534] a “smith’s prize” paper; solutions. 477 
that is, 
26<f) + 2a/3 - (6 + <f>) (a + /3) = 0, 
26<f> + 278 — (6 + <£) (7 + 8) = 0, 
or, what is the same thing, 
26(f) : 2 : 6 + cf> = — a/3 (7 + 3) + 78 (a + /3) 
: 7 + 3 — a — /3 
: 78 — a/3, 
viz. we have thus the values of 6cf>, 6 + (f> (and thence of 6, (f>) corresponding to the 
relation 2 (a/3 + 78) = (a + /3) (7 + 8) of the roots. And by cyclically permuting /3, 7, 8 
as before, we have the values of 6(f), 6 + </> corresponding to the other two forms 
respectively of the relation between the roots. 
3. If in a plane A, B, G, D are fixed points and P a variable point, find the 
linear relation 
a. PAB + /3. PBG + 7 . PCD + 8 . PDA = 0 
which connects the areas of the triangles PAB, &c. 
Taking (x, y, 1), («!, y 1} 1), &c. for the coordinates of P, A, B, C, D respectively, 
we have 
PAB = 
x, y, 1 
Xi, y I, 1 
x 2 , x 2 , 1 
PBC = 023, &c 
012, suppose, 
(where the values of the several determinants fix the signs of the several triangles). 
The identical equation then is 
a. 012 + /3.023 + 7.034 + 8.041 = 0; 
(that such an equation exists appears at once by the consideration that a, /3, 7, 8 can 
be determined so that the coefficients of x, y, and the constant term shall severally 
vanish); and in order actually to find the values we may make P coincide with the 
points A, B, C, D successively. We thus have 
/3.123 + 7.134 = 0, 
7.234 + 8 . 241 = 0, 
8.341 +a .312 = 0, 
a. 412+ /3.423 = 0, 
/3.123 + 7.341 = 0, 
7.234 + 3 . 412 = 0, 
8.341 +a . 123 = 0, 
a .412+/3.234 = 0, 
or, what is the same thing,