1887.] A. Mukhopadliyay — Memoir on Plane Analytic Geometry. 303 

 III> General invariants. 



,,. a-'fb — 2h cos oi ah — li^ 



<•> ii^.. ^"> liiT^^T 



("0 -A-- 



It is clear that since any function of an invariant is an invariant, various 

 invariants may be deduced from these by combining them in different 

 ways or by imposing limiting conditions on them. Thus, for rectangular 

 axes, A is a general invariant ; and, if we examine the equation 



ax2 + 2hxy + hy + 2fy = 0, 

 which denotes a conic referred to a tangent and normal as coordinate- 

 axes, we see that it has the three general invariants, (a + 6), (ab — h^), 



We have shewn above, hy actual calculation, that the discriminant 

 is a translation-invariant ; it is interesting to note that the same result 

 may be obtained as an illustration of Dr. Boole's method. Thus, if 

 by translation-transformation the equation 



ax''-^'2.hxy-\-hy''-\-2gx-\-2fy^c=0 

 is transformed into 



the same transformation changes 



x^-^y^"\-2xy cos w 

 into 



(X-^,r-h(Y-t/,r + 2(X~^J(Y-2/,)cosc., 

 whence we have 



ax''-\-z}ixy^hy^ + 'lgx-\-2fy-irc-^\{x^ + y'' + 2xy cos o>) 

 = aiX» + 2/t,XY-f &,YH2^,X-t-2/,Y + c, 



.f\|(X-arJ='-f(Y-2/,)^+2(X-;rO(Y-^Jcosa,|. 



Equating the discriminant of the left hand side to zero, we have 



c sin^o). X^->r I c{a-\- h - 2h cos w) - {ap -f- hg^ - 2fg cos w) I A 



+ A = (48) 



If we equate to zero the discriminant of the right hand side, the equa- 

 tion in X apparently comes out to be a cubic ; but the coefficient of X^ 

 is found on calculation to be zero, while, in the coefficient of X2, the 

 terms involving x^'\ x^y^, y^^, a^, y^ separately vanish, and the constant 

 is c sin^w ; hence the equation may be written 



csin^o). \2 + RA+Ai =0 (49). 



Therefore, equating coefficients, we have 



which shews, as before, that A is a translation-invariant. It may be 



