Page:Philosophical Transactions - Volume 014.djvu/187

Rh the rest be en creased or diminished, either Arithmetically, or by multiplication and diyiiiou in a known Rana? certainly regular Progreilions will arise, though as yet, we cannot cncrcaie create the true roots of an IEquation without as much di miuilhing the Negative. nor can we multiply or divide the roots without we alter all of them. and consequently cannot reduce ooeilicients to such habitues as are' delirabllr. 14. It is a pleasant concmnity out ot a root to raise a Reiblvend iblvend without railing any of the Powers of the root, and at the same time without a thorough binomial Division to depress the Equation a degree lower. EXAMPLE Let the Equation be a 1* to a 'l'6a 'l' zoa;: -1072.

Let the root be 4, the resolvent is thus raised by adding the coefficients as you go, and multiplying by the root, thus -I-45' I 0', § .1fQ.I1[4, X4 "7"i'§ 6.§ .6. ', § .~62X4- »»2f§ Xl' 2o“'~;.(>8;»{4, 1',7v,, with the same work the Equation may be depressed without Division. EXAMPLE Let the Equation be as before, and place the root with the former products underneath respectively, the summ is the depressed Equation. ai. + lou; J, gal” zca- ~I'~ 71.7170 T4. 'l'5'6'l'24f8 l'1C77.

The sum 8.4-'l'I4 h 'ii 6 z d+z6 S:i::;o. that is divided by a. 14:1 1' 62 a1'z6 5321* o. which is the under ZEquation fought found without Division

25 It's conceived that al1 Equations maybe so regulated as robe reduced tons mam/Arithmetical Progrellion sol multi pliers in whole numbers, asthelF uation hath dimensions, whereof one of the progressions shall be a Seriesot'roots= hence the raifiiiglleiolveiitlsby tentative work is rendred Logarithmetical mb e c, .., N

J:'orExam plc write down any; arithmetic all'rogrefiioi1s, 'w%-R R lr

1X6X3

2X7Xf

3X8X7' fl§

fliiifyo Fhmigenm Rc'omp¢wmw1g golces of the Rank R. for out of the Rank R in any Æquation proposed raise separately the respective powers (with regard to their Cocllicients) and out of the three ~ ~ "'i t n

ag  ranks so raised compose their rcipective 1 erenccs, fl 21.3 they shall be the same with the differences of the rank 11 5 of Resolvends or Homogenea Campamrivnw here noted by H. If