146 Mr Swan's Formulcefor constructing 



are supposed to be known both before and after reflexion, it is 

 evident that tangents to the arc may be drawn at these points. 



Let the co-ordinates of the two extremities of the arc be 

 ic'y, of' y", then the equations to the tangents at these points 

 will be 



y— y=rtan ^(a; — a/) ; ^— y = tan y (x—x") 

 where and 7 are known. 



Also the equation to the circular arc will be 



where r the radius of the circle, and a, b, the co-ordinates of 

 its centre, are to be found. 



The equation to the tangent to the circle at the point x'l/ is 



and since this line is the same as y— y=tan (x — oif) we 

 have evidently 



-; — ^-=tan 6 



y'-b 



But since x^ y' is a point in the circle {x! — of ^-{i)' ^1)f=T^^ 



Therefore substituting for a:'— a 



y' — b= drzr cos Q 



In like manner we obtain 



y"''b=^ db r cos 7 



v'—v" 



Therefore r= — —,-^ 



cos ^— cos 7 



Similarly it may be shewn that a;' — a = =p r sin d ; 



a:"—«= q=r sin 7; from which, along with the equation to 



the circle, it will be found that 



xf-x" 



r—— ; — - 



sm 7— sm 



From this and the value of r already obtained, we have 



y-/=tanit/(a/-a:0 



an equation of condition between the co-ordinates yf y', x!' y" ^ 

 from which, if three of them be assumed, the fourth may be 

 found. 



Next, to obtain a and b, we have 



«/--«= — /• sin ^ 2/' — 6 =7* cos ^ 

 from which, by substituting the values of r, 



^ xf sin y—x" sin 6 ^ . __ y" cos — y' cos 7 

 sin 7— sin ' cos Q — cos 7 



