72 Mr. De Morgan on the General Equation of Curves 



From the suppositions made respecting the axes 



sin(0-d>) , sin (9-^) , 



x = v . a Y ' x + — \ a r ' y' + m, 



sin 9 sin ^ 



(3) 

 sin <b - sin y > 



y sin 9 sin ^ 



If we substitute in (1) these values of x and y, the resulting 

 coefficients of jf*, x'y, &c. must be proportional to a', 6' 3 &c. Pre- 

 serve this condition and make the substitutions and developments 

 as follows : 



Let A = a-b cos + c cos 2 0, 



B = sin 9(b- 1c cos 9), 



C = c sin 2 9, , . N 



D= 2aw + iwi + rf-cos 0(2cm + ijz + e), 



£ = sin 9(2cm + bn + e), 



F = ari ! + bmn + cni i + dn + em+f. 



In which it is important to observe that if the primitive 

 co-ordinates be rectangular A = a, B = b and C=c If in addition 

 the origin be not changed D = d, E = e, F=f. Also that if the 

 axes of x' is parallel to that of x and x'y' = 90°, the equation 



becomes 



Ay* + Bxy +Cx* + Dy+Ex + F = o. 



The substitution above indicated will now give the following 

 results : 



Xa' sin 2 9 = A sin 2 -y + B sin ^ cos >|/ + C cos 2 ^, 



\b' sin 2 0=2 A sin ^sin 0+ B (sin ^ cos + c'os >|/ sin 0) + 2Ccos >/, cos0, 



Xc'sin 2 = ^sin 2 + £sin0 cos <£ + C cos 2 <£, .^ 



Xd' sin 9 = D sin ^ + JE cos >fr, 



Xe'sin0=.Dsin0 + .E cos 0, 



X/ = F, 



where x is any quantity whatever. 



