28 
Proceedings of the Royal Society of Edinburgh. [Sess. 
that the equations by which astronomers had, for over a century, been 
accustomed to calculate the motions of the heavenly bodies, do not, in their 
ultimate elements, converge, but diverge. Such a solution is accepted 
pro tanto and made the basis for a more accurate one. The quadratic 
C 
Ax 2 — Bx + C = 0 will have a root belonging to the operand g if the 
operator Dp • Dp . D B 2 acts convergently, or if B' 2 > 4AC. But let us 
suppose that we now introduce another root a, intermediate between the 
two roots of Ax 2 — B^ + C = 0, then this root a will make the quadratic 
operator of the smallest root act divergently, unless it be greater than 
twice this root, and if we introduce two roots a very close to each other, 
then they must be greater than four times the smallest root if the latter is 
to retain convergently-acting operators. None the less, this root belongs 
to the same operand of the new equation, and may be — though less 
rapidly — approximated to by using an operator with ultimately divergent 
action. 
Divergence among elements infinitesimally small and differing in 
sign is an indeterminate case, to be settled only by conducting the 
investigation into the immediate neighbourhood of the suggested root 
and examining whether the new operators persist in their divergent 
action. 
If, however, the operators — or any one of them — be steadily divergent 
in their action, the root obviously must be sought elsewhere. 
IV. Proof that these are Solutions. 
It may be considered advisable to give a more formal demonstration 
of the validity of these solutions than that based merely on continuity 
with the similar solutions of the quadratic, that is, solutions springing 
from operands with integral indices. 
In the first place, then, these solutions may be established by Lagrange’s 
Theorem. Lagrange himself, as is well known, applied his Theorem to 
such forms as x — a-\ -bx 11 , but the scope of the Theorem, 
“=/( z ) + | }'{ + §.. 
m/'r 
2 A etc., 
91 
where u=f(x) and x = z + y<j)(x), is wide enough to embrace every solution 
— belonging to integral-indexed operands — of every algebraic equation by 
merely altering the opening term and the function </>. 
