96 



JAMES CLERK MAXWELL. 



[CHAP. IV. 



Scholium. Join AC, join BC, 

 and produce till AC : CD : : n : m. 

 Join AD, bisect ACD by CO. 

 Draw OE perpendicular to AO. 

 Make OCE = EOC. It is evident 

 from the proposition that ECF is 

 the tangent. 



If a line AE be cut in C and B, so that AB : BC : : AE : CE, and 

 two semicircles BOB, BXE be described on BE, and AO, CO be drawn 

 to in the circumference, and perpendiculars DH, DL, be drawn from 

 the centre, DH : DL : : AB : BC. 



For AB : BC : : AE : CE .'. AB : AE : : BC : CE .-. (AE + AB) = 

 (2 AB + 2 BD)= 2 AD : AB : : (CE + BC = ) 2 BD : BC and AD : AB : : 

 BD : BC .-. (AD - AB = ) BD : AB : : (BD - BC = ) CD : BC .% 

 BD : CD : : AB : BC. 



Draw DT perpendicular to OB, then as it bisects the base it 

 bisects the angle ODB and ODT = BDT, then in the triangles OTP, 

 DPL, OTP = DLP and OPT = DPL .-. PDL = POT. But POT = 

 BOA (6 F Cor.) and BOA = SOH = PDL, and in the triangles SOH, 

 STD, SHO = STD, and OSH = TSD .-. SOH = SDT and SDT = PDL. 

 But ODT = BDT /. CDL = ODH and DHO = DLC .'. HOD, CDL are 

 equiangular, and HD : DL : : (DO = ) BD : CD, but BD : CD : : AB : BC 

 .-. HD:DL::AB:BC. 



Cor. Sine HOD : Sine DOL : : AB : BC. 



PROPOSITION 9 THEOREM. 



If lines be drawn from the foci to any 

 point in the oval, the sines of the angles 

 which they make with the perpendicular 

 to the tangent are to one another as the 

 powers of the foci. 



Sine DCE : sine ACE : : power of A : 

 power of B. 



For describe the circle CXT as in 

 Prop. 7, so that DT : TA : : DX : AX, 

 then SC the tangent to the circle is a 



