A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter.

  • Isaac Todhunter
Date:
1865
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Licence: Public Domain Mark

Credit: A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter. Source: Wellcome Collection.

    527/648 (page 507)
    We might suppose that zx,y is the coefficient of Pr* in the ex- pansion of a function of t and r; then it would easily follow that this function must he of the form (t) +']r (t) rt 1 T t a T b t where $ (t) is an arbitrary function of t, and yjr (t) an arbitrary function of t. Laplace, however, proceeds thus. He puts 1 a rt r and he calls this the equation generatrice of the given equation in Finite Differences. He takes u to denote the function of t and r which when expanded in powers of t and t has zx, y for the co- 'LL efficient of fr*. Then in the expansion of ^—y the coefficient of ?t° will be zx, y. Laplace then transforms thus. By the equation generatrice we have a i c + ~ 1 T * l-b T therefore, u <V M(M+;)Tc+ai+a(;~*) - - b Develope the second member according to powers of b; thus