142 A. Mukhopadhyay — Differential Equation to all Conies. [No. 2, 

 By a similar reasoning, we can prove that the equation 



dx' dec* [dx^J » 



which denotes all parabolas, the equation 



\dx^/ dx^ dx\dx^/ dx dx'' dx*^ 



which represents all conies referred to co-ordinate axes through the 

 centre, the equation 



/d,y\^ d^y dy 



which represents all conies referred to principal axes through the centre, 

 the equation 



V dx yjdx ~ ^^^ dx^' 



which represents all parabolas referred to two tangents as axes, and 

 the equation 



I '^\dx) I dx^ dx\dxy' 



which represents all circles, have permanency of form. Of course, by 

 actual calculation we can establish that, if in each of these equations 

 we make y the independent and x the dependent variable, we have 

 simply to interchange x and y. We subjoin below the formulae neces- 

 sary for such a verification. 



dx dx 



dy 



d'^x 

 d'y _ dy^ 

 dx* Ax.^ 

 \dy) 





/d^'xy^ dx d^x 

 d^y \dyV dydy^ 





dx^ "" /dx\^ 

 \dy) 





dxd^ofd^x idx)/^ d*x 

 d*y dy dy" dy^ \dyf dy"^ 



"MS)' 



dx'' ~ /dx\' 





\dy 



