IN THE HIGHER ALGEBRA. 139 
u=x(4,, O,, O35, O4, 95; 4); 
we find 
—— ft ———————————— 
ay ewe 
and, hence, conclude that 
is a constituent of y. If we were conducted to unsym- 
metric or, by the assumption w=F(@), to general functions 
of 6, it would become necessary to inquire into the solvi- 
bility of the equation in @. 
§ 24. 
A third mode, also involving that inquiry, would be to 
seek the values of uv in terms of those of @ by elimination. 
And here, as it would not be difficult to show, Mr. Harley’s 
investigations, as yet unpublished, substantially change 
the data into four quadratics, involving four unknowns, 
but with certain relations existing among the coefficients. 
§ 25. 
Writing for a moment 6(u) in place of 9, the transition 
from 6(u) to 6{R(u)}, where R(w) is a rational and general 
or unique function of w—i.e. a rational function of any 
and only one value of ~—though perhaps interesting, 
does not seem to demand our attention at present. 
§ 26. 
A finite algebraic solution of the general equation of 
the fifth degree may be impossible or unattainable. If 
the hypothesis of its existence lead to contradictory or 
inconsistent conditions, or to a self-contradictory result, 
such a solution is impossible or unconditionally unattain- 
able. If the hypothesis conduct to the conclusion that 
every known process for solution leads to an illusory re- 
sult, e.g. to vanishing fractions unassignable in value, or 
VOL. XV. u 
