146 A Demonstration of one of the 



DEMONSTRATION. 

 First, it is evident that if all the permutations of any number of letters 

 expreffing figures be put down ; and thofe in the firil place to the right 

 hand be multiplied by unity ; thofe in the fecond place by ten ; thofe in the 

 third place by 100, and fo on; then the Sum of all thefe, will be the Sum 

 of the permutations required. 



Secondly; fuppofing the different permutations to be put down one 

 under another, it w 11 reaiy ap pear, from the manner in which permuta- 

 tions are generated, that all the letters occur an equal number of times in 

 each perpendicular column ; and alfo that the number of times of 

 occurrence in the permutations of n letters, is equal to the permutations of 

 n— -i letters ; but the permutations of n — 1 letters is equal to 1.2.3... ( n — 1 )» 

 or 1x2x3 carried to n — 1 terms ; and confequently if there be n letters 

 in the given number, each letter in the Columns aforefaid will occur 

 1.2.3... (n — tmies » 



Thirdly j Let 1.2.3.. (n — i)=mtheii, 



m (a + b + c-f ...n) i=Sum of numbers in the units place or firft Column^ 



m (a + b + c + ...n) 10= Sum of numbers in the tens or fecond Column. 



m (a + b + c + ...n) 100= do. third Column. 



m (a + b + c + ...n) ioo...to (n — 1) Cyphers = ditto in then Column; and 

 the Sum of thefe is evidently equal to m(a + b + c-f-...n).{l + 10+ 100+.. .to 

 n terms); and putting for (1 -1-10+ 1 00 + ..n) its value 111...11, the exprefTion 

 becomes (i.2-3..(n-i) ) x (a + b + c+..n) x (1 11. ..n) ; but i.2.3...(n— 1) 

 is equal to l -"~i- n and therefore the expreffion for the Sum of all the 

 permutafidiis is [ z j±±±J x(a + b + c + ...n)x(m...n), which is the Hindoo - 

 rule when the figures of the given number are all unlike* 



