi^^yj + KA^)2 = f-= hc^. 



Hence, the squares of the mean errors of the observed angles x, y, z, their weights 

 being/, q, r respectively, are 



he'' he" hf 



T' T' 



respectively. 



To get the mean error of a correction, Ajc for example, formula {a) gives 



A K= A(A;c) = - -(e^ -\- e, -\- c), 



and the corresponding expressions for the actual errors of A^ and A2 are found 

 from this by replacing p hy q and r respectively. Thus by {b), observing that 

 the mean errors of x, y, z are given above, there result 



Square of mean error of A:c = {hejpy, 

 " " " AjT = {hclqf, 



" " " A0 = {helrf. 



Likewise, the formula for the actual error of x -[- ^x is 



A V= A(x + A^) = ^i -^^y, -y, - ^U, 



and the corresponding expressions for the actual errors oi y -\- \y and z -^ ^z 

 are found by interchange of q and r with/. Thus the squares of the mean errors 

 of the adjusted angles are : — 



for(^ + A^), ^'^i-|^, 



for (,+ ..), f(.-^^), 



