• . . (XVIII.) 



SIO Cambridge Philosophical Society. 



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

 to the cases wherein PJ " , or P|m, o-} 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 m=3. 



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

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

 tions ; viz. 



P{«,^}=S ;[^^'p(.,r}. ?{«-„, cr-r}]. (XIX.) 



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

 art. 7, Sect. II., he shows that P{(r-1, o'}=P{(r, tr} ; that is to 

 say, that 



l«l'.l/3|l.iy|l. &c. 



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

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

 when (T— I elements, as when c 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 the composition of a polynome function of 5 

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

 ApE'iO', and of every dimension from to m inclusive. Let [*]" 

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

 number of its terms. Then, 1st, 



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

 either A*, or B^, or C, &c., he expresses by 



N W-N[«]«-« -f-N[s]«-*-/^- &c. 

 -NM«-^+ &c. 

 — &c. 



