422 TREMBLEY. 



reverse the order which De Moivre has taken. Trembley does not 

 supply general demonstrations ; he begins with a simple case, then 

 he proceeds to another which is a little more complex, and when 

 the law which governs the general result seems obvious he enun- 

 ciates it, leaving to his readers to convince themselves that the law 

 is universally true. 



780. Trembley notices the subject of the summation of a cer- 

 tain series which we have considered in Art. 460. Trembley says, 

 M. Euler remarque que dans ce cas la somme de la suite qui donne 

 la probabilite, pent s'exprimer par des produits. Cela pent se de- 

 montrer par le calcul integral, par la methode suivante qui est 

 fort simple. But in what follows in the memoir, there is no use of 

 the Integral Calculus, and the demonstration seems quite unsatis- 

 factory. The result is verified when a? = 1, 2, 3, or 4 and then is 

 assumed to be universally true. And these verifications them- 

 selves are unsatisfactory; for in each case r is put successively 

 equal to 1, 2, 8, 4, and the law which appears to hold is assumed 

 to hold universally. 



Trembley also proposes to demonstrate that the sum of the 

 series is zero, if ti be greater than rx. The demonstration how- 

 ever is of the same unsatisfactory character, and there is this ad- 

 ditional defect. Trembley supposes successively that n = r (a? + 1), 

 7i = r{x + 2), n = 7^ {x+S), and so on. But besides these cases ?i 

 may have any value between rx and r (x + l), or between r {x+1) 

 and r {x+2), and so on. Thus, in fact, Trembley makes a most 

 imperfect examination of the possible cases. 



781. Trembley deduces from his result a formula suitable for 

 approximate numerical calculation, for the case in which n and x 

 are large, and r small ; his formula agrees with one given by La- 

 place in the Hist de V Acad.,.. Paris 1783, as he himself observes. 

 Trembley obtains his formula by repeated use of an approximation 

 which he establishes by ordinary Algebraical expansion, namely 



('-3'=-(-S)- 



Trembley follows Laplace in the numerical example which 

 we have noticed in Art. 455. Trembley moreover finds that in 



