567 
of Edinhurcjli^ Session 1865-66. 
any path, we must have EP + PF constant, because the axis major 
depends on the velocity, not the direction, of projection. Hence (1) 
tlie locus of F is the circle AFO. Again, since, if F be the focus 
of the path which meets PR in Q, we must have FQ = QS ; it is 
obvious that the greatest range Vq is to be found by the condition 
Oq = qs. 0 is therefore the second focus of this trajectory, and 
therefore (2) the direction of projection for the greatest range on 
PR bisects the angle APR. If QF = QF' = QS, F and F' are the 
second foci of the two paths by which Q may be reached ; and, as 
^ FPO = F'PO - we see the truth of (3). If Q be a point reached 
by the projectile when moving in a direction perpendicular to 
PR — we must evidently have PQF' = ’^PQF = ^ SQR = ^EQP ; 
i.e., EQ passes through F'. This case is represented on the other 
side of the diagram — where fg = gh=fg> The ellipse whose second 
focus is / evidently meet Pr at right angles : and that whose 
second focus is f' has (4) its vertex at g. The locus of g is evi- 
dently the envelop of all the trajectories. Now 
P^=P0 + 0^ = PA + 0^, 
Eg' = Es — sg = E A — Og. 
Hence 
P g -f- Eg' — PA -f- AE, 
or (5) the envelop is an ellipse, whose foci are E and P, and which 
passes through A. 
5. On the Fairy Stones found in the Elwand Water near 
Melrose. By Sir David Brewster, K.H., F.R.S. 
On the banks of the Elwand Water, which runs into the Tweed, 
about two miles above Melrose, there is a picturesque glen called 
the Fairy Dean, which has become a favourite place of resort, from 
its association with the incidents in “ The Monastery” by Sir 
Walter Scott. It has acquired an interest of a different kind 
from certain mineral concretions which have received the name of 
Fairy Stones, from their being found in that part of the rivulet 
which runs through the Fairy Dean. 
When the Waverley Novels were not acknowledged by their 
author, facts or incidents to which they referred, were always wel- 
4 E 
VOL. V. 
