78 A. Mukliopadhyay — Bernards on Monges Equation. [Feb. 



which, again, we recognize to be a first integral of 



which is the differential equation of all circles. Guided by these two 

 analogous cases, we guess that the Colonel's interpretation in the case 

 of the conic may belong to a first integral of the Mongian equation, and, 

 this point we now proceed to examine ; the process will consist of two 

 parts, viz., we shall first form the differential equation whose interpreta- 

 tion is that the eccentricity of the osculating conic of any conic is con- 

 stant, and, secondly, we shall examine whether this differential equation 

 is a first integral of the Mongian equation. 



The Mongian equation being one of the fifth order, it is clear that 

 it has five independent first integrals, and, curiously enough, gentle- 

 men. Col. Cunningham's interpretation does not belong to any of those 

 first integrals which may easily be derived from the equation. Consi- 

 der, now, the osculating conic of any conic ; the equations of the two 

 conies are identical, viz., either being 



ax^-i-2hx2j + 'by^-{-2gx-\-2fy-\-c = 0, 

 we have 



where 



= P^ + Q + -v/A;p2 + 2H^-i-B, 



P— -^ Q--f 

 ^- &'^- 6' 



Hence, as usual. 



dy AiB-fH 



d^y AB - H2 



^ = T2 = ± 3 



'^'^ (Aa;H2H^-i-B)^ 



dz__dPy_ _ 3(AB-H2) (A^^+H) 

 dx'dx^' "■ (Ax2^.2Ha.-l-B)^ 



dh__d^j^ -^(^^ - H2) [4(A^+Hf - (AB - H^)} 

 dx^ -dx^~- " (A:«2 + 2Hx-fB)^ ' ' 



Now, as shewn in my previous paper {Journal A. S. B. Vol. LVI, 



Part II, 140), if we employ z - V as an integrating factor, a first integral 



is obtained from 



d^z ^ -%dz dh . 40 - V , ~ , 



\dx) 



