196 



An abi'idged mode of finding divisors may be more simply 

 Jeduced from the general expression, in the same manner as 

 Gravcsand has done in his example for the cubical divisor, 

 ior, the following being the form of the divisor, viz. 



fl n — 1 T-v"— ' n TTJ A "~- 



AxD+A xD a + b + c+&.c. + A 'xD ' xnb + ac + Scc. 

 + &C... + A X Dflic&c.+«cf/&c.+&c.+flftcf/e &c. If we there- 

 fore begin the operation by subducting the sum of the first 

 and last members, or A" x D" + abcde Scv. (A" x D" being found 

 iis before, and abccle Sec. being found opposite to cypiier) 

 and, if we then divide the remainder by A, we shall thus have, 

 depressed the indices of the powers of the substituted num- 

 bers by 2, and therefore the differences to be taken will be 

 iewer. Hence, in the case of a divisor of three dimensions, 

 or a divisor of four dimensions, whose second term is wanting, 

 the quotes or depressed remainders are in arithmetical series, 

 and thus, two co-efRcients are discovered together, the second 

 co-efficient being the common difference of the arithmetical 

 series, and the penultimate being the term of that series 

 which should correspond to cypher. This term, however, is 

 not immediately discoverable opposite to cypher, and al- 

 though in the cubical divisor it is always to be known, being 

 the basis of the arithmetical series, yet the term is not really 

 expressed, (for the divisor abcde 8cc. being subducted from 

 itself would leave cypher, and as this corresponds to the sub- 

 stitution of cypher, the division of the remainder by the 



substituted 



I 



