48 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and 
cf)* 
, 1 (-6) 1° 1 174 U 
+ 2 1 129 + 2 ' 4 ‘ 1291 
1 3 (-6)8 1* 1 9 ( - 6) ^ (174) V 1 (174)8 
2 2 2! 1291 2*"' 1 1 129 3 2 2! 
1 2 _ 
129“ 4 
4- 
1 
3.2. 
( - 6) 3 l 1 1 7 5 (- 6)2 (174) II 
3! 
129 4 2 2 2 2! 1 
17^ 
2! 129“ 
+ 1.9.7.H). (17f Ji_ 1 . n . 9 . (174) 3 _li_ 
1 2 2 2 1 2! iv,9¥ + 9 9 , ° 1 o n \ i_5 5 •, etc. 
129? 
9 
2 ‘ 3! (129)2 
or 
129V 
1 
giving 
1 + — + - • -A- , etc. (the other terms are very small) 
2xl29 2 8 129 3 v J \ 
— ? — -f very small terms, 
129 129 
2 = - '02325 ± (1~V( 1-00522). 
For the imaginary roots we write 
« 2 = 
174 6 , 1 4 ■/ 
c — z + — z . or with z = i\ 
129 i 29 129 ’ VI 29/ 
^ = 
174 . 6 /174M , 1 /174V" 
+ i, 1 ' 1 ' ' 
i ./174V1 , 1 6 174° .1 174? 
= ii — I + • — i- • 
129 + 4 129 x 129 / 129 x 129 / J \ 129 ) ' 2 1 129 1 129 ? 
The operators are therefore —iD~+ 6) . Dh ? -11 • At and —Dp 1 . D ^] . D 1S J , 
and 
129/ 
1 6 
hr 
174° 
.1 1 
1741 
129 
L — 
2 1 
129? 
.1 
1 6 2 
174-? 
1 
- • 2 * 
2 
6 1 
1741 , .1 
-{- X — • 
129? 2 
l 2 
174? 
X - 
2 
*2*2! 
129?“ 
1 1 * 
2! 
‘ 129? 
1 
6 3 
^ 0 3l 
174- 1 , .1 
5 3 
6 2 _ 1 
174* 
+1-4-3 
Li 
. 6 
I 2 
129- + 2 
2 2 
2! 1 
129? 
1 
2! 
174? 
lonio 
.1 11 9 l 3 174? , 
l 2 ‘ 2 ’ 2 ’ 3f ' 1 29V 5 ’ 6 C ' 5 6 C ' 
= + i 
129/ 
V 1 174 . 
1 — — • etc. 
2 129' 2 
+ A, etc. = -0232 + if— Y x -99478. 
129 ~ V 1 29/ 
The integral part of the root being soon apparent by this method, we 
can readily find a commensurable root, when there is one. 
When the integral part is found, we may pause a moment to recoil- 
