VOL. XVI.] PHILOSOPHICAL TRANSACTIONS. SQQ 



z^ — bz' + pz -\- q ^ O he given. In this case, that there may be three roots, 

 the centre of the circle should be found somewhere within the space PNa, 

 terminated by the right lines PN, Pa, and the curve of the paraboloid Na ; 

 therefore, since E F is = ^i 6, p must be less than -i-bb; now to determine 

 the quantity q, d being = -i-i^ — ^p as before, y/d? + Vt^^ — ibp should be 

 always greater than -jj', that so the centre of the circle may be posited in the 

 foremen tioned space PN A; which when it is so, such an equation has two 

 affirmative roots, and one negative : but if p be greater than ^h b, or \q greatei 

 than y'ff + ^7-^* — i^pj it is explicable only by one,, and that a negative 

 root. 



Now let the equation 1? — bz^ — pz — q = be proposed. That this 

 equation may have three roots, the centre of the circle must be found some- 

 where in the indefinite space between the right line DPD and the curve of the 

 paraboloid PX: the quantity p is not here liable to limitations; but ^q should 

 always be less than ^d^ — ~^b^ — -^bpy supposing d to he =. -^b b -^ -^p : by 

 this means there are given two negative roots, and one affirmative ; but other- 

 wise, if 4-^ be greater than »/ d^ — -^^b^ — \bp, the equation is explicable 

 only by one affirmative root. Fourthly, let the equation z' — bz^ — pz -\- q 

 ^ O, be proposed, which has two affirmative roots and one negative, if the 

 centre of the circle be found in the indefinite space between the right lines 

 Pa, PD, and the curve of the paraboloid aL; that is, (putting d = -^bb 

 + -hP) if \q be less than »/ d^ -\- ^'^bbb + -i-bp ; but if ^q be greater than this 

 quantity, there is but one negative root. 



Now the four remaining equations, in which we have -f b, do not difi^r from 

 those that have been mentioned already, as to the limitation of the number of 

 the roots, if the sign of the last term be changed, retaining the sign of the 

 third term ; but then the roots that were affirmative in the former will be nega- 

 tive here, and contrarywise. Thus, in the equation z' — bz^ -\- pz — q ■=. o, 

 the affirmative roots were either one or three; but in this equation z^ -j- bz^ + 

 pz -\- q = 0, there is either one or three negative roots under the same con- 

 ditions, but no affirmative root at all. So also in the equation 7? -\- bz^ -{■ pz 

 — ^ := O, there are two negative roots and one affirmative, if p be less than 

 Af^l)b, and 4^9 be less than -v/cP -f- Vt^^ — i^P ; just as in the equation z" — 

 iz'^ + pz + 9 = O, there were two affirmative roots and one negative root; 

 but p and q exceeding those prescribed measures, there is only one affirmative 

 root in this latter, which in the former was negative. After the same manner, 

 in the equation i^ -\- b:^ — pz -\- q =■ O, either there are two affirmative 

 roots and one negative, or one negative only. Lastly, for the same reason, in 

 the equation z^ -|- ^z*^ — pz — q =: O, there are two negative roots and one 

 affirmative root, or one affirmative only ; while, in the equation z^ — bz^ — 



