52 Transactions of the South African Philosophical Society. 



4. Compute the co-ordinates of P either from triangle P C A or 

 triangle PB C. 



Second method. 



1. Compute the length and angle of 

 direction of C from the co-ordinates of 

 A and B. 



2. Compute the co-ordinates of 

 from the triangle O A B in which 

 OAB = /3 and OBA = «. 



3. Compute the angle of direction 

 of O C from the co-ordinates of 

 and C. 



4. Compute the co-ordinates of P 

 either from triangle POA or PBO. 



Both these methods are avoided by 

 many surveyors on account of their length. A shorter method will 

 now be given, with a numerical example showing the arrangement 

 of the computation. 



Taking the middle point C for origin, put x' y' and x" y" for the 

 co-ordinates of A and B. The equations to the circles (1) through 

 A and C and containing the angle a (2) through C and B and con- 

 taining the angle /3 are 



tan a -J y (y-y') + x(x-x') \-—xy'+yx' = 

 tan j3 -j y (y-y")+x(x-x") \--yx'[+xy" = 



reducible to 



Where 



Then 



y 2 + x--\-ky — B,£ = 

 y 2J rx 2 - Cy + Dx = 



A = X' COt a — if' B — //' COt a + X' 



C = x" cot p+y" T> = y" cot p-x" 



y B+D 



iii 2 x-\-x — 'B — ni\ 



B-wA 



x = - y — mx 



