ANALY1K' 9XOMBTBT Cn. \. 



-, by subtraction, 



hfllCe 



Therefore, supplemental chords are parallel to a pair of 

 diameters. 



i the special case when a = b, the product of the slopes becomes 

 mm' = 1, and therefore the supplemental chords are perpendicular ; in 

 other words, the angle inscribed in a semicircle is a right angle. 



158. Equation of the ellipse referred to a pair of conjugate diameters. 

 In the simplest form for the equation of the ellipse, viz., 



g + g-l, . . . (1) 



the coordinate axes are the axes of the curve. These axes are conjugate 

 <li;iineters, and they are the only pair which are at right angles to each 

 other (cf. Art. 155, ft). It is desired now to find the equation of the 

 curve referred to any pair of conjugate diameters, as Pf and P,'P, , in 

 ill. With the notation of Art. 154, let and & be the angles thn 

 new x-axis, CP V and the new y-axis, C/\', make with the old ar-axis, re- 

 spectively; they satisfy the relation [64], 



Unftan^-g. ... (2) 



The lengths of the conjugate semi-diameters are of = CP l and 

 V f'P ' 



Then, by Art. 73, the equations for transformation to the new axes are 



x = if cos -f \f cos &, y = * sin $ -f y* sin 9 t . . . (:i) 

 and after transformation equation (1) becomes 



/cos 2 6 , sin 9 0\ ,, , /cos 6 cos & , sing sin &\ ,. 



\* w \ ^~~ ~* / 



1. ... (4) 



