478 



Mr. WARBUETON, ON THE PARTITION OF NUMBERS, 



from which expression, however, every factorial expression is to be excluded which has a negative 

 quantity for its first factor. 



4thly. By operating on formula (xiii), with y^ in a similar manner, and on the result of that 

 operation with 5^, and so on in succession, until there remain no factors undisposed of, we shall 

 obtain for the coefficient of the developement of [l - a;]"' x [l - .r"] [l - aP'\, &c. the following 

 expression, subject to the same rule as before, of omitting every factorial which has a negative 

 quantity for its first factor : 



I"?//-'!' -[m -aj'-i|'+ [,«^-a, -/3J-'|' -&c.- 



- K - 7,r t + [«. - /3, - xr T - &<=• 



- &c. + &c. 



1,-1 1 . ■» 

 .5. Now if a, fi, 7, &c., are all equal, that is to say, if the required coefficient is that of ,r" 



(XIV). 



in the developement of I J*, formula (xiv) will become 



<"T -«[w -«,]■'"' I' + 4TrK-2a,]'"'|'-&.c. 



(XV); 



+ (-0'r^-K-e«j-T+8^ 



&C.1 



where, for any determinate value of u, the maximum of Q is the integer nearest to, and not 



exceeding — ; but if u attain its maximum (which is sa), then the maximum of is the 



a+l ' 



«a + 1 



integer nearest to, and not exceeding . 



a + l 



Example of formula (xiv). 



How many different combinations can be formed by taking 2, or 8, at a time, of the 10 elements, 

 of 4. different kinds, 



A, BB, CCC, DDDDi 



Answer, for u = Z; fS . 4 . 5 - 1 . 2 . sl = Q. 



1.2.3'- -■ 



Answer for m = 8. 



f9.10.11- 7.8.9 + 4.5.6, 



-6.7.8 + 3.4.5 



-5.6.7 + 2.3.4, = (). 



-4.5.6+2.3.4 



+1.2.3 



Example of formula (xv). 



How many different combinations can be formed by taking 2, or 8, at a time, of the 10 elements, 

 belonging to 5 different kinds, 



A A, BB, CC, DD, EE? 



1.2.3 



