52 



P{«,o-}=D«ri«|i[i+*+|J] = l 



)r J l 2112 — J' j 



(XVIII.) 



This is the only addition which the author has been able to make 

 to the cases wherein P , or P-f w, crj. is expressed by an explicit 



function of u, symmetrical in form. 



Example. Let there be five kinds of elements, and two of each 

 kind. Let w = 3. 



Dr , 1.5.4.3 3.2x5.4 lor> 

 P{ M ,<r} = _ + = 120. 



5. The author gives the following theorem, which is precisely 

 analogous to that of art. 6, Sect. II., formula (XL), in Combina- 

 tions ; viz. 



F ^'"> = ?o[Sr lp ^ r >- P ^- u ' cr ~ T 0' (XIX-) 



6. By a mode of proof precisely analogous to that employed in 

 art. 7, Sect. II., he shows that P{<r — 1, «r}=P{(r, <r} ; that is to 

 say, that 



i«|i.l*|i..irl l . &c. 



denotes the number of permutations that can be formed with a ele- 

 ments A, /3 elements B, &c. (where [a+/3 + y+ &c.] = <r), as well 

 when <r— 1 elements, as when er elements, are taken at a time. 



Since correcting his paper for publication, the author has had his 

 attention called to the work of Bezout on Elimination (4to. Paris, 

 1779, p. 469), as containing a formula similar in structure to that 

 numbered VIII*. in the present abstract. 



Bezout investigates rlie composition of a polynome function of * 

 quantities, A, B, C, &c, consisting of terms which are of the form 

 AP B? 0\ and of every dimension from to u inclusive. Let [s]" 

 denote such a polynome, complete in all its terms, and N[s] M the 

 number of its terms. Then, 1st, 



N W «= &+W 1 ; 



and 2nd, the number of the terms in [_s] u which are not divisible by 

 either A a , or B' 3 , or C^, &c, he expresses by 



N[5] M -N[s] M - a +N[s] M - a -^- &c. 

 -N[s] M ~^+ &c. 

 — &c. 



