LAMBERT. S37 



the above identity. This last remark is in fact the novelty of 

 Lambert's memoir. 



Lambert gives the general law which holds among the quan- 

 tities -4j, A^, ... , namely 



A,= rA,_^+(-iy (2). 



He does not however demonstrate that this law holds. "We 

 have demonstrated it implicitly in the value which we have found 

 for (f) {n) in Art. 161. 



We get by this law 



A, = 9, X=44, ^, = 265, ^7=1854, A= 14833, ... 



We can however easily demonstrate the law independently of 

 Art. 161. 



T \V — 1 ) 



Let A** I stand for \r — r r — \ -\ ^^^ — ^ 



r-2- 



so that the notation is analogous to that which is commonly used 

 in Finite Differences. Then the fundamental relation (1) sug- 

 gests that 



^. = A-[0; (3), 



and we can shew that this is the case by an inductive proof For 

 we find by trial that 



A"Lo = Lo = i = A> 



A^ [0 = 1 -1=0 = ^,, 

 A^[0= 2 -2 + 1=^,; 



and then from the fundamental relation (1) it follows that if 

 -4^ = A** [0 for all values of r up to w — 1 inclusive, then A^ = A" [0. 

 Thus (3) is established, and from (3) we can immediately shew 

 that (2) holds. 



628. We now come to another memoir by the writer whom we 

 have noticed in Art. 597. The memoir is entitled Sur le Calcul 

 des Prohahilites, par Mr. Mallet, Prof. d'Astronomie a Geneve. 



This memoir is published m the Acta Helvetica ... Basilece, 

 Vol. VII. ; the date of publication is 1772 : the memoir occupies 

 pages 133—163. 



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