LOC 
in the formula, and substituting m for e, we 
shall get yy — 9,ry — px-\-rr-\-ps=.o. 
Again, by comparing the correspondent 
terms — ‘2r y and — ^ay, as also — p x 
and — b X, VIC have r = a, and p — b; and 
comparing the terms wherein are neitlier of 
tlie unknown quantities x y, we get rr-}- 
ps = cc; and substituting a and b for r and 
p, then will s = — • which is a ne- 
' 0 
gative expression when a is greater than c, 
as is here supposed. There is no need of 
comparing the first terms y y and y y, be- 
cause they are the same. Now the values 
of n, r, p, s; being thus found, the sought 
locus may be constructed by means of the 
construction of the formula, and after the 
following manner. 
Because B E = n = o (fig. 9), the points 
B E, do coincide, and the line A E falls in 
A P ; therefore through the fixed point A 
draw the line A D = r = a parallel to P M, 
and draw D G parallel to A P, in which 
take D C = ^ — — s ; then with 
DC, as a diameter, whose ordinates are 
right lines parallel to P M, and parameter 
the line C H = p = 6, describe a parabola : 
then the two portions O M M, R M S, con- 
tained in the angle P A O, formed by the 
line AP, and the line AO drawn parallel 
to P M, will be the locus of the given equa- 
tion, as is easily proved. 
If in a given equation whose locus is a pa- 
rabola, a; a;' is without a fraction ; then the 
term of the second formula must be com- 
pared with those of the given equation. 
Thus much for the method of construct- 
ing the loci of the equations which are conic 
sections. If, now, an equation whose locus 
is a conic section, be given, and the parti- 
cular section whereof it is the locus be re- 
quired : all the terms of the given equation 
being brought over to one side, so that the 
other b^ equal to nothing, there will be two 
cases. 
Case I. When the rectangle a;y is not in 
the given equation. 1. If either yy or xx 
be in the same equation, the locus will be a 
parabola. 2. If both x x and y y are in the 
equation with the same signs, the locus will 
be an ellipsis, or a ciifcle. S.lfxx and y y 
have different signs, the locus will be an 
hyperbola, or the opposite sections regard- 
ing their diameters. 
Case II. When the rectangle x y is in the 
given equation. 1. If neither of the squares 
XX ov y y, or only one of them, be in the 
LOG 
same, the locus of it will be an hyperbola 
between the asymptotes. 2. It' yy and x x 
be therein, having different signs, the locus 
will be an hyperbola regarding its diame- 
ters. 3. If both the squares xx and yy 
are in the equation, having the same signs, 
you must free the square y y from fractions ; 
and then the locus will be an hyperbola, 
when the square of i the fraction multiply- 
ing X y, is equal to the fraction multiplying 
X X, an ellipsis, or circle, when the same is 
less; and an hyperbola, or the opposite 
sections, regarding their diameters, when 
greater. 
LOCUST. SeeGRYi-Los. 
LODGMENT, in military affairs, is a 
work raised with earth, gabions, fascines, 
wool-packs, or mantelets, to cover the be- 
siegers from the enemy’s fire, and to pre- 
vent their losing a place which they have 
gained, and are resolved, if possible, to 
keep. For this purpose, when a lodgment 
is to be made on the glacis, covert-way, or 
in a breach, there must be great provision 
made of fascines, sand-bags, &c. in the 
trenches ; and during the action, the pio- 
neers with fascines, sand-bags, &c. should 
be making the lodgnjent, in order to form 
a covering in as advantageous a manner as 
possible from the opposite bastion, or the 
place most to be feared. 
LOEFLINGIA, in botany, so called 
from Peter Loefling; a genus of the Trian- 
dria Monogynia class and order. Natural 
order of Caryophyllei. Essential charac- 
ter : calyx five-leaved; corolla five-petal- 
led, very small ; capsule one-celled, three 
valved. There is but one species, viz. 
L. hispanica, a native of Spain. 
LOESELIA, in botany, from Joseph 
Loesel, a genus of the Didynamia Angios- 
permia class and order. Natural order of 
Convolvuli, Jussieu. Essential character : 
calyx four-cleft ; corolla with all the seg- 
ments directed ope way ; stamina opposite 
to the petal ; capsule three-celled. There 
is but one species, viz. L. ciliata, found at 
La Vera Cruz in South America. 
LOG, in naval affairs, a machine used to 
measure the rate of a ship’s velocity through 
the water. For this purpose, there are se- 
veral various inventions, but the one most 
generally used is the following, called the 
common log. It is a piece of thin board, 
forming the quadrant of a circle of about six 
inches radius, and balanced by a small plate 
of lead nailed on the circular part, so as to 
swim perpendicular in the water, with the 
greater part immersed. The log-line is 
f • 
