VOL. XXXIV.] PHILOSOPHICAL TRANSACTIONS. 14Q 



of half the last term. Suppose that the roots of the equation are -\- a, — b, 

 — c, or — a, -\- b. -\- c, and that a = b -\- c, then the second term in the 

 equation will be wanting, and the other terms will be expressed thus: 

 f ^ — b'^y ±bc X T+l: 



— bey 



— c'i/ 



The square oi b — c is always positive, since b and c are real quantities. 

 Suppose it, (viz. h^ — Ibc + c^) equal to d, then Z;'^ + Z;c + c' = d + Zbc, and 

 ZT+^' = D + Abe. Therefore ^-t^ til = g + £l^ + oi^c^ 4. ^,3^3^ gnj 



b\'^ X ^ = ^ + hW Now it is obvious that j. + °'g*^- + dZ)V + 

 Z)V is greater than — — + b^(?, since d is positive, and be also positive, 

 b and c being roots having the same sign. Therefore the cube of 4- of the third 

 term having its sign changed ( = ^~~ ) is always greater than the square 



of half the last term (^ = P c^ X — j^ j . In the cubic equation x^ :^ -\- qx 

 + r = 0, if 9 be positive, or if it be negative and + -5^.7^ be less than \r^, 

 it appears that two roots of the equation must be impossible, from this corol- 

 lary, and from Cor. 3, Prop. 1, taken together. 



Pkop. 4. In a biquadratic equation, all whose roots are real quantities, \ of 

 the square of the second term, is always greater than the product of the first 

 and third terms; and \ of the square of the fourth term, is always greater than 

 the product of the third and fifth terms. 



1 . Let the equation be x^ — px* -f- qx'^ — rx -\- s = 0; and since the roots 

 are supposed to be all real, let them be represented by a, b, c, d, then p = a 

 -\- b -{■ c -\- d, and q =: ab + ac -\- ad -\- be -\- bd -{- ed. But it is plain 

 from Lemma 3, that 3a^ + 3b'- + 3c^ + 3d- is greater than lab + 2ac + 2ad 

 + 2bc + Ibd + 'i.ed; and consequently, by adding Gab + 6ac + 6ad + 6bc 

 + 6bd -f 6cd to both, we shall find that 3 X a -\- b -\- c + d' must be greater 

 than Sab + 8ac + Sad -\- Sbc + Sbd -\- Scd; that is, 3p- greater than Sq; 

 and therefore f /j'V greater than qx^. 



2. Since r =■ abe + abd + acd -\- bed, and s = abed; and since qs is equal 

 to a^b'^cd + a^c'^bd + a^d^bc + b'^c'^ad + b'd^ac + c^d^ab, which are the 

 products that can be made of any two of the quantities abc, abd, acd, b<:d, 

 whose sum is r, multiplied by one another; it follows, that 3r^ is always 

 greater than Sqs: so that f of either the square of the second term, or of the 



