Again: 



ON INVERSE PROBABILITIES. 63 



1+4 5 1 7 l' + 4 + 9 1 5 



8 8 3 24 27 3 27 



S42 30 1 13 S52 1 





64 or 43 64 3 9(5 125 or 5 3 3 75 



^03 1000 3 3000 



In this way, it appears that the sum of all the 

 squares of numbers is nearly one third of the cube of 

 the last number, and that the greater the number of 

 squares taken, the greater the proximity in question. 

 This proposition is general, namely, that the sum of the 

 nth powers of numbers is nearly the (n -f l)th part 

 of the (n + l)th power of the last of the numbers : 

 thus, the sum of all the 13th powers I 13 + 2 13 -f 

 up to 1000 13 , is very nearly the 14th part of 1000 14 , 

 This proposition, never absolutely true, may be made as 

 near the truth as we please, by taking the number of 

 terms sufficiently great; and the error made by the 

 substitution, is nearly such a fraction of the whole as 

 has one more than the index of the power for its nu- 

 merator, and twice the number stopped at for its 

 denominator. Thus, if the tenth powers of all num- 

 bers were summed up to 1 0,000 10 , the substitute for 

 this sum given by the theorem, namely, -fa of 10,000 1] , 

 would be too small by about 



10+1 or ll of the whole. 



2 x 10,000 20,000 



PROBLEM. A lottery contains 10,000 balls, each of 

 which may be white or black. A ball is drawn and 

 then replaced, and 100 such drawings give nothing but 

 white balls : what is the chance that the five next 

 drawings shall all be white ? 



S 10,000*00+ s 



This chance, by what precedes, is 



10,0005 S 10,CKX)ioo 



But S 10 3 OOOioo+5 = J_ x 1 o,000i <5 very nearly. 

 106 



