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



cal property which belonged to every curve of the system covered by 

 the differential equation, and the inherence of which property was 

 adequately represented by the equation ; take, for example, the case 

 of the circle, of which the differential equation is 



l'-(: 



dy\^ ) ^ _ 3 % /^ "* 



,dx/ ) dx^ dx \dx^/ 



Boole points out that this equation represents in an "absolute 

 character " the geometrical fact of the invariability of the radius of 

 curvature of all circles ; in fact, I may remark in passing that this 

 equation represents the vanishing of the angle of aberrancy at every 

 point of every circle ; for, if 8 be the angle of aberrancy, and, 'p, q, r, 

 the first, second, and third differential coefficients of y in regard to x, 

 we have the formula 



tan 8=^--^-^. 



(See Salmon's Higher Plane Curves, p. 369, Ed. 1879). Hence, 

 when the angle of aberrancy vanishes, we have 



which is the differential equation of all circles. We see, then, that the 

 differential equation of a circle is the " absolute " analytical representa- 

 tion of some geometrical property which belongs to all circles, and 

 the existence of which is manifested by the differential equation. But 

 Professor Sylvester's interpretation of the Mongian equation is of an 

 entirely different character ; it does not furnish us with some property 

 common to all conies ; it is simply regarded as the representative of 

 a sextactic point on any curve. What Boole wanted was some intrinsic 

 property, that is, a property belonging to the curve whose differential 

 equation we are interpreting ; what Professor Sylvester arrives at, 

 is, if I am allowed the expression, an extrinsic property, that is to say, 

 9, property belonging not to the curve in question, but to some extra- 

 neous curve which has six-point contact with the given one. If 

 Professor Sylvester's interpretation were the one contemplated by 

 Boole, nothing would be easier than to interpret a differential equation. 

 Thus, for example, with reference to the differential equation of a circle, 

 we might simply say that it is satisfied at a quadruple point on any 

 curve. Again, the equation 



3 A^ __ 5 /(^\ 

 dx^ dx* \dx^) 



when integrated, is found to represent all parabolas ; but would it be 

 a sufficient geometrical interpretation to say that the equation is 

 satisfied at a quintic point on any curve ? The whole point of the 



