of the Second Degree. 73 



From which we find 



\-{b'--A a'c) sin 4 9 = (Br- 4 AC) sin" ty- <{>) = {& -4 ac) sin 2 9 sin" ff 



or A*ti^=*lz^£ (6) 



sin-6>' sin 2 (0) 



of which one particular case is that the parallelograms described 

 about conjugate diameters are always the same. 



Again we find that 



a' + c' - b' cos ff _ a + c-b cos 



sin 2 ff sin 2 V' 



of which when divided by (6) one particular case is that the 

 sum of the squares of conjugate diameters is always the same. 



When 



a + c — b cos 9 = 0. 



it indicates an equilateral hyperbola. Again, we find that 



x$ c'd' t + a'e' i -b'd'e\ ^cd 2 +ae --bde-(b--4ac)(an°+bmn+cm-+dn+em) 



sin 2 ^ sm*9 ( 8 > 



from (8), (6) and the last of (5) we deduce that 



A 



fc'd'- + a'e' i -b'd'e „\ cd 2 + ae"-bde 



. „\ ca+ae- — bae „ . . 



+f ) = b"-4ac + f ® 



b'°--4a'c r 



If each side of this be nothing, the equation represents, either 

 two straight lines which intersect, a point, or is impossible. 



If the new co-ordinates be such that 6=0 we find from the 

 second of (5) 



2 A lan<£ tan >// + B(tan <p + tan ^) + 2C=0. 



If in addition to this the new axes must be rectangular, in 

 which case tan <j> tan ^ + 1 = o, we find the following equation 

 . „ B sin 9 \b-2ccos9\ ,, . 



tan2v=-p- ; = ; '- 5 (10 



* C-A a-6cos0 + c(cos a 0-sin 2 fl) K ' 



from which two values of x are found differing by 90", either 

 of which may be the value of (p, and the other of f. 

 Vol. IV. Part I. K 



