304) A. Mukhopadhyay — Memoir mi Plane Analytic Geometry. [No 3, 



noted that, from a comparison of (48) and (49), it is clear that the value 

 of R in (49) is 



j c(a + 6 - 2h cos w) - (ap + hg^ - 2fy cos co) | , 



as, indeed, may be verified by direct calculation. 



§.11. Lengths of axes and area of conic— We have shewn 



above that the semi-axes a, /8 of a conic are given by (39) and (40), viz^ 



a^"" Q' ^^- Q' 



and, from the theory of invariants explained above, we have further 



, _. a-\-b — 2k cos (li .^ ab — h^ 

 A+B = -i — — , AB = -— -- (60), (51). 



Hence, if p be a semi-axis, we have 



p^-(a^-\-P^)p^'^a^P^=0 (52) 



where 



Substituting in (52) from (50) and (51), and putting from (41) 



A 



we get 



A (a -f- 6 — 2/2, cos (o) ^ . A* sin^w ^ ,^„^ 



''*+^^6rr»F-V+(^^rii)i=o. • (53) 



which is, accordingly, the equation furnishing the semi-axes of the given 

 conic ; and, as it is a quadratic in p^, it shews that there are four semi- 

 axes, which may be grouped into two pairs, the two axes in each pair 

 being equal in magnitude but opposite in direction. It follows from 

 (53) that, if Pi*, Pg* ^® *^® roots of the quadratic in p*, the area of the 

 conic is 



TT A sin oi .^ .. 



^PxP^= 1 (64) 



(ab-h^)^ 

 Again, it is clear that A and B will have the same sign or different signs, 

 according as AB is positive or negative, that is, according as AB is 

 greater or less than zero ; hence, since A and B in the equation (37) 



are connected by the relation (51) 



sm'^w 

 it follows that A and B, and thence necessarily a^, /S^ in the equation (38) 



a2 ^ y32 "^ * 



