Cambridge Philosophical Society. 



305 



4. If «=/5=y=&c., formula (IX.) becomes 



{"■•■} =-i;4ip s<i;)[(-i)''^C»,-K]-"]. (X.) 



Example of formula (X.) Given seven kinds of elements, and 

 three of each kind ; and let m=4. Then 



^"'''>'" l.2.3^4.5.6 [^-^-^-^-^-^Q~^-^-^-^-^-^-^]= 



203. 



5. If it is required to determine many, or all, of the terms of the 

 series {0,<r}, (l.o"}. {2, <r},..., {(T, cr}, formulas (VIII.) sug- 

 gest the following process for the determination of those terms. 

 An example will best explain the process. 



Given 1 element of one kind, 2 elements of a second kind, and 3 

 of a third kind. How many combinations can be formed from these 

 elements, when taken 0, 1 , 2, 3, 4, 5, 6 at a time, respectively ? 



6. Let a set of elements, S, such as we have been previously con- 

 sidering, consist of two similar sets, T and T', which do not contain 

 in common any elements of the same kind. If S consists of tr ele- 

 ments combining m at a time, and T consists of r elements combining 

 V at a time, T' will consist of {v—r) elements combining (u — v) at 

 a time. Consider u as constant, for the moment, and v as variable. 

 The author then shows that if by the process described in art. 5, the 

 whole series of terms {u, t'} and the whole series of terms iu—v, 

 c — rj, have been determined, we can thence determine the whole 

 series of terms {m, c} by means of the formula 



{«,cr}=S^[{t;,r}.{M-v,(r-T}]; . . (XI.) 



and of this he gives examples. 



7. In formula (XI.) substitute (cr— «) for u ; and develope iu, <t\ 

 and /<r — u,<t\ in the manner indicated by that formula. By com- 

 paring the 1st, 2nd, 3rd, &c. terms respectively of [u, cr} with 

 the last, last but one, last but two, &c. terms of icr—u, <r\, and vice 



Phil. Mag. S. 3. Vol. 3 J . No. 208. Oct. 1 84.?. X 



