1887.] A. Mukhopadhyay — Differential Eqiiation to all Conies. 143 



\dhif dii \dii^/ dii^ 



d^y _ ^dhjf dy ^dy^/ dy' 



^dx^ 

 <dy> 



dx^ /dx\^ 



\di/) 



t^id^K^ i d*xd*oe /d^x\' \ /dx\^ d^x 



\hi) [ df d^ ^ \ d^) ] ~ \d ^) dP 

 /dxT^ 



From the character of these formulae, it must be evident that con- 

 siderable calculation is unavoidable even in the simplest cases of 

 verification. 



§. 6. Geometrical Interpretation. 



We shall, in the last place, refer to the geometrical signification 

 of the Mongian equation. It will be seen from the passage quoted 

 above (§. 1) from Boole, that he regarded this as a case where our 

 powers of geometrical interpretation fail. With respect to this pas- 

 sage, Professor Sylvester, in his Lectures on the Theory of Reciprocants 

 already mentioned (Amer. Jour., Vol. IX, p. 18), remarks, " The theory 

 of reciprocants, however, furnishes both a simple interpretation of 

 the Mongian equation, and an obvious method of integrating it " ; 

 and the geometrical interpretation which the learned professor arrives 

 at, is that the differential equation of a conic is satisfied at the sextactic 

 points of any given curve. With regard to this geometric interpre- 

 tation, it may be remarked that it was not necessary to call in the 

 aid of the theory of reciprocants to establish this theorem. The 

 theorem is self-evident from the very definition of a sextactic point 

 as one where an infinite number of conies can be drawn having five- 

 pointic contact with the given curve j the integral equation of a conic, 

 with its five available arbitrary constants, denotes a determinable conic 

 for any given values of the constants, while the differential equation, 

 being free from constants, appropriately and adequately represents all 

 conies ; and, as an infinite number of conies may be made to pass 

 through a sextactic point, the Mongian must be satisfied at such a 

 point. But, apart from this, and with all deference to the opinion of 

 Professor Sylvester as that of one of the greatest of living mathematicians, 

 I believe that his geometrical interpretation is not the one contemplated 

 by Boole. A careful examination of the section on "geometrical 

 illustrations ' ' (Boole, pp. 18 — 20) will make it clear that, by the 

 process of the geometrical interpretation of the differential equation 

 of a curve, Boole meant the determination of some particular geometri- 



