422 Olbers' Eamy on Co))iets. 



§27. 

 Let us now put ^" = M^', and ^"' r: N ^' ; we shall then 

 have z = 5' tang. /?', z" zz M ^' tang. /3 ', and z"' ~ N ^ ' tang. 

 ^"' ; hence the three equations may be thus expressed : — 



g' ^ y-^' t^"g- h _ y"'-x" tang, h. _ y'" -x'" tang, h 



COS. h tang, i ~" tang. |3 — M tang. &' ~~ N tang. /3"' 



Consequently (y' — a;' tang. K) M tang. /3'' n: {y" — a:" tang A) 

 tang. /3' ; and (y' — x' tang. A) N tang. &" = {y" — x'" tang. A) 

 tang. ^ : and if we substitute in these equations the values of 

 X and y, we obtain two equations containing the unknown 

 quantities ^ and h only ; either of which may be found by 

 means of an equation of the second degree. If we prefer the 

 formula for h, the solution will greatly resemble that of Professor 

 Hennert; if for ^', we shall obtain expressions analogous to 

 those which Mr. Dusejour has invented, and which he consi- 

 ders as extremely convenient. 



§28. 

 It will be sufficient at present to give an example of the latter 

 method, and to find the value of j'. If we exterminate tang, h 

 from the two equations, we obtain 



y" tang. $'-^Uy' tang B" _ y" tang, g'— Ny tang. ^"' 



x" tang. ^' — Ma;' tang ^'' ~~ x'" tang. /S — N a:' tang. 0" 



consequently tang. /3' {y" x'" — ij" x") -f M tang. B" [y" x — 



y' x"') + N tang. 0" (x" y' — x' y") =: ; which is an equation 



of the second degree. Now we have, from § 7, a?' ~ ^ cos. a 



— R' COS. A', x" =: M ^' cos. a!' — R" cos. A", x"' z: N ^ cos. a!" 



— R'" cos. A'", y' =z ^' sin. a! — R' sin. A', y" — M^' sin a!' — 

 R" sin. A", and y'" — ^q' sin. a'" — R"' sin. A'". Substituting 

 these values, we obtain, after some easy reductions, making 

 P = M tang, r R' R'" sin. (A'" — A') — tang. /3' R' R'" sin. 



(A'"— A") — N tang. &'" R' R'" sin. (A" — A') 

 Q = M tang. /3" (R'" sin (A"' — a') + N R' sin. [«'" — A']) 



— tang. $' (M R'" sin. (A'" — a') + N R" sin. [a!" — A"]) 



— N tang. 0" (R" sin. (A"~.«') + MR' sin. [«"-- A']) 

 S = M N (tang. /?' sin. («'" — a') — tang. ^ sin. {«!" — «") — 



tang.^"sin. [a" — a']) 

 the quadratic equation S^'* — Q?'+ PzzO; whence 



