Theorem of Cauchy on Arrangements. 379 



instead of 1 we write p"+P+ • • -+ x in the numerator of the quan- 

 tity under the sign of summation (p being any quantity what- 

 ever), the sum becomes expressible as a known function of p. 

 Nothing can be easier than the proof. 



Let the a, /3, 7, ... X in the preceding statement be supposed 

 subject to the further condition that their sum is r; then for any 

 assigned value of r (a positive integer) it is easy to see that the 

 sum of the terms within the sign of summation in Cauchy's 

 theorem is 



s ( 1 . p r Y 



\a?! x z . . . x r !!(/•)/' 



where x v x 2) . . . x r mean every system of values of x Xi %%,... x r 

 (permutations admitted) which satisfy the equation 



x 1 + x 2 + ... -f-a? r =ra. 



(It should here be observed that a, (3, 7, . . .X; a, b, c, . . .1 are 

 the systems which satisfy aa + fib + yc+. . . +Xl=n, permuta- 

 tions being excluded; that is to say, if, for example, «, /3, 7 

 should happen to be equal for any partition of n, the values 

 a, a; ct,b; «, c would figure only once, and not six times, among 

 the systems included under the sign of 2.) Hence then we see 

 that 



Ux.a* n/3.bP...Il\.l x *=" U(r) 



(t t q t? v 



y + 2 + o- + & c - adinfin. J, 



i. e. in (logK — -J) ; and the total sum designated by 2 will 

 be consequently the coefficient of t n in 



I°s(ra>+(l°ST^) 2 + &c '' 

 G=i),«V e .in( T i^)'. 



P log 



* For if we take a system of values satisfying the above equation, con- 

 sisting of cc equal values a, (3 equal values b ... X equal values I, such a 

 ...... 1 7r(r) 



system will give rise in 2 to -, — ; ; a \ TTS repetitions of 



J ft x x a? 2 . . . x r {nee) (irp) . . . {n\) l 



the term — 7c, and consequently in 2—— to a total 



value - — ; — -. , condensed into a single term in Cauchy's 



theorem. 



2C2 



