197 



substituted quantity, or of cypher by cypher, will give any 

 finite quotient, and the number being indeterminate is un- 

 derstood till it is discovered by continuing the law of the 

 series.) But in finding divisors of higher dimensions where 

 those depressed remainders or quotes are not in arithmetical 

 seriesj and where, in order to obtain arithmetical series, the 

 differences of the quotes are to be taken, no such quotes can; 

 be understood, and therefore the substitution cannot then be 

 continued beyond cypher. But if we substitute the terms of 

 the natural series, from the index of the polynome to unit, 

 when the arithmetical series shall be obtained from the higher 

 order of differences of the quotes or depressed remainders, 

 the term opposite to cypher is found by continuing the law 

 of the differences, and thence the law of the series for one step 

 farther. From those principles, as in the following rule, we 

 discover two co-efficients of the divisor at once. 



Substitute for the unknown letter in the proposed the terms 

 of the natural series, descending from the index of the divisor 

 required to unity ; adding the divisors opposite to cypher to- 



th 



the divisors of the highest term, multiplied into the n powers^ 

 of the natural numbers, let the sums be respectively taken 

 from the divisors of the results of substitution of such corres- 

 ponding natural numbers ; divide the remainders by those 

 corresponding substituted numbers, and if the difference of 

 the quotes be common, Avhen the number of orders is n — 2, 

 the common difference divided by 1.2.3. ..n — '2 is the numeral; 

 ^o-efficient of the second term of the divisor of n dimensions ;. 



2 D 2 leC 



