A Treatise on the Calculus of Finite Differences |
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algebraical analogous apply approximate arbitrary constant assume Au₂ ax+b B₁ Bernoulli's numbers C₁ Calculus of Finite CHAPTER complete primitive condition corresponding Crelle curve deduce degree denoted determine Differential Calculus differential coefficients differential equation divergent series dy dx equa equal expansion expression factor Finite Differences formula functional equation given equation gives Grunert Hence independent variable indirect integral infinite Infinitesimal Calculus integral function involving Lagrange's formula limits linear difference-equation linear equation method notation obtain operation ordinates P₁ periodical function quantity rational and integral represented result roots satisfied Schlömilch second member shew singular solution solving substituting successive differences successive values summation supposed symbol Taylor's Theorem theorem tion u₁ u₂ unity Ux,y Ux+1 Ux+2 v₂ whence Δα Их
Popular passages
Page 119 - A SERIES is said to be convergent or divergent according as the sum of its first n terms approaches or does not approach to a finite limit when n is indefinitely increased.
Page 5 - Jacobi polynomials will be found in the exercises at the end of this chapter.
Page 80 - The successive orders of figurate numbers are defined by this ; — that the **" term of any order is equal to the sum of the first x terms of the order next preceding, while the terms of the first order are each equal to unity.
Page 185 - ... (1) and also by n — 1 , other differential equations, of the second order, to which the calculus of variations conducts, as supplementary to the given equation (1), and which may be thus denoted : /(*.)- «*/'(«**,)_ _/'(*.)-«*/'(«**.).
Page 315 - OS of the angle which the normal makes with the axis of x...
Page 315 - Hence integrating we find ...... ..... (3), Ex. 3. Required a curve such that a ray of light proceeding from a given point in its plane shall after two reflections by the curve return to the given point. The above problem has been discussed by Biot, whose solution as given by Lacroix (Diff. and Int.
Page 229 - ... 0. The solution of these equations, which are linear, can be made to depend upon that of a linear equation of the second order having t = 0 for a singularity : it appears that the integrals are normal in the vicinity of t = 0. Their full expression is...
Page 101 - I numerically. aa; (1 — e ) x=0 [Schlomilch, Grunert x. 342.] 10. Shew that the sum of all the negative powers of all whole numbers (unity being in both cases excluded) is unity ; 3 if odd powers are excluded it is 7 . 11.
Page 45 - The two radii which form a diameter of a circle are bisected, and perpendicular ordinates are raised at the points of bisection.