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) ,