A Treatise on the Calculus of Finite DifferencesMacmillan, 1872 - 336 páginas |
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A³u algebraical analogous apply approximate arbitrary constant assume Au₂ ax+b B₁ Bernoulli's numbers C₁ C₂ Calculus of Finite CHAPTER complete primitive condition corresponding Crelle curve deduce degree denoted determine Diff 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 Lagrange's formula limits linear difference-equation linear equations 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 supposed symbol Taylor's Theorem theorem tion u₁ u₂ unity Ux+1 Ux+2 V₁ v₂ Vx+1 whence Их
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Página 271 - Hence, that of the given equation is 7. Partial difference-equations are of frequent occurrence in the theory of games of chance. The following is an example of the kind of problems in which they present themselves. Ex. 6. A and B engage in a game, each step of which consists in one of them winning a counter from the other. At the commencement, A has x counters and B has y counters, and in each successive step the probability of A's winning a counter from B is p, and therefore of B's winning a counter...
Página 218 - A seed is planted — when one year old it produces ten-fold, and when two years old and upwards eighteen-fold. Every seed is planted as soon as produced. Find the number of grains at the end of the xth year.
Página 84 - 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.
Página 317 - Hence integrating we find + c) ...................... (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. Gale. Tom.
Página 231 - ... 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...
Página 232 - Now let va, v1} vt &c. vn be a series of quantities, the successive terms of which are connected by the same relation as that which we have seen to exist between the successive values of xm + — and of 2 cos md, viz. x x" Also as in those cases let vt = 2, but let i\ be any quantity whatever, thus we have &c.
Página 187 - ... (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 : /(*.)- «*/'(«**,)_ _/'(*.)-«*/'(«**.).
Página vi - I have strictly adhered to the principle of inserting all that was contained in the First Edition.
Página 49 - The two radii which form a diameter of a circle are bisected, and perpendicular ordinates are raised at the points of bisection.
Página 103 - Br-l = -^-r b-~—( numerically. [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 j- . 11.