302 PHILOSOPHICAL TRANSACTIONS. [anNO 1717. 



1st, rr-, when x -{- xv -\- -— &c. = 0. 



2* 



2d, — - — rr-, when — aj -|- .fv 4- ^ &c. = O. 



' XX 2 



i + 



2x 



3d, — - — r-, when x — xv •\- ~ &c, = 0. 



XX ' 2 



4th, ^^^ —. when — jt — it; + ^ &c. = O. 



' XX 2 



* +• 



Ux 



This formula will also triple the number of true figures in z. And the cal- 

 culation may be repeated, after every operation, taking for a divisor 



i* + J V + T-T^ + TTnn + &c. instead of x -\- %. 



— 2 ' 1.2.3 ' 1.2.3.4 ' ' 2x 



Dr. Halley has fully explained the manner of using both these formulas in 

 equations of the common form ; wherefore I shall be the shorter in explaining 

 two or three examples of another sort. 



Ex. 1. Let it be proposed to find the root of this equation y^ -|- i]'/* -|- y 



— l6 = O. In this case, for y writing z, and for O writing x, we have 

 z2 _^ j|>/* ^2 — i6 = 37. Whence, by taking the fluxions, we have 



jh = 2 /2 X z X z^ + \\^^~^ + ^y and J? = 2\/2 X 8 — 4v/2 z^ X 

 z2 _|_ j|V2— 2^ Pq^ findin g the fi rst figures of the root y, for ^2 take -§-, 

 and we have the equation y^ -\- l\^ + J/ — l6 = O. which being expanded 

 gives / + 3/ + 2/ + 32z/ — 255 =0. 



By this equation I find that for the first supposition we may take z = 2. 

 Therefore in order to find v, let us now make \/2 = 4-, (which is nearer 

 than before) and we have a? = z* + l]"^ + z — l6 = 2^ -f- l\^ — 14 = 5t 



— 14 = —• 4,48; X = 10,66; x = 4,72. Whence bv the second rational 



form V = 



4,448 



4,72 X 4,48 

 10,66 + 



2 X 10,66 



= 0,38; which must be too large, because 4- < \/2, and therefore will require 

 a larger value of j/ to exhaust the equation, than where -/ 2 is exact. For 

 the second supposition therefore, let us take z = 2,3, and make \^2 

 = 1,4142136, and by help of the logarithms we shall have z^~+~l]'^ 

 = 13,47294, whence a? = — 0,22706; i = 14,9342Q, ahd x = 5^1 841 9. 



