1887.] A. Mukhopadhyay — Differential Equation to all Conies, 137 



From these, we have 





and 



= Constant, 



HS)i 



whence the differential equation of all circles is 



/ d^ 



/d\ \ d x^ 



c^+ mi 



= 0. 



But, again, from the above values of -r-, - — , we may derive 



dx dx"^ 



/dy\ _ (d'^y\\ _ B.-x 



dy 



whence ( — j < — > = 0, 



\dx} /d^y\\ 



or 



\dxy 



3 /^y ^ . 4 ^ {^Y _ o dy d^y d^y 

 \dx^/ dx^ dx\dx^) dx dx'' dx^' 



and this would also be the differential equation of all circles. Bat, 

 then, if we examine this equation for a moment, we see that it con- 

 tains ---|, and, therefore, the integral equation corresponding to it 



contains four arbitrary constants ; hence the equation not only includes 

 all circles, but something more, viz., it denotes a certain family of 

 conies which include all circles. In fact, if we integrate the equation, 

 the result comes out in the form 



Aa72 + B2/^ + 2Ga;+2F2/+C = 0, 

 which represents conies referred to the centre. 



§, 3. First Method of integrating tlie Mongian. 



We shall next proceed to integrate the Mongian equation by 

 ordinary methods. As far as we are aware (and the same view is 

 apparently held b}'- Boole and Sylvester), no direct integration of the 

 equation has as yet been performed, except Professor Sylvester's 

 solution by the Theory of Reciprocants (Amer. Jour. Vol. IX, pp. 18- 

 19). 



18 



