1817.) On Cubic Equations. 381 
nious remarks which have been founded on such a supposition are, 
therefore, quite irrelevant. Only contemplate the number of sides 
as infinite, and a simple consideration will establish the truth of the 
idea which has been combated. The versed sine which bisects an 
are, and is the measure of its deviation from the state of lying 
“* evenly between its extreme points,” is a third proportional to the 
diameter, and the chord of half the are. When this chord, then, 
becomes evanescent, or null in comparison of the diameter, the 
bisecting versed sine becomes also null in comparison of the arc. 
So that an evanescent or infinitesimal arc is in geometrical strictness 
a right line. 
Deviation from rectilinearity is not exactly the sense in which the 
term curvature is to be understood in the fluxionary analysis ; but it 
is the only interpretation the word can receive when applied to 
different portions of the same circle. On the principle just 
assumed, the curvature of any arc ¢ may then be estimated as 
ver.sin. 3 > 
( sin. 3) 
unit of comparison. 
=) tan. 3 ¢, the curvature of the semicircle being the 
If S,,S,,S,,5,, represent three successive sums of the powers 
of the roots of the equation x? — lx —c = 0, the theory of 
recurring series gives S, = )S, + cS,. Hence the following 
upinomial functions of ) and c, in addition to those in this paper :— 
ie? =k (RO 7® 7 6") ee ee ee ee ee oe (27) 
ges (i — 75 eu seer eeescerese ees eeeerer ee ee ee (28) 
whence ) = 7 == Ot bps eek PR a os a a8) 
ArticLe VIII. 
Queries respecting the Probability of Mit € from the Island of 
Spitzbergen the North Pole, by Means of Rein-deer, during the 
inter ; and answered by Persons who wintered there. By Col. 
Beaufoy, F. R.S. 
(To Dr. Thomson.) 
MY DEAR SIR, Bushey Heath, Feb. \1, 1817. 
Some years past I was impressed with the idea of the possibility 
of reaching the North Pole from Spitzbergen during the winter by 
travelling over the ice and snow in sledges drawn by rein-deer. 
Therefore, with the view of determining how far this plan was 
practicable, I sent several queries, and requested answers to them 
from Russians who were at that time living at Archangel, and had 
wintered in those remote islands, Those queries, together with the 
