480 
ON THE DEMONSTRATION OF A THEOREM &C. 
[291 
which proves the theorem. The integral may also be written 
jj x 2 +2f- + z 2 , xx' + yy + zz' , xx" + yy" + zz" 
x'x + y'y + z'z, x' 2 + y 2 + z' 2 , xx” + yy” + z'z" 
x”x + y”y + z'z, x'x + y'y' + z'z', x” 2 + y" 2 + z' 2 
The conditions in the second problem are 
dmdm. 
(b + c) + (c + a) + (a + &) — +, 
(c +a) (a + b) + (a + b) (b + c) + (b + c) (c + a) —f 2 — g 2 - k 2 = +, 
(a + b) (b + c) (c + a) — (b + c) f 2 — (c + a)g 2 — (a + b) h 2 — y'gh = +, 
the first and second of which are respectively equivalent to 
ci + b + c = +, 
(a + b + cf 4- be + ccl + ab — f 2 — g 2 — h 2 = +, 
which are already proved. The last may be written 
(a + b + c) (be + ca + ab —f 2 — g 2 — h 2 ) — (abc — af 2 — bg 2 — cli 2 + 2fgh) = +, 
which, putting for shortness, 
A = x 2 +y 2 + z 2 , B = x 2 + y 2 + z 2 , C = x' 2 + y" 2 + z' 2 , 
F = xx” + y'y" + z'z", G = x”x + y"y + z"z, II = xx' + yy' + zz', 
is by what precedes expressible in the form 
ijf {A (BG- F 2 ) + B(CA - G 2 )+ G{AB - H 2 ) - (ABC- AF 2 - BG 2 - CH 2 + 2FGH)} dmdm' 
= i [J (ABC- FGH) dmdm', 
or, since VBC > F, */CA>G, \/AB>H, we have ABC > FGH, or ABC-FGH = +, 
and therefore the value of the integral is also positive. 
2, Stone Buildings, W.C., 6th March, 1860.