260 A GEOMETRICAL CONSTRUCTION RELATING TO IMAGINARY QUANTITIES. [756 
where 
A = P + m 4 + n i — 2m 2 n 2 — 2nH 2 — 2 Pm?. 
It is to be observed that A, = (P — m? — rPf — 4mV, is negative ; hence, calling 
the factors fA + gB + hC, f'A + g'B + h'C respectively, the coefficients f g, h, and 
/', g', li are imaginary; moreover f+ g + h = 0, f' + g' + ti = 0. 
The values of X thus are 
(l + m + n) X = IA + mB + nC ± *J{(fA +gB+ hC) (f'A + g'B + h'C)}, 
and then passing to the geometrical representation, we have 
1/A "I - 7YlB -|- TlC 
represented 
1 + m+n 
by the point which is the C.G. of weights l, m, n at the points A, B, G respectively ; 
on account of the imaginary values of the coefficients the construction is not immedi 
ately applicable to the factors 
fA+gB + hC, f'A + g'B + h'C ; 
but a construction, such as was used for the factors 
A + ft)B + oPG, A + o) 2 B + to (7, 
might be found without difficulty.