1888.] A. Mukho-psLdhjeij— Differential Equation of all Parabolas, 319 



— = 3 tan (0, 

 P 

 which gives the required formula 



tan = - — . 

 3 p 



The above formula in the case of a central conic follows also from 

 the properties of conjugate diameters, viz., if rj be the semi-diameter 

 conjugate to r, we have 



pr^^ = ab 



Hence 

 and 







r,B 











'- ab' 









rdr-^-r^dr^ -. 



= 





dp 



3r/ 



dr. 



3r, 



rdr 













ds 



ab 

 3r 



ds 

 dr 



ab 



ds 



z:z 





^ = 3 tan 8. 







P 



[ds 







since 



Therefore 



dr p 



-— - = — sm 0, - = cos 5. 

 ds r 



tanS-^''"-^''' 



3 ds 3 p* 

 as before. 



We now proceed to express the elements of the osculating conic in 

 terms of the differential co-efficients. For this purpose, we remark that 



_ M|)T _ (sy 



reduces the equation 



to __^ 



d<i> dx^ 



dx / d.fi \ 2 



\ax/ 



and we have also 





dhj 



dhJ 





dx^ 



dx^ 





ds 



ds dx 





^~ (^0)~ 



dx doi 





d'^y 



d^y 



dm 



doc^ 



dx^ 



dx 



- /ds\^ - 



. . idyV 



