1888.] A. Mukhopadhyay — Remarks on Mongers Equation. 81 



and, from 



dz ^ , . _ . . __ . . _ — 4 



dx 



= + 3 (AB - H^) (Aaj + H) (AajH2H^ + B) ^ 



= + 3 (A^ + H) z^^ (AB-H2; ■" 3, 

 we have 



^ 



z ^ dz 



Substituting in 



A;^ + H=+ J ^ . 



(AB - H?0 ^ 



ax ~ 



A^ + H 



(Aa32_i.2Ha? + B)2 

 we get, after reduction 



^_p^_^ (AB-H2)*d% 



Now, it is clear from the mode of genesis of this equation, that if we 

 differentiate it twice and eliminate 



P, (AB-H2)i, 

 we should obtain Monge's equation ; but, that would also be the case, if 



P, (AB - H2)i 

 are replaced by any two constants ; hence, it follows that P and 

 (AB — H^) are constants. Now, as we have already shewn that 



h^ " b^ b 

 is a constant, we see, by remembering that 



» 

 that - is also a constant. But, the equation of the eccentricity is 



(2 



-e^f (a+&)2 \6 / 



l_e3 ab-h^ a__'h^ ' 

 b b^ 

 so that it follows at once that the constancy of the eccentricity is the 

 geometrical interpretation of a first integral of the Mongian equation. 



We now proceed to form the actual differential equation whose 

 geometrical meaning is the constancy of the eccentricity. The above 

 investigation shews that 



'd^y\^ . T^ d^y dy /dhjy 



vr ryy^^ d^y _dy nPy\ 

 ^1 W/ '^ 'da;^-d:.W) 



