Prof. Everett on the Optics of Mirage. 257 



dec CL 



(2) (x oc Vyt y- — — 9 the differential equation of a 



a y vy — a 2 

 parabola, with the axis of oo for directrix, and a 2 for distance of 

 vertex from directrix. 



The cases /x cfc v'y + b and jx a \/b—y are reducible to this. 



(3) ix oc -> ~r = ■ . =. = ;- — , the differential 



equation to a circle of radius - having its centre on the axis 

 of x. This result agrees with Section XVI. The cases /x oc 



y±b 



and ll a ; are reducible to this. 



b-y 



(4) '" h ^^zr^fr y ' the diff f n - 



tial equation of a cycloid generated by a circle of diameter 



-5- rolling along the axis of x. The cases /x oc and 



a yy + b 



ix a — , are reducible to this. 



Vb-y 



(5) h a:VW=f, |=^— |^ ?;a; + c= as in-. vp L ?) 



a result which agrees with Section XV. The case //. oc Vq+py — y 2 

 is reducible to this. 



(6) M acvy±i», ^y—^P whence 



or. determining c so as to make -~ vanish with so. 



The ordinates of the curve have therefore the constant ratio 



Va 2 ± W- / 



to those of a catenary. The case \x oc Vy 2 i:pyi:q is 



reducible to this. 



(7) \x Q£e iy , or~—~-==b*j so that the curvature for a given 

 Phil. May. S. 4, Vol. 45. No. 300. April 1873. S 



