The Collected Mathematical Papers of Henry John Stephen Smith

The Collected Mathematical Papers of Henry John Stephen Smith

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1894 edition. Excerpt: ... ai+1 = l, mod 12. Caeterum, ut ex inventa sequatione 4X==of_j_1 + 3&f + 1, ipsius X reprsesentatio habeatur, fiat ai+1 + bi + 1, =0, mod 4: quo pacto erit contra, si data fuerit sequatio X = A2 + SB2, fiat A = QjuL + 2e1, B = 2v + e2, designantibus Uteris el3 e2 unitates, quarum altera ipsa sequatione definitur, altera ita capiatur ut /a + v par evadat. Quibus rite animadversis, habebuntur sequationes 4: X = (3e2Be1A)2 + 3(e1A + e2B)2, 3e2B1-61A = (-l)iai+1 = (-l)i+lai+2, =1, mod 12 e1A + e2B=(-iy+1bi+1; nam (--1)+1 bi+1 certo numerus positivus est, cum debeat esse X cujus Veritas facile colligitur ex aequivalentia formarum (--4X, 0, 3), (-12 X, 0,1), Tres isti numeri 3e2B--e1A', 2.4, 3e2B + e1A, qui repraesentant totidem valores indeterminati x in sequatione 4k = x2+.3y2y eo etiam nomine nobis memorandi sunt, quod, evoluta radice /3 in fraetionem continuam vulgarem, inter denominatores quotientium completorum uno tractu veniunt. Atque, si est A3B, erit 2 A medius, si vero A 3B, erit idem vel primus vel postremus. Nam, si A 3B, colligimus sequationes 3B-A-2(3B + A)x + 2Ax2 = 0, 2A-2(3B-A)x-(3B + A)x2 = 0 alteram alteram excipere in periodo aequationis--3 X + x2 = 0; utramque enim sequationi--3 X + x2 = 0 sequivalere per demonstrationem theorematis praecedentis evincitur; utraque autem eharaeterem aequationis periodicae habet propter insequalitatem A 3 B. Similiter, si A 3B, sequationes-2A-23B-A)x + (3B + A)x = 0 (3B + A) + Ax-(3B-A)x2 = 0 deinceps occurrunt in eodem periodo. Hinc nanciscimur methodum non inelegantem solvendi aequationem = A2 + 3B2 Namque in evolvenda radice x/3X, inveniemus tres juxta denominatores, quarum medius sequatur extremorum summae, idemque exsuperat radicem /3; ex bis unus erit 2 A, reliqui duo habebunt...show more

Product details

  • Paperback | 214 pages
  • 189 x 246 x 11mm | 390g
  • Rarebooksclub.com
  • United States
  • English
  • black & white illustrations
  • 1236967518
  • 9781236967510