LOG 



LOG 



a') X* __ 



which is an equation to the ellipfe whofe axes are 

 b'a , ^ 



and —. -yr- 



6- - a" tJ V Ci' 



"1 



Again, if l> = a, the eU'pfe becoires a circle, and the 



equation for the value of C P becomes C P' = —jr> 



which is an equation to the parabola, whofe parameter is 

 Pa 



I. For the hyperbola (^ff. 21.)' ^^'^ ^^^^ notation re- 

 maining, D C = - a X + - ;r'' by the property of tlie 



curve J confequently C P' = -7^ X B D' = 



7- " •* + —d 



b 



d^ la- + i + " J 



irhich expreffes an equation to an hyperbola, whofe axes are 



r 3na — 7—. — ; — jr • 



•'- + i' d ^/ [a' + * ) 



3. For the parabola [Jig. 12.) ; put the parameter = p, then 



C D' = px, and B D' =/«• + x"; therefore C P' = -- 



{p X + H^) the equation to an hyperbola whofe axes are p 



and —J. 

 a 



PROB. II. 



If on any given right line, A B, there be taken any va- 

 riable diftance A L, and from L, in the fame direftion, any 

 given invariable diftance L M ; and if with the centres L and 

 B, and radii L A, BM, arcs be defcribed, it is required to 

 determine the nature of the curve, which is the locus of P, the 

 point of interfeftion. 



Let A B = a (/^. 23) ; L M = i ; B M = B P = (p, 

 and having drawn P O perpendicular to A B, put B O ^= *. 

 Then BL = « + i;LO = ?i+«-*; LP=:AL 

 -a-i-0i and becaufe L P^- L O' r;: B P' - B O', 

 we have in fymbols (a — i — ip)" — (tp + b — x)' = 

 ^' — X ; whence a' — 2 {a — x) b = 9' + 2 (a — x) $ ; 

 and adding — b' + b' + [a — x)' to one fide, and its 

 equal (a — *)' to the other fide, there refults a^ — A* 4- 

 (a -b- x)'- = (a ~x + q>y. 



Now take A C := L. M =: 3, draw C D perpendicular 

 to A B, and make A D = A B = a ; then C D" =:; ar 

 - b'-; C 0'= (a- b - xy,znd (O A + B P)^ =: (a- 

 M 4- ?)' whence we have DO=AO+PB;orPB=: 

 DO-AO. 



Hence it will be eafy to derive aR algebraical equation for 

 the rectangular co-ordinates of the curve ; for we have only 

 to put P O — _y, to fubititute ^/ (jr* + y ) for J>, and to 

 clear the equation of radicals. The equation thus found 

 ■will {hew the curve to be of the fourth order ; but the 



curve and its principal properties may be more readily de- 

 duced from the property above invcftigati d ; w'l. P B = 

 D O — A O. The curve will confift of two equal and fimi- 

 lar parts, lying on different fides of A B, it will be a fort 

 of oval, encloling the point B on every fide. 



The following are fome of the fimpleft cafes of the higher 

 order of loci. 



1. The bafe, and fum of the fides of a plane triangle 

 being given, the locus of its vertex is an cllipf!. 



2. The bafe and difference of the fides of a plane trian- 

 gle being given, the locus is an hyperbola. 



3. The locus of that point, from which, if perpendicu- 

 lars be drawn to tliree right lines given in pofition, and fuch 

 that the fum of their fquaies fliall be equal to a given fpace, 

 is an ellipfe. 



And the fame is true, whatever be the number of lines 

 given in pofition. 



4. If a triangle given in fpecies have two of its angles 

 upon a ilraight line given by pofition, and the fide adjacent 

 to thofe angles pafling through a given point, the locus of 

 the angle, oppofite that fide, is an hyperbola. 



5. Let A, B, be two given points in the right line A B, 

 given in pofition; let C, D, be two given points with- 

 out that line ; and alfo let C V, D V, be drawn meeting 

 in F and G, fo that the redangle A F x B G is given, 

 the locus of the point will in all cafes be a conic feSion. 



6. Let A B be a given ftraight line, and P a given point 

 without it ; let C P D be drawn, meetiugJA B in C ; and 

 let C P be to P D as A C to C B ; the locus of the point 

 D is a given hyperbola. 



7. When the bafe of a triangle is given, and one of the 

 angles at the bafe doubles the other, the locus of the vertex 

 is an hyperbola. 



8. The locus of the angles of parallelograms, formed by 

 drawing tangents at the vertices of any two conjugate dii- 

 meters of an ellipfe, is alfo an ellipfe cocentric ncilh the 



former. 



The above cafes, and feveral other curious properties of 

 this kind, the reader will find inveftigated in Leybomrn's 

 " Mathematical Repofitory." 



The method of conflrufting geometrical loci, by reducing 

 them to equations a« compound as poffible, we owe to Mr. 

 Craig, who iirll pubhibed it in his Treatife of the Quadra- 

 ture of Curves, 1693. It is explained at large in the fe- 

 venth and eighth books, of the Conic Keftions of the marquis 

 de I'Hofpital. This fubjed is particularly illuftrated in 

 Maclaurin's Algebra, part lii. See alfo Des Cartcs's Geo- 

 metry ; Stirling's lUuflratio Linearum Tertii Ordinis ; De 

 Witt's Elementa Curvarum : Bartholomxus Juhari, in 

 his Aditus ad nova Arcana Geometrica delegenda, has 

 fhewn how to find the loci of equations of the higher order. 

 See alfo the other writers mentioned in the preceding part of 

 this article. 



LOCUST, LocusTA, in Entomology, a genus of infedSy 

 referred to that oi gryllus ; which fee. 



Under that article the reader will find a particular account 

 of the devaftations occafioned by fwarms of locufts in their 

 marches, and he will perceive the propriety of the frequent 

 allufions to them that occur in the facred writings. Dr, 

 Shaw, Niebuhr, Ruffell, and many other travellers into the 

 eaftern countries, reprefent their tafte as agreeable, and in- 

 form us that they are frequent ly uied for food. Dr. Shaw 

 obferves, that when they are fprinkied with fait and fried, 

 they are not unlike, in tafte, to our frefh-water cray-fifh. 

 Ruflell fays, that the Arabs fait them, and eat them as a 

 delicacy. We learn alfo from Niebuhr, that they are ga- 

 thered by the Arabs in great abuudaace, dried, acd kept 



for 



