WARING. 447 



is there is no power of r higher than this power, and the coefficient 

 of this power is unity. Waring refers to another work by himself 

 for the demonstration ; the student will see that it may be deduced 

 from the elementary theorem in Finite Differences respecting the 

 value of A"a;*^, when n is not less than m. 



Waring does not apply his Lemma until he comes to the 

 part of the work which relates to Annuities, which forms his 

 pages 27 — 59. 



831. Waring now proceeds to his propositions in the Theory 

 of Probabilities ; one of his examples will suffice to indicate his 

 method. 



It is identically true that ^ — ^^^ — = — — -^ . Suppose -^ 



to represent the chance of the happening of an assigned event in 



N — a 

 one trial, and therefore — :^ — the chance of its failing : then the 



identity shews that the chance of the happening of the event in 

 the first trial and its failing in the second trial is equal to the dif- 

 ference between the chance of the happening of the event once 

 and the chance of its happening twice in succession. 



882. There is nothing of any importance in the work respect- 

 ing the Theory of Probability until we come to page 19. Here 

 Waring says, 



Let the chances of the events A and B happening be respectively 



and J ; then the chance of the event A happening r times 



a + h a + b 



more than B in r trials will be 



in r -f- 2 trials will be 



or 



a' 



{a + by ' 



■l+..-^li 



in r + 4 trials will be 



or 



{a + by\ {a+bf 



ah r (r + 3) a%- 



{a-vby\ (« + 6/ 2 (a + 6) 



and in general it will be 



