the nature of the parabola, DE โ DG = {EF = DF from above, 2y2which equation reduced [; gives y3—6ry2+1cr2y 2ry-y therefore 13: E H2; FI2 5 3′′y 32. Confeq. 3y22 -2730; the root of which is y230708rHE. Hence IN= 2y379921r the base of the para-s bola, and its abfcifs D F — GE=Xis='416693 ►. 2ry- -y2 ry Solutions were also given by Meffrs Amicus, Bre berick, Cock, Dalton, Gooch, Hornby, Howard, Mudge, Rowe, Todd, White, and Williams. Ꮓ XI QUESTION 873 anf. by Mr Geo Sanderson, of London. Let м and s represent the true places of the moon and star or fun, and m and s the apparent places of the fame; alfo z the zenith of the place. Put A = fine of the half sum, and R mainder, in rule I laft Diary; true distance мs; dfine of true zenith distances z M, zs; fine the half rexfine of half the half the diff. of the and v vers. z. finx +dx finx-d fin z MX fin z s Then A X a zs, Therefore = x + dxx -- fin z m X fin zs Put N half fum log. fines of the factors in the numerator added to the arith, comp.log. fines of the factors in the denominator, from which fub =log. of x, or log. fine of half the true diftance, as was to be inveftigated. Ingenious demonflrations were also given by Messrs Amicus, Dalton, and White. XII QUESTION 874 anfwered by Mr Abel Whitehoufe, Wolverhampton. Take the fluxion of the given equation, making ỳ conftant, and by reduction it is 2axyy; the fluents of which give 4ax=y2; an equation to the parabola, the parameter being 4a, Much in the fame manner is the folution given by Meffrs Amicus, Afpland, Cunliffe, Dalton, Gould, Howard, Mudge, Rowe, Surtees, Terril, Todd, and White. XIII QUESTION 875 anfwered by Mr John Dalton, Kendal. If a falling body move with a uniform velocity, it must neceffarily meet with a refiftance, in the medium it is moving in, equal to its weight. Now it has been proved (Emerion's Mechan. prop 108 cor. 3) that the refiftance to a cylinder, moving in a fluid in the direc tion of its axis, is equal to the weight of a cylinder of that fluid, of the fame bafe, and its length equal to the height a body falls in vacuo to acquire its velocity. Put now g 32 feet, v 10 the velocity, then the altitude falien to acquire the given velocity, which 72 g:2 2g Saltitude call a; put also p7854, 6·075 lb the weight of a cubic foot of air, m 200lb the man's weight, allo x = the diameter of the parachute. Then 502x250x2 the weight of the fame, which added to 200 or m, must be equal to the refiftance, namely abpx2, that is 3x2+m=abpx2; and hence x = 80 feet nearly, the diameter fought. g Answers to this question were also given by Meffrs Amicus, Asplană, Howard, Mudge, and Rowe. XIV QUESTION 876 anfwered by Mr John Farey, London. Let ABC be the three given points; join AB, which bifect in E, and draw EC, parallel to which draw KB, AL; alfo through E draw KI perp. to EC. Let p be a point in the curve, and join AP, BP, CP; alfo draw PL perp. to A L meeting EC in G. 2 4 m 2112212 5r4 — 5 n4 † 14r2 n2 m4122m42 · 2 m2 n4—2m2 r4 + 4m2n2p2 — 7n4r2 + 7n2r4 + n® — x6 4 m 3 which equation re- Sm2nr + n22 duced gives y = 2 m2 x n2r ±√c ? where c is the bi , Squadratic equation under the vinculum. When co, the four roots or values of x are it is evident that if m2+2 be lefs than n2, the curve will be Sir I. Newton's 33d fpecies, as in the figure; if m2 + r22, it will be the 34th fpecies; and if m2 2 be greater than n2, it will be the 37th fpecies. Alfo if rn, it will be the 38th fpecies; if n2 m22, it will be the 40th fpecies; and liftly if no, it wilkbe the 45th fpecies. it Corol. When ro, (or the three points are in the fame right line, as in queft. 696), and if m2 be lefs than 2 n2, the curve will be the 30th Tpecies; if m2 2n2, it will be the 41ft fpecies; if m=n, will be a right line with a conjugate point; if m2 be greater than 2 n2, it will be the 45th fpecies; and lastly if mo, it becomes the conic equilateral hyperbola. Mr George Sanderfon, after giving the folution nearly as above, makes the following obfervations: If o be the centre of a circle paffing through the given points, and co perp. to A B (c being always confidered as the point from which the mean proportional is drawn); then if co be less than a third proportional to 2 AD and E B, the curve will be Newton's 33d fpecies; if co be equal to the 3d proportional, it will be the 34th fpecies; and if co te greater than the fame, it will be the 37th fpecies. If co=EO, will be the 38th fpecies; and if E0=0, or ACCB, the 45th fpecies; the afymptote being parallel to AB, and its diftance from E equal to AD. the If coo, or the three points in the fame right line (A B'), afymptote is perp. to A B; and if E c2 be greater than 2 x B2, the locus of the point P is fill the 45th fpecies; if c2 2E B2, the 41ft fpecies; and if it be lefs, the 39th. If EC EB, the afypmtote paffes through E, and is the locus of the point P, and B is a conjugate point. Lafly, if c fall in the point E, or ACCB, it becomes a conic equilateral hyperbola whofe foci are A and B. This question was also answered by Mers Amicus, Cock, Howard, and Plus Minus. XV Or PRIZE QUESTION anf. by Mr Geo Sanderfon, London. LEMMA. If -m, -n, -r be the three roots or values of x in the @quation ax + bx2+ex+d=0; then ift, if r=m+n±2√mn, C in then is b24ac; 2d, if lie without m+n+2mn and m+n mn, that is greater than the former or less than the latter, then b is greater than 4ac; 3d, if r lie between m++ 2 √mn and m + n − 2√mn, then bis lefs than 4ac. 4th, In the first cafe, where r = m+n±2√mn, if m4n, the two leaft roots, n and y, are equal. 5th, If the two leaft roots be equal, and the third greater than 4 times. the leaf, (or equal root), then b' is greater than 4ac. Laftly, either the first or fecond, cates, namely when does not fall between m+n±√mn, then half the fum of the roots is lefs than the greateft of the three. All the cafes of this lemma will be evident to every one who confiders, that in the equation x3+px2+7x+r=0, p is equal to the fum of the rcors, their signs being changed, q the fum of the produis of every two, &, &c. SOLUTION. Fig. 21 to Sir 1 Newton's 15th fpecies, confile of two ambigenous hyperbolas at d and 8, and one infcribed at D, as in the annexed fig. but without oval or conjugate point. Here it appears that Sir Ifaac confidered the two leaft roots (AT, A1) as ( 4ad impoffible when 22 7AB= 4ac Sx (the correfponding ab cia to the ordinate BC at the point c where the afymptote cuts the B curve) is affirmative, or when 62 is greater than 4ae. But whoever confiders the lemm, will find that examples are numerous in which the equation ax + bx2 + cx + d. las three real and unequal nega. tive roots, (or three real negative roots with the two leaft equal), and 2 greater 4ad a pofitive quantity, alfo A D lefs than the than 4ac, orb2 4ac greatest of the three roots; and confequently 'ax2 + bx2 + cx + dy confifts of an inferibed Shyperbola at D, the equation y±1 and two ambigenous ones at d, 8, with an oval in the triangle Ddd, or the fame with a conjugate point. Here the two greater roots cannot become equal, as in the 11th fpec'es. Whence it is manifeft that these two fpecies fhou'd follow Newton's 14th, fo as to make the 15th and 16th, and not the 11th and 15th as Mr Stirling has placed them at pages 99 and 100. nad 4ac By the firft cafe of the lemma examples may be found in which the eq. ax 3 + bx2 + cx+do has three real unequal roots (or three real roots with the two leaft ones equal), and b24ac, and A D lefs than the greatest root A t. S will be infinite; or the afympConfequently A B = x = b2 Stote cut the curve at an infinite distance; and the figure confifts of three infcribed hyperbolas with three diameters, and an oval in the triangle D dd, or the fame with a conjugate point. If the two leaft roots be impoffible, the curve becomes Newton's 22d fpecies; and therefore thefe two fhould make the 24th and 25th fpecies, as Stirling has placed them at page 102. As to Mr Stone's difcoveries; I find that Sir I Newton's 59th figure to his 55th fpecies is the locus of the equation xy2bx2. - cx+d; where the two roots or values of x, when yo, or bx2 ex + d=o, are both pofitive. 1 But as no notice is there taken of the locus of the equation xy2 = bx2 + cx+d, let us defcribe it here; taking for an example the numeral equation xy2 = 4x2 + 56x+160, cry •4x2+56x+160, where the two roots or values of when (y = 0, or 4x2+56x+1600, are 4 ánd - 10. x Draw two indefinite right lines a B and as cutting each other at right angles in A, (fig. 2). Let A be the beginning of the abfciffa A B, and As the first ordinate; then if AB be reprefented by + x, and Bc its correfponding ordinate by y; on AB, but on the contrary fide of A, take AT-4, and A-10. If xo, then BC is infinite, and the curve runs on infinitely towards s; therefore As is an afymptote to two hyperbolic legs equally diftant from AB. If + x be infinite, then 4x2 + 56x + 160 ? becomes y=2x, or y2 x 4x, an equation to the conic parabola, whofe vertex is A, and parameter 4: whence the curve has two parabolic legs joined to two hyperbolic ones, meeting the parabola at an infinite diftance towards B. If -x be taken - -5, then y +2: if -x be taken 4 or — 10, then yo, and the curve palles through the points and 7: if. -x be taken less than 4, or greater than 10, S ‚4x2 — 56x+1607 it is impoffible, because the quanthen y -X Stity under the vinculum is nega tive, and no part of the curve can fall between As and r, or beyond 7; therefore the part of the locus lying on that fide of As is an oval, and the curve confifts of two hyperbolo-parabolic figures on one fide of the afymptote a s, with an oval on the contrary fide. If the two roots be equal, the oval becomes a conjugate point; which is Stone's fecond fpecies. If the two roots be impoffible, the curve becomes Newton's 57th figure to his 53d species; and therefore My Stone's two fhould be the 57th and 58th fpecies in the enumeration, the catalogue being deficient without them; that is, they fhould be the 538 and 54th in Newton's, and the 57th and 58th fpecies in Stirling's enumeration. The fame anfwered by Amicus. As, in fig. 2d of Sir Ifaac Newton's 1ft fpecies, where AD bis greater than the greatest root of the equation for the limits, zal when the conic hyperb. that bilects the ordinates of the curve, coincides with its afymptotes, and the term ey of confeq. is wanting, coincides with, 7 with, and t with p, and that fig. 2d becomes the fame with fig. 17th, or the roth fpecies, and when the oval vanishes, or the two lefs limits are equal, it becomes fig. 20 and fpecies 13th: So, in |