1888.] A. Mukhopadhyay — Differential Equation of all Parabolas. 327 



A^ _ J_ _ _ 729g^ 

 (h^~ah)^ ~ c^ ~ US • 



Therefore, the equation for the lengths of th^* axes reduces to 



9^2 T„ 729j5 



<^* + -u^ -' + xj^ = 



where Tq = is the differential equation of all equilateral hyperbolas, 

 and U = of all parabolas. 



If the roots of this equation be o--^^, a-^, the area of the conic is 



U^ 

 a result I have obtained before.* 



We may similarly consider the osculating parabola and the osculat- 

 ing equilateral hyperbola at any point (x, y) of a given curve. Thus, if 



a^2 + ojixij + by^ -\- 2gx -Jf 2fy -\- c = 

 where 



h^ = ah 

 be the osculating parabola, and m its principal parameter, we can easily 

 calculate m in terms of the differential coefficients from the formula 





2 " (a + 5)-! 



For, solving for y, 



we have 





2/ = Pa; + Q + \/ 2H» + B 



where 











yr_M-bg j,_P-ic 



Hence, as usual, 







p = p+—^^^ 



q = 





SO that 



and 



^r — Sq^ = 



(2Ha3 + B)2 

 -_H2 



i2}Ix + B)f 



(2H^ + B)i 

 3PH3 



(2Hx- + B) 

 * P. A. S. B. 1888, p. 84. 



