353 
of f'z, so as to reduce it to the form I have mentioned, we will 
divide both sides by 
(2° + az? + bz +c) x’, 
and thus we will have a new form of the equation in these 
terms, viz. :— 
, 327+2az+b \ , 3z+a ; 
aw 8 + | —————_—__ Jv-?. +. | ———_______ ]q""1 
23+az*+bz+e 23+az*+bz4+¢ 
+ (@4ae'4+bz40) 
which is an equation the roots of which are the reciprocals of 
those of fz. Now, let me compare the model equation 
e+ Av?+iA'a'+e 
to the above, and we see that 
3274+ 2az+6b . 
~ 4az4¢bz+0e. 
also 
3z+a 
ie Ss es 
23+az*+bz+¢€ 
3 
or 
2-3 32+a@ _( 32 4+ 2az+6 ae 3z+a ‘ 
—“\estaz?tbzte) \etaz*+bz+e) ~\2+a2+bz+0e)’ 
or, by development and clearing of fractions, we get the fol- 
lowing function of z, viz. :— 
(327+ 2az +b)? =3 (33+ a)(2°+ a2? + bz +0) 
9z! + 12az*+ (6b+4a*)z?) (924+ 12az° + (9b + 3a*)2*) 
+ 4abz+ 0? J (+ (9e+ 3ab)z+ 3ac J 
- Hence, by contraction or transposition, we get the following 
- quadratic :— 
(3b— a) 27+ (9e- ab) z+ (Sac - 5") = 0. 
2L2 
