1888.] A. Mukhopadhyay — BemarJcs on Mongers Equation. 79 



to be 



-% d^z 5 -%/dzy „ 



The value of the left hand side is found on calculation to be 



- 3A (AB - H2) ~ ^. 

 Hence, 



where A is the discriminant of the conic. But, as the area of the 

 conic is 



vA 



(ab-h^)^ 

 we have 



(Area) ^ = . 



It follows, therefore, that the geometric meaning of the above first 

 integral of the Mongian equation is the constancy of the area of the 

 osculating conic. 



Another first integral may be obtained as follows, viz., employing 



-_ 1_3 



z ^ as an integrating factor, we have from 



^ dx^ V3 "*■ 3/ ^ dx dx^ "^ "9" ^ \dx) " 



the first integral 



' 5^-3^ U) =^'- 



The value of the left hand side is found on calculation to be (always 

 taking the upper sign) 



- 3 (AB - H2) ~ ^. 



Hence, C3 = -(AB-H2) " ^ 



A 

 But, as AB - H2 = -, 



we have 



C3 = - 6A ""^, 



so that this first integral shews the constancy of 



A 



It may be noted that both the above first integrals may be obtained 



