1888.] A. Miikhopadhyay — Remarhs on Mongers Equation . 85 



We now proceed to verify that the differential equation 

 (2 - e^f _ 1 T^ 



l-e2 "" /(%y U 



whose geometrical meaning is Col. Cunningham's theorem about the 

 constancy of the eccentricity, is a first integral of the Mongian equation. 

 Thus, putting 



dy dhj d^y d^y dhj 



we have, 



^"dx' ^ "■ dx^' "" " dx^' ^ ~ du*' ^ ■" dx^' 



(e8_2)2 T3 



1 - e2 9 f/ U' 



so that taking the logarithmic differential and remembering that 



(?T 



— = (H-^2)(3 (^fh rs) +10 qr (3 r/ - 2pr) 



S-»'-' 



we get 



2^ (3 ^2 _ 2 |;r) (45 ^r5 - 9g 2^ - 40 r^) 

 = {l-\-p^)(qs--2 r2) (45 qrs -9qH~ 40 r^), 

 which proves that the equation of the eccentricity leads on differentia- 

 tion to the equation 



9 22^-45gr5 + 40rS = 

 and is, therefore, a first integral of the Mongian equation. 



Gentlemen, I have now examined the subject as completely as was 

 necessary to shew the erroneous nature of Col. Cunningham's interpreta- 

 tion. I have explained to you, as lucidly as I could, the true meaning 

 of geometrically interpreting a differential equation, and I have shewn 

 you that the Colonel's interpretation signally fails to satisfy the funda- 

 mental tests which every geometrical interpretation ought to satisfy ; I 

 have, further, pointed out to you that the Colonel's interpretation really 

 belongs to a differential equation which is quite distinct from the Mongian 

 equation, and, by actually forming that equation (as I have never seen 

 done before), I have proved it to be a first integral of theMongian equation. 

 But, gentlemen, as this first primitive contains an arbitrary constant, it 

 denotes any member of the given family of curves, while the differential 

 equation itself indiscriminately denotes all the members of the family. 

 Col. Cunningham's interpretation, therefore, involves a quantity, which 

 remains constant as we pass from point to point on the same curve, but 

 varies as we pass from one curve of the system to another. In reality, 

 therefore, he failed to perceive the fundamental difference between a 

 differential equation and its first primitive ', he did not notice that while 



