490 G. H. KNIBBS AND F. W. BARFORD. 
The elimination of Land M from the first three of these 
last equations gives 
Lo Pegi Sy Oia ee (42) 
a B Yp 
a® B ys 
And the elimination of L and M from the last three gives 
UE Mi 1 tgp = Ose eee (43) 
a? 8? ys | a B ys 
a? By, | a” BP 4 
the second form being obtained by dividing the first column 
by « and the second column by Pf. 
From equation (42) is obtained the equation 
Y3 — Ye (a + B) - ya = 10) 4. AU (44) 
and from equation (43) is obtained the equation 
Y, — ¥3 (a + B) + Yap = 0 o..0c8 (45) 
Also « and f are the roots of 
@ f(a = B) + eOReNO Ri Lee (46) 
Eliminating « + 6 and of from (44), (45) and (46) it is 
evident that « and f/ are the roots of the equation 
Since « and / are respectively k? and k’‘, this gives a formal 
solution for p and q since kis known. It must, however, 
be carefully examined. Since p and q are, in practical 
computations, restricted to real values, it follows that k? 
and k* must be real and positive. The equation (47) must 
therefore have realand positive roots. If written at length 
the equation takes the form 
SE (wYs — Ys) + E(Y%Ys — YrYs) + (YoYs — Ys) = O......(47a) 
The condition for real roots will be first investigated. 
The condition is 
(Yo¥s — WiYs) — 4(%Ys — Yo) (Yo¥s — Ys) = O5 or 
YsVi — SYYsys — 3y2¥3 + 4NY3s + 4Hsy2 > O. 
This may be written in the form 
yayi — 2ygyo(B ys — 2y2) + Ys(4HYs — 3 y2)> 0 
