Adams. — On Triangulation by Least Squares. 205 



is done readily on the Brunsviga calculating-machine without 

 any intermediate record, the result in this example being — 



2 (a 2 + b 1 + c 2 ) = + 126-790. 

 Let k = i 2 (a 2 + b 2 + c 2 ) 

 „ h = c x -+- c 2 — c 3 — c 4 (from column No. 8). 

 „ i — the number of triangles. 

 Next form the following equations : — 

 /iP + 2<iQ + e = o 

 2kP + hQ + £ = o 

 and solve them for P and Q. 



With these values of P and Q calculate the corrections to 

 the observed angles thus — 



x x = a x P - Q x 2 = a 2 P - Q 



y x = b 1 P-Q y 2 = 6 2 P - Q 



z x = Cl P+ 2Q £ 2 = c 2 P + 2Q 



x 3 = a s P + Q x i = a 4 P + Q 



y 3 = 6 3 P + Q */ 4 = 6 4 P + Q 



^ = c,P-2Q 4 = c 4 P-2Q 



where x lt y u z lt &c, are the corrections in seconds, and fche 



corrected angles are — 



A x + ajj A 2 + x. 2 A 3 + x 3 A 4 + x 4 

 B 1 + y 1 B 2 + y, B 3 + y, B 4 + y t 

 Ci + *! C 2 + z 2 G, + z 3 C 4 + ar 4 



where the angles A u B lt &c, are taken from column No. 4. 



Columns 9, 10, and 11 show the calculation of the 

 corrections. 



Column No. 12 gives the final angles (seconds only), and 

 is equal to column 4 + column 11. 



Column No. 13 gives the natural sines of the angles in 

 Column No. 12. 



This completes the calculation of the least-square correc- 

 tions. 



In practice it is always desirable to check the results 

 obtained, consequently the two following checks are ap- 

 plied :— 



(a.) By forming the products of — 



Sin Ai sin A 2 sin A 3 sin A 4 (from column 13) ; and 

 Sin Bi sin B 2 sin B 3 sin B 4 „ 



which are equal (see schedule). 



(/3.) By comparing the values of — 



Ci + C 2 (from column 12) ; and 

 C s + C 4 „ 



which are equal. 



The triangles are now solved, using the angles from 

 column 12, and the results are — 



