30 
Proceedings of the Royal Society of Edinburgh. [Sess. 
We may now extend this remark of Lagrange’s by adding that if, 
in the equation 
x n +p 1 x n ~ l +p 2 x n ~' 2 . . . +p n -‘>x 2 +p n -iX+p n = 0 , 
there be roots belonging to the operands 
Vn _ Vn-l _ _ Pi 
Vn-l Vn - 2 ’ 1 
then these, as also the operands from which they spring, are, irrespective 
of sign, in ascending order of magnitude. For, take two consecutive 
operands, 
-Pr±? and - h- ■ 
Pr Pr - 1 
T) 
If there be a root belonging to the operand — , this implies, as we shall 
Pr 
see, that its quadratic operator D^ 1 ^ . D . D ~ acts convergently, and 
this again means that p? > 4*p r+1 . p r - x >p r +i • Pr-i • 
Hence 
Pr ^ Pr+l 
Pr- 1 Pr 
As it is not axiomatic that the roots follow in magnitude their operands, 
we shall show this to be true for <p(x) = Arr 3 + Bx 1 + Cx + E = 0 by a proof 
that is easily seen to be general. Let the graph of this function be drawn, 
values of <f> being indicated along the y - axis as usual. Let now two 
indicators, starting from the origin, move with equal velocities to the 
right and left along the x-axis, both pausing for a moment whenever 
either passes through a value of x that makes <p vanish. Let this occur 
first when x = a v then the origin being transferred to (cq, 0) and the 
function transformed, the absolute term vanishes, the coefficient of the 
original function which disappears is E, and the root which has disappeared 
— the least root thus — is the one which had E in the numerator of its 
E 
every element, that is, the one whose operand was — ^ . The indicators 
now resuming their journey, let the next halt occur when x has a value £ 
which causes Af 2 + (3Acq + B)£ -f 3 Acq 2 + 2Bcq + C to vanish. The origin 
being again shifted, and the function transformed, the absolute term again 
vanishes, and now it is the original coefficient C that disappears. The root 
which has disappeared, then, is one having C or E (but not E alone) in the 
numerator of its every element, that is, the one whose operand was 
C 
— . The root £ which has disappeared being cq^cq, the algebraic 
