492 G. H. KNIBBS AND F, W. BARFORD. 
In this case the curve degenerates into the curve y=ba? 
and there is only one possible value for y;, viz., the fourth 
term of the geometrical progression of which y y,and ¥; are 
the first three terms. [Since y ¥2 ys; ys, are in geometrical 
$ uea 0 
progression the expression "4 assumes the form = 
Y2¥3 — Yiys 
whose limiting value has been proved to be @ ori 
41 2 
This limitation of the possible values for y; can be illus- 
trated by an example. Suppose that three points are taken 
whose abscissae are 1, 2 and 4 (and consequently in geo- 
metrical progression) and whose ordinates are 13°7, 22°4 
and 24°0. It is required to find the limits for y, when «=8, 
In this example k = 2, and 2°, 2° are the roots of the 
equation 1°43°7 22°40) =O 
€ 22°4 24°0 
& 240 y, 
which when expanded becomes 
172°96 & — € (537-6 — 13:7 y,) + (576 — 22°4 y,)=0...(49) 
In this example y, = 13°73 yo = 22°43; y; = 24°03; conse- 
quently, referring to condition (48) already established, 4, 
cannot lie between the values —2o-+ = #9" 
187°7 
22°2 and — 26°28. 
Also, since the roots of (47a) must be positive, it follows 
that 576 — 22°4 y, must be positive. Consequently either 
y, is negative, or, if positive, cannot be greater than 516 
22°4 
or say 
say 29°7. 
From these conditions it is evident that the positive 
values of y, are limited to the region between 22°2 and 25°7. 
All negative values are admissible which are numerically 
greater than 26°28. 
We now proceed to investigate the curve of three terms, 
iz 
y = be? + cw + dx 
