42 Scientific Intelligence. (Jury, 
V. Further Observations on Mr. Lockhart’s Imaginary Cube 
Roots. 
(To Dr, Thomson.) 
SIR, 
Mr. Lockhart seems to have made a mistake in one of the signs 
_of the root connected with ¢: when corrected, his roots will stand 
thus: 
x = b x a b 
or 7 eae es A)? Tees 
‘hed b ne, ? 6 
esi, NY tes a -7tV/4-4 
b 
But, as Mathematicus justly observes, any quantity “ in which the 
square root enters admits of two values;” the above roots may be 
more conveniently expressed thus— 
\/ = i fee, eae 
2 4 27 
x =? b 
ae Bee yh 
es 4 3 
i e b 
ri + AA 4S 
Th 
» fv? b 
"2 Mab 
In which formula, if the upper sign be used in the first, the upper 
sign must likewise be used in each of the roots, and the contrary. 
It may be proper, before I make any further observations, to point 
out the part of Mr. L.’s demonstration where the error appears to 
at bt 
have originated. In extracting the square root of © a ne 
_ ae he has overlooked the ambiguity of ie quantity 
basi (ie — =) 5 for — os f-fs= 
Ce eS 5 (os (ta 
/t-t —— > But as the quantity originally squared was + a ef; 
~~ 
oe ee a 
