4 7 
1916-17.] Operators applied to Solution of Equations. 
The equation 7V + 20;r 3 + 3V — 16# — 8 ==0 has, as already stated, three 
roots belonging to 
and a negative one beginning 
20 
7‘ 
The conver- 
gency is very slow, however, so that, though it be possible to locate the 
roots from the formula belonging to the equation as it stands, we may use 
it to exemplify the transformations that are often necessary in calculating 
roots belonging to operands with fractional indices. In the first place, it is 
always an improvement to have the opening coefficient 1. This coefficient 
becomes at least once, and perhaps twice, a denominator — a denominator, 
however, that soon becomes a numerator. In this case, then, multiply the 
roots by 7. Thus we have, with y = 7x, 
i/+ 20// 3 + 21// 2 - 7S4</ - 2744 = 0. 
The next transformation must aim at making the absolute term and one 
other coefficient in the centre predominate over the others, and, if it be 
possible, at removing a term, and with it an operator. In the present case, 
if we increase the roots by 5, we have, with z = y + b, 
- 129z 2 + 6z — 174 = 0, 
/129V 
and now we shall have two real roots from ±( as °P eran d, and two 
imaginary ones from zb as operand. 
For the development of the former we have 
2 2 = 
129 6 1 174 1 
1 1 
1 z 2 ’ 
giving 
and taking first 
2= ± 
?)* • 
129 6 / 1 
T I 1 129 
+ 
174 1 
1 129. 
or 
129y , 1 (-6) 1° | 1 .174 1* 
1 ) + 2 1 
129 2 1 1294 
A 5 
giving operators D,^, . D r J . Dj , D„J . D, 5 . Dj 
