146 PHILOSOPHICAL TKANSACTIONS. [aNNO J 726. 



must be greater than the sum of the products m;ide by multiplying any two of 

 them. 



Lemma 3. — The triple sum of the squares of four quantities is greater than 

 the double sum of the products, that can be made by multiplying any two of 

 them; for 3a^ + Si^ + 3c^ + 3(P — lab — 'lac — 'lad — ibc — ibd — led 

 = a" - lab + b"" + «' - 'lac + c^ + cr - lad + d' + U' — ibd + d' + 

 i' _ 2/t + c!" + c- — 'led + d- — a — /-' + a — c + a — / + /!> — c' 

 -f i- — f/ -f c — f/ , the sum of the squares of the differences of the four 

 quantities a, b, c, d. Therefore Sa^ + 3b'^ + 3c- + 3d^ is greater than lab + 

 lac + lad + i/jc + ibd + 2cc/, the excess being always positive. 



Lemma 4. — Let the number of the quantities a, b, c, d, e, &c. be m, the sum 

 of their squares a, and the sum of the products made by multiplying any two 

 of them B. Then shall -^ — X a be always greater than b. 



For by adding together the squares of the differences a — b, a — c, a — d, 

 h — c, b — d, c — d, &c. we add a" as often to itself as there are quantities 

 more than a ; the same is true of Z'-, c', d', e', Sec. But the rectangles — 2ab, 

 — lac — "lad — '2hc — ibd, &c. arise but once each. Therefore the sum of 

 all the squares a — b , a — c , b — c , b — d , &c. = m — 1 X cr -\- m — I 

 X b"^ + m — \ X C-, &c. — lab — lac — ibc, &c. = m — \ X A — 2b. 

 But a — /' + « — c -\- a — d , &c. is always a positive quantity ; therefore 

 711 — \ X A — 2b is positive, and consequently ^~;- X a greater than b. 



Cor. — It appears from the demonstration, that the excess of ?« — 1 X A 

 above 2b is always equal to the sum of the squares of the differences of the 

 quantities a, b, c, d, &c. and that when the quantities a, b, c, d, 8cc. are all equal, 

 then m — 1 X a — 2b = O, and with this restriction the preceding lemmata 

 must be understood. 



It is to be observed, that though we have supposed, in these lemmata, the 

 quantities a, b, c, d, &c. positive, they are, a fortiori, true of negative quanti- 

 ties, whose squares are the same as if they were positive, while the sum of their 

 products is either the same, or less than it would be, were they all positive. 



Prop. I. — In a quadratic equation that has its roots real, the square of the 

 second term is always greater than the quadruple product of the first and third 

 terms. 



Let the roots of the quadratic equation be represented by + « ;"itl + '''; 'ind 

 if X be the unknown quantity, then shall x'^ — ax -\- ab = O, 



- bx 



Now since a' + b'^ is greater than 'lab, by lemma 1, therefore d^ -\- b'^ + 



