( B52 ) 



fig 'f 



a siuiple fUiAe. Siniplei- are the loi'i ot" I lie poles of tlie diaiiieter 



(>[i, and I lie iionual directed on to it 

 out of () which we can take as axes 

 for the calculation. 



Let therefore (fig. 4) Oii be the 

 A-axis, the normal OV on it the 

 3^-axis, then we find the equation of 

 X- as follows : 



Let O A = a, / ACii = <p, thus 

 / ^4^>/?=: 27. As /- passes through 

 0, ,? and touches circle {()) in 1\ its 

 e(puition can be wi-itten: 



y{.r. cos 2<p -f ,V sin 2<fi -f a) -j- m {y — o- tfj 2if>) {y—-v t(j tf -f- a it) </) = ; 



the coefficient m is determined by the condition that the poijit 

 y( — a cos^ (f, — a cos if sin <f) lies on X". By substitution of the coor- 

 dinates of y for .V and // and after reduction we get : 



in = cos if) cos 2(f sin (f 



and the equation of X'^ becomes : 



a/n' <f sin 2</i . .r' -\- {cos 2<f — cos rp sin <p sin 2<p — sin'* <f cos 2<f) xy -\- 



{sin 2<f -\- cos <f sin (f cos 2(f) y" -|- ^ ( ' ~h •*""" *f ^■'^*' "*/ ) .'/ — '* *'"' 'f ''"^ 2(f .x = 



or shorter: 



2 si?i* (f . A'* 4" ^^^ ^P {^ ^^^"^ *P — ^) '*'// "I" 



sin (f (3 — 2 sin"* (f) //'■' — 2a sin^ <f . ./• f- a cos </) (3 —2 cos'* <f) y =r- 



The three derivatives become : 



4 sin* (f . .)• -|- COS' if (4 cos" if — 3) y — 2a sin* tf ^=z 0. . . (1) 



cos if (4 cv.s' if — 3) x -\- 2 sin (f{3 — 2 sin'- f/) y \- a cos if {o — 2 cos^ (/^) = 0. (2) 



2 sin* if . iv — cos if (3 — 2 cns^ y) (/ = (3) 



If we eliminate out of these equations two by two the value if, 

 we get the three loci. 



Finally we shall deduce the simplest of these loci, namely the 

 locus of the poles of the axis OX which is obtained by eliminating 

 if out of (1) and (3). 



From (1) and (3) we deduce by subtraction the following two 

 simpler equations : 



3 y cos if z=z 2a sin* if , (4) 



2 sin* if . X — cos r/) (3 — 2 cos'' tp) y z=z 0\ . . . . (5) 



eifter substitutiun of the value sin* tf out ot (4j into (5) we tind : 



