78
ANALYTICAL RESEARCHES CONNECTED WITH
[114
we have
p-g’-hjp(i + ra)
= 4 (l - ( j) |(P - (g - h)‘) [(/»-gh) (P - (g - hf) - 2gh (g - h)> - 2gh ( ^=^ y ']
4g 2 h 2 (g - h) 2 (J 2 - gh))
J 2
K 2 [K (F + £) - 2f 2 - 2g 2 - 2h 2 + 4J 2 }
= 4 (l -1) | p - (g - hp [(/* - gh) ((g+hy - p - iö!)] - Clz£ h >J.
Also, since
we have
(p-(g-h)») + (( g +h)*-p-
4g 2 h’
>
, , (J 2 — gh)
*g h —jr-
f>_g«_h>+^£^)^(l + }'g) + A' 1 fiT(7+2)-2P-2g ! -2h s )
= 4 (l - i) (P - (g - h.y) 2ghJ i (l -1) (l - £ a ) ,
and the values obtained above give also
2 g h y/i - vr+r° vrr^
- 4 (l - 3) (P - (g - hp 2ghJ* (l - |) (l - jl) ,
which shows that the relation between F and Z is verified by the assumed values of
these quantities, and the other two equations are of course also verified. The solution
of the problem will be rendered more complete if the equations of the required sections
and of the auxiliary sections made use of in the geometrical construction are expressed
in terms of f, g, h, J.
7.
First, to substitute in the equations of the required sections or resultors. Writing
the first equation in the form
K 2
2Vi3<£
the coefficient of x will be
aXx + (hX + V(2D) y + (gX + V23) -z + V - ap Vl + X 2 w\ - 0,
f 2 \ (2fgh
^ + (-f+g + h )2 -2J ( -f+g + h) ,