LOC 



LOG 



ing 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 fix- 

 ed point A draw the line A D = r = a 

 parallel to P M, and draw DG parallel to 

 A P, in which take D C = afl ~ cc _ 



s , then with D C, as a diameter, whose 

 ordinates are right lines parallel to P M, 

 and parameter the line C H = p = b, 

 describe a parabola: then the two por- 

 tions O M M, R M S, contained in the 

 angle P A O, formed by the line A P, and 

 the line A O drawn parallel to P M, will 

 be the locus of the given equation, as is 

 easily proved. 



If in a given equation whose locus is 

 a parabola, a: a: is without a fraction ; 

 then the term of the second formula must 

 be compared 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 giv- 

 en, and the particular section whereof 

 it is the locus be required, all the terms 

 of the given equation being brought over 

 to one side, so that the other be equal to 

 nothing, there will be two cases. 



Case I. When the rectangle x 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 circle. 

 3. If x x and y y have different signs, the 

 locus will be an hyperbola, or the oppo- 

 site sections regarding their diameters. 



Case II. When the rectangle xy is in 

 the given equation. 1. If neither of the 

 squares x x or y y, or only one of them, 

 be in the same, the locus of it will be an 

 hyperbola between the asymptotes. 2. If 

 y y and x x be therein, having different 

 signs, the locus will be an hyperbola re- 

 garding its diameters. 3. If both the 

 squares x x and y y 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 5 the fraction multiplying 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. SeeGRTLi.rs. 



LODGMENT, in military affairs, is a 

 work raised with earth, gabions, fascines, 

 wool-packs, or mantelets, to cover the 

 besiegers from the enemy's fire, and to 

 prevent their losing a place which they 

 have gained, and are resolved, if possible, 

 to keep. For this purpose, when a lodg- 

 ment 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 pioneers with fascines, sand-bags, 

 &c. should be making the lodgment, 

 in order to form a covering in as advanta- 

 geous a manner as possible from the op- 

 posite bastion, or the place most to be 

 feared. 



LOEFLINGIA, in botany, so called 

 from Peter Loefling; a genus of the 

 Triandria Monogynia class and order. 

 Natural order of Caryophyliei. Essen- 

 tial character : calyx five-leaved ; co- 

 rolla five-petalled, 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 An- 

 giospermia class and order. Natural or- 

 der of Convolvuli, Jussieu. Essential 

 character ; calyx four-cleft ; corolla with 

 all the segments directed one way ; sta- 

 mina 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 several various inventions, but 

 the one most generally used is the fol- 

 lowing, called the common log. It is a 

 piece of thin board, forming the quadrant 

 of a cii'cle of about six inches radius, and 

 balanced by a small plate of lead nailed 

 on the circular part, so as to swim per- 

 pendicular in the water, with the greater 

 part immersed. The log-line is fastened 

 to the log, by means of two legs, one of 

 which is knotted through a hole at one 

 corner, while the other is attached to a 

 pin fixed in a hole at the other corner, so 

 as to draw out occasionally. The log- 

 line being divided into certain spaces 

 (which are in proportion to an equal 

 number of geographical miles, as a half, 

 or quarter' minute, is to an hour of 

 time), is wound about a reel. The whole 

 is employed to measure the ship's head- 

 way in the following manner: the^reel 

 being held by one man, and the half mi- 

 nute-glass by another, the mate of the 



