820 LAGRANGE. 



with ease and interest, and at the time of pubhcation its value 

 must have been great. The promise held out in the introduction 

 that something would be added to the labours of Laplace is 

 abundantly fulfilled. The solution of the general problem of the 

 Duration of Play is conspicuously superior to that which Laplace 

 had given, and in fact Laplace embodied some of it subsequently 

 in his own work. The important pages 231 — 233 of the Theorie 

 . . . des Proh. are substantially due to this memoir of Lagrange's. 



589. We may notice a memoir by Lagrange entitled Me- 

 moire sur une question concernant les annuiies. 



This memoir is published in the volume of the Memoires de 

 V Acad. ... Berlin for 1792 and 1793; the date of publication is 

 1798 ; the memoir occupies pages 235 — 246. 



The memoir had been read to the Academy ten years before. 



590. The question discussed is the following: A father wishes 

 to pay a certain sum annually during the joint continuance of his 

 own life and the minority of all his children, so as to ensure an 

 annuity to his children after his death to last until all have attained 

 their majority. 



Lagrange denotes by A, B, G, ... the value of an annuity of 

 one crown for the minority of the children A, B, G ... respectively. 

 Then by AB he denotes the value of an annuity of one crown 

 for the joint minority of two children A and B ; and so on. Hence 

 he obtains for the value of an annuity payable as long as either 

 ^ or ^ is a minor, 



~A + B- AB. 



Lagrange demonstrates this ; but the notation renders it almost 

 obviously self evident. 



Similarly the value of an annuity payable as long as one of 

 three children A, B, G remains a minor is 



A + B + C - AB - AG - BG + ABG. 



De Moivre however had given this result in his Treatise of 

 Annuities on Lives, and had used the same notation for an annuity 

 on joint lives. 



Lagrange adds two tables which he calculated from his 

 formulae, using the table of mortality given in the work of 

 Sussmilch. 



