468 merwin: equations with one unknown constant 



Equations I and la represent the same rectangular hyperbola.^ 

 If a definite value of x is given, equation I should be used, other- 

 wise la. 



x^-x ^ ^^-Xi ^ ^ ^^_^^^ J 



2/2-?/ ?/2-2/i 



yizy^yi.zyi^C{y-y.) la 



3/2 — X X2 — Xi 



II and III are parabolas- with principal axes parallel to the 

 X- and y -Sixes respectively. 



x^^ = ^^ + C {y-y,) II 



2/2-2/ 2/2 - 2/1 



yizy. = y^jzi} + c (x-x,) m 



X2 — X X2 — Xi 



Exponential forms^ are as follows: 

 2/2-2/ 2/2-2/1 



2/2-2/ _ Vi-yi Qy-y. y 



/v. /y* /y» 'V* 



X2 — X X2 — .t 1 



Equivalent to IV is 



2/2-2/ _ 2/2-2/1 (^:.-xx lY2i 



X2 — X X2 — ^1 



Equivalent to V is 



•^2 X X2 Xl ^y^y^ -y 



2/2-2/ 2/2-2/1 



1 On expansion equation I becomes A+ Bx — Cxy + Dy = 0, where 



A = y^ (?i^=^ - Cxi") -X2, B = l + Cy2, D = Cn - ^^^^l^. 

 \y2 — yi / yi-yi 



2 By expansion II becomes A -\- By — x + Cy^ = 0, where 



A=x,- ?y.(^^^^ + Cy:), 5 = ^i^£i - C (y^- y.). 

 \2/2 - 2/1 / 2/2-2/1 



3 IV may be written A + x+ B {D — y) C^-^i =0, where A = — Xi 



X*} — Xl fl T 



B = , D = yi. Or it may be written in the form y = ^7- + -^ + d. 



2/2 - 2/1 C^ Cx 



