1887.] A. Mukhopadhyay^ — Memoir on Plane Analytic Geometry* 301 



belong to this class (Salmon's Conies, §. 159, Ed. 1879, p. 159). Again, 

 as we have seen that a, b, h are translation-invariants, it follows that 

 a-\-h — 2h cos 0) ah—h^ 

 sin^to) ' sin^o) 



are invariants for the componnd transformation as well, and may, accord- 

 ingly, be called General Invariants. We shall now proceed to investi- 

 gate, by a process analogous to that employed by Dr. Boole, certain in- 

 variants which include as particular cases those noticed above. 

 Suppose that by a rotation- transformation the equation 



assumes the form 



AX^4-2HXY+BY»+2aX+2FY -1-0 = 0. 

 Then, by the same transformation 



ix^-\-y^-\-2xy cos w 

 is altered into 



X»+TH2XYcos,S3, 

 because each of these expressions denotes the distance of the same 

 point from the fixed origin. Hence, we have 



(a -\-\)x^ +2(h-\->^G0S o))xy ^ (h ■^X)y'' -\-2gx -\-2fy -|-c 

 = (A+X)X2 + 2(H + Xcos S8)XY + (B + X)Y3 + 2GX + 2FY + C. 

 Each side of this identity will resolve itself into linear factors for the 

 same value of A. ; hence, equating the discriminant of each side to zero, 

 we have the two equations 



c sinV \^-\r { c{a+b — 2h cos w) — (/^4-/ — 2fg cos w) j A 



+a6c + 2/^/t - of - hg* - ch''=0 



Csin»g3.X"+ |'c(A-l-B-2Hcos S3 ) - (F« + G^- 2FG cos Q)|x 



+ ABC + 2FGH - AF^ - BG»- CH=^ = 0. 

 As these quadratics in X must be identical, we have, by equatino* the 

 coefficients of corresponding terms, the two relations 

 a-i-5 — 27i cos o) f^-}-g^ — 2fgcos<o 

 sin^w c sin'^w 



A + B-2HC0S Q F"4-G'»-2FGcos Q 



(44) 



sin* Q C sin^ S3 



alc-\-2fgh-af-hg^-ch?' _ ABC + 2FGH- AF'-BG'- CH' 

 c sin'w C sin^S3 ' 



If /=0, ^=U, these equations furnish Dr. Boole's invariants. As we 

 have noticed that c is a rotation-invariant, these results shew that the 

 functions 



j c{a-Vh-2h cos 0)) -(/*+<;* -2/^ cos to) | -r- sin^w (46) 



