1 00 Mr . J. R. Christie on the Extension gj Budan's Criterion, fyc. 



-2-1 2-1 l ! 



* ni ~ 2+4- * m " * + -J- i ^"-T + l 1 

 2 1+ 2- 2+ Y : 



whence x — 3'4, .r = 3*2 14, a: = 3*227. 



As a still more difficult example of the separation of roots 

 very nearly equal to one another, let us take the equation 



-^—82 ^ + 2404 a 3 — 26394 w 2 + 61 32^ — 360 = 0, 



and it will be found that the following transformations will be 

 obtained, viz. 



360 TOviii 5 + 8268 TOvni 4 + 60570 Wvm 3 + 1 1 9564 Wvm 2 



— 156270 WVHI + 40335 = 0... w = — , 



v 



40335 Xu 6 4- 247080 x„ 4 + 482804 x n 3 + 254274 x^ 



—88524tfii + 6088 = x = 4—,', 



TOvm 



60883/vn 6 + 124556 ,vvn 4 + 758712 3/vn 3 + 678342j/vn 2 

 — 3983874^11 + 2418165 = Q...y — — , 



Xu 



241 8 165 z x 6 + 8106951 V + 8924496 V + 30721 44 z? 



-167665*, + 1989 = z = — , 



1989 * 5 — 167665 t 4 + 3072144 1 3 + 8924496 t* + 8106951 1 



+ 2418165 = t= — . 



Now it will be found that one root of this equation is between 

 30 and 31, the other between 50 and 51, so that we have the 

 continued fractions, 



= — 1 = 2_ 



30+- + S0+2 



which give the values 



v — -118057 and Vm -11805649: 

 the actual approximate roots to twelve places are 



•118056983866 and '118056440257. 

 The number of reciprocal transformations necessary to effect 

 the separations of the roots, will of course primarily depend 

 upon the number of figures in them which are identical; but 

 there are certain points in the scale between and 1, from 

 which, if the roots differ, in however small a degree, the one 



