80 A. Mukhopadliyay — Bemarlcs on Mongers Equation. [Feb. 



from Roberts' Theorem that 



is a second integral of the Mongian equation, viz., differentiate this 

 equation, then, eliminating c, we have one first integral, and, eliminating 

 c', we have the other. 



The constancy of the quantities shewn above may also be shewn in 

 another way, viz., as Dr. Wolstenholme has shewn by actual calculation, 

 (Educational Times Reprint, t. XXIV, 105) the equation of the conic 

 leads to 



/dW_9h_ /d^\y 9(Ag - ah ) /d^\^ 



But, if we have 





where c, c' are any two constants, we see that by differentiating 

 twice and eliminating c, c', the Mongian equation is obtained ; hence, 

 the quantities 



h h' - ah 



A*' 



A^ 



e we have 





2 



92A~^ 



c,= (h^- 



ah) A 



2 



-ab 



62 • 



that 



These relations, however, do not shew the constancy of the eccentricity ; 

 but, as the Mongian being an equation of the fifth order has five in- 

 dependent first integrals, the fact of the eccentricity of the osculating 

 conic being constant is probably the geometrical interpretation of one 

 of the other three first integrals ; before, however, actually proceeding 

 to form that equation, we shall show how the constancy of the eccentri- 

 city may be otherwise established. Thus, we have 



(AB - H3) 



(Aa^H2Ha;+B)^ 

 or, 



