353 
of f'x, so as to reduce it to the form I have mentioned, we will 
divide both sides by 
(23+ a2? + bz+c) 2°, 
and thus we will have a new form of the equation in these 
terms, viz. :— 
; 32°+2az+b \ , 3z+a ; 
ee 2 ie 
23+az?+bz+e 23+az*+bz+e 
1 
"(@aae*sbzic) 
which is an equation the roots of which are the reciprocals of 
those of fa. Now, let me compare the model equation 
w+ Av?+iAa'+e 
to the above, and we see that 
32724+2az+b . 
~ 234 a2%4bz+0. 
also 
3z+a i 
~ 3 4azt+bz+e’ 
sa 
or 
B=3 32+@ _ ( 327+ 2az +b fe 3z+a ; 
~°"\ 4az?tbz+e) \etazt+bz+e) \z8+a2*+bz+c)’ 
or, by development and clearing of fractions, we get the fol- 
lowing function of z, viz. :— 
(327 + 2az +b)? =3 (32+ a)(2°+ az? + bz +0) 
(924 + 12az*+(6b+4a7)2*) [924+ 12az* + (9b + 3a*)z?) 
+ 4abz + 0? J (+ (9e+ 8ab)z+ 3ac J . 
Hence, by contraction or transposition, we get the following 
quadratic :— 
(3b — a?) 2? + (Qe — ab) z+ (3ac - *) = 0. 
2L2 
