47 

 4. If a=/3=y = &c, formula (IX.) becomes 



x ^ L. "J 



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

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



^ ^ = i.^.s.e E 5 - 6 - 7 - 8 - 9 - 10 - 7 - 1 - 2 - 3 - 4 - 5 - 6 ]^ 



203. 



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

 series {0,<r}, {1,°"}» {2, try,..., {<r,<r\, 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 <r ele- 

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

 v at a time, T' will consist of (tr— 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, 

 cr—r}, have been determined, we can thence determine the whole 

 series of terms {w, er} by means of the formula 



{u,<r}=$l[{v,r}.{u-v,<r-r}-] 



(XI.) 



and of this he gives examples. 



7. In formula (XI.) substitute (<r — u) for u; and develope iu, <r} 

 and {c— u, <r} in the manner indicated by that formula. By com- 

 paring the 1st, 2nd, 3rd, &c. terms respectively of {w, <r\ with 

 the last, last but one, last but two, &c. terms of {or— u, cr}, and vice 



