44 
Proceedings of the Royal Society of Edinburgh. [Sess. 
sponding quintic however, which still remains algebraically intractable. 
This of course involves no contradiction with Abel’s results, but is, on 
the contrary, in direct fulfilment of them, for these also are based on the 
private properties of the natural numbers. 
Before giving two or three numerical examples, it may be formally 
stated that from the convergent action of all the operators at work 
follows the convergence to a finite magnitude of the root itself. The 
values we have investigated as the limiting values of the ratio of each 
element to its predecessor in the same ray are also the maximum values 
of that ratio. The ray as a whole, therefore, is less than the G.P. with 
the same initial element as first term and this maximum value for common 
ratio. In this way it may be shown that the whole solution (a), e.g. in 
§ HI, is 
1 
-i , 27 AE 2 
4 C 3 
1 
4 BE X 
1 TB 
and that the first solution of the quartic in the same section is 
< , _ 256 FI? . ^ 27 AE 2 X n 4 BE X V C 7’ 
27 C 4 4 IV 1 C 2 
and so on. 
It may be worth while perhaps to repeat that this condition, though 
sufficient, is not absolutely necessary ; that is to say, a slight divergence, 
or even a more pronounced divergence among elements differing in sign, 
should not deter us following a development for some distance till 
it be thought expedient to decide the matter definitely, and at the 
same time accelerate the solution by shifting to a new origin. Finite 
roots, e.g., are finite because the elements, after a certain point, all 
cancel each other out, or it is possible to re-cast the operators to make 
them do so. Only when the development is flagrantly and ah initio 
divergent must we abandon it as utterly worthless in giving informa- 
tion about the roots. 
§ VII. Solution of Numerical Equations. 
The form in which the operators first give off a solution is fairly well 
exemplified by the root of the equation, x ?> — 49a? 2 + 658^ — 1379 = 0 be- 
1379 
longing to Both the operators Dy 1 . D“b . D 658 2 and D” 1 . Df^ 9 . D 658 
