450 WARING. 



In the second example we observe that the event may occur m 

 the first % + 1 trials, and the chance of this is P ; or the event may 

 have failed in the first n + 1 trials and yet may occur if we proceed 

 to « + 1 more trials. This second case may occur in the following 

 ways : B may hajDpen twice in the first n+1 trials, or twice in 

 the second w + 1 trials ; while A happens in the remaining 2)i 

 trials. Thus we obtain 



2 {n + 1) n ^'V 



2n+2 ) 



2 [a^hf 



which must be added to P to give the chance in the second ex- 

 ami^le. 



In the third example we observe that the event may occur in 

 the first 2n + 2 trials, and the chance of this is Q ; or the event 

 may have failed in the first 2n-\-2 trials, and yet may occur if we 

 proceed to w + 1 more trials. This second case may occur in the 

 following ways : 



Jj may happen three times in the fii^st n + 1 trials, or three 

 times in the second n+1 trials, or three times in the last n + 1 

 trials ; while A happens in the remaining S?i trials. 



Or B may happen twice in the first n + 1 trials and once in the 

 second n + 1 trials, or once in the second n + 1 trials and twice in 

 the third n + 1 trials ; while A happens in the remaining 3/^ trials. 



Thus we obtain 



:^ {n + l)nin-l) ^ ^ {n + 1)' nl a'^'P 



371+3 ) 



[3 2 ) {a + by 



wliich must be added to Q to give the chance in the third ex- 

 ample. 



834. * The following specimen may be given of Waring's imper- 

 fect enunciations ; see his page 21 : 



Let a, h, c, d, &c. be the respective chances of the happening of 

 a, /5, y, 8, &c. : in one trial, and 



(ax'^ + hx^ + cxy + doc^ + &c.)" = a^x'"^ + . . . + Nx^^ + &c.; 



then will iV be the chance of the happening of tt in ti trials. 



Nothing is said as to what ir means. The student will see that 

 the only meaning which can be given to the enunciation is to 



