equations of all orders, by continuous approximation. 321 
that the simultaneous value of the second derivee will be 
affirmative. But as the principal result has evidently con- 
verged and subsequently diverged again in this interval, no 
conclusion relative to the simultaneous sign of that result 
can be immediately drawn. We will return to complete the 
transformations. 
For x= 1.0 
1.1 
1.2 
1 -3 
1.4 
1-3 
1.6 
1.7 
1000 
— 400 
30 
1 
631 
—337 
33 
1 
328 
— 268 
3 6 
3 
97 
— 56 
—125 
—104 
*3 
Here the first column was formed from that under #=1, 
by annexing ciphers according to the dimensions of the 
functions ; the 2nd and 3rd columns and the number 97 were 
found as in the former Example ; the remaining numbers by 
differencing and extending the series 1000,631,328,97. We 
have no need to continue the work, since the changes of signs 
in the principal results indicate the first digits of the roots in 
question to be 1.3 and 1.6. But if we proceed by farther 
differencing to complete all the lines, the columns standing 
under these numbers will give the co-efficients of 0(1.3 4- z) 
and cp (1.6+ z) without farther trouble. 
23. Assuming, then, that R has been determined, and 
R -f- z substituted for x in the proposed equation, thereby 
transforming it to 
A = az bz? -{• cz 3 dz 4 -f- 
it is to this latter equation that the analytical part of our 
theorem is more immediately adapted. Now the slightest 
degree of reflection will evince, that our method is absolutely 
identical for all equations of the same order, whether they 
mdcccxix. T t 
