302 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



-4- (4^7) 



are rotation-invariants. 



In order to see if any of these is a general invariant, we must 

 examine whether they are translation-invariants. It will be found on 

 examination that the first is not a translation-invariant, while for the 

 second we know that, by a translation-transformation, the equation 



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

 is transformed into 



a V -f 2h'xy + hy + 2g'x + 2f'y + c' = 0, 

 where 



a' = a, A' = Ji, h' = &, 

 which, by the way, shews that the part of the second degree in the 

 general equation is a covariant for translation-transformation, 

 and g' = ax'-]rhy'-{-g 



f = hx' + hy'4-f 



c' = Point-function, 

 from which, by actual calculation, we fiud that the coefficients of x^, ccy^ 

 2/', x.yin 



a%'c' + 2fg^K - a'/ - h'g'^ - c'h" 

 all vanish, and the absolute term is A. Hence, we infer that A is a 

 translation-invariant, and so also is 



A 

 sinV 

 since <o is unaltered by translation- transformation ; thus, from what 

 precedes, we have finally that 



A 

 sin'o) 

 is a general invariant of the conic. To sum up, we enumerate below 

 the principal invariants of the general conic. 



I. Translation-invariants. 



(i). a. (ii). h. (iii). 6. (iv). A. 



II. Rotation-invariants. 



(i) Absolute term, (ii) ^-^ 



sin CO 



,...^ ah-h^ 

 (ill) . a 



^ ^ sin^o) 



,. . a-{-h — 2h cos (ii f -{• g^ — 2 fg cos (a 



(iv) r-5 = ~ 



Sin 0) c sill to 



sm w c Sin CD 



