LOC 
LOC 
■Ward, shank, the pot or bread, bit, and 
bow-ward. The importation of locks is 
prohibited. 
Lock, or Weir, in inland navigations, 
the general name for all those works of 
wood or stone, made to confine and raise 
the water of a river; the banks, also, which 
are made to divert the course of a river, 
are called by these names in some places. 
But the term lock is more particularly ap- 
propriated to express a kind of canal in- 
closed between two gates ; the upper called 
by workmen the sluice-gate, and the lower 
called the flood-gate. These serve in arti- 
ficial navigations to confine the water, and 
render the passage of boats easy in passing 
up and down tlie stream. See Canal. 
LOCUS geometrkus, denotes a line, by 
whicli a local or indeterminate problem is 
solved. See Local Problem. 
A locus is a line, any point of which may 
equally solve an indeterminate problem. 
Thus, if a right line suffice for the construc- 
'' tion of the equation, it is called locus ad rec- 
tum ; if a circle, locus ad circulum ; if a pa- 
rabola, locus ad yaraholam ; if an ellipsis, 
locus ad ellipsin ; and so of tlie rest of the 
conic sections. 
The loci of such equations as are right 
line.s, or circles, the ancients called plain lo- 
ci ; and of those that are parabolas, hyperbo- 
las, &c. solid loci. But Wolfius, and others, 
among the moderns, divide the loci more 
commodiously into orders, according to 
the numbers of dimensions to which the in- 
determinate quantities rise. Thus, it will be 
a locus of the first order, if the equation is 
a; = ; a locus of the second or quadratic 
order, if = a x, or z=a^ — x^ ; a locus 
of the third or cubic order, if a^ x, or 
y' = ax^ — x^, &c. 
The better to conceive the nature of the 
locus, suppose two unknown and variable 
rightlines A P, P M (Plate VIII. Miscel.fig. 
4 and b) making any given angle A P M with 
each other ; the one whereof, as A P, we 
call X, having a fixed origin in the point A, 
and extending itself indefinitely along a right 
line given in position ; tlie other P M, which 
we call y, continually changing its position, 
but always parallel to itself. An equation 
only containing these two unknown quan- 
tities, X and y, mixed with known ones, 
which expresses the relation of every varia- 
ble quantity A P, (x), to its correspondent 
variable quantity P M, (y) : the line pass- 
ing through the extremities of all the values 
of y, i.e. through all the points M, is called 
a geometrical locus, in general, and tlie locus 
of that equation in particular. 
All equations, whose loci are of the first 
order, may be reduced to some one of the 
four following formulas : 1. y = ^. 2. y: 
bx , bx ^ bx 
• 1- c. 3. y =: c. 4. y = c . 
a ' ^ a ^ a 
Where the unknown quantity, y, is sup- 
posed always to be freed from fractions, 
and the fraction that multiplies the other 
unknown quantity x, to be reduced to this 
expression and all the known terms 
to c. 
The locus of the first formula being al- 
ready determined : to find that of the se- 
h X 
cond, V = — — }- c ; in the line A P, fig. 6, 
take A B = a, and draw BiE = 6, AD = c 
and parallel to PM. On the same side 
AP, draw the line AE of an indefinite 
length towards E, and the indefinite straight 
line D M parallel to A E. Then the line 
D M is the locus of the aforesaid equation, 
or formula ; for if the line M P be drawn 
from any point M thereof parallel to A Q, 
the triangles ABE, and APE, will be si- 
milar : and therefore A B (a) ; BE (i) s; 
A P (x) P F = ; and consequently P M 
(y) = PP(^=^)+FM(c). 
To find the locus of the third form, y = 
~ — c, proceed thus ; assume A B = a 
(fig. 7); and draw the right lines B E = A, 
AD = c and parallel to PM, the one on 
one side A P, and the other on the other 
side : and through the points A E, draw 
the line A E of an indefinite length towards 
E, and through the point D, the line D M 
parallel toAE; then the indefinite right 
line G M shall be the locus sought ; for we 
shall have always P M (y) = P F = 
Lastly, to find the locus of the fourth for- 
h X 
mula, y = c- — — ; in A P (fig. 8) : take 
A B = a, and draw B E = 6, A D z= c, and 
parallel to P M, the one on one side A P, 
and the other on the other side; and 
through the points A, and E, draw the line 
A E indefinitely towards E, and through 
the point D draw the line D M parallel to 
A E. Then D G shall be the locus sought ; 
for if the line M P be drawn from any point 
