104 
PROFESSOR DONKIN ON THE 
should have, for determining a 15 &c., either of the systems 
a i — 
( 40 .) 
We might obtain in this way an indefinite variety of sets of elements for the case 
of elliptic motion, beginning with those given at the end of art. 30. But it will be 
better to defer this illustration till after the discussion of the Method of the Variation 
of Elements, which will form the subject of a future Section. 
30. It results, from the investigations of this and the preceding Sections, that if a 
set of n integrals a„ a 2 , ... a n be given, satisfying the — ^ — conditions [a ; , aj\ — 0, 
the determination of n more integrals b x , ... b n , constituting, with the given ones, a 
complete normal set, is a determinate problem , admitting of a unique solution, and 
always reducible (setting aside algebraical difficulties) to quadratures. 
But if, out of a complete normal set, n be given of which one or more pairs are 
conjugate, then the completion of the set is no longer a determinate problem, since 
the remaining n integrals, containing also one or more conjugate pairs, admit, to 
some extent, of arbitrary transpositions and combinations, as is evident from con- 
siderations similar to those employed in arts. 13 and 35. Hence we should expect 
a priori that the problem would require the solution of partial differential equations. 
It appears, indeed, at first sight, that having any n of the elements given functions 
of the variables, the relations established in art. 9, with the others included in the 
formula (21.), art. 9, would furnish more than a sufficient number of equations to 
determine explicitly all the partial differential coefficients of the remaining elements 
in terms of the variables*, at least in the case in which the principle of vis viva sub- 
sists, and the given integrals do not contain t. But it is certain from the above con- 
siderations that this cannot be the case, and therefore that the equations furnished by 
those conditions cannot be all independent. I have not at present attempted to show 
this directly, though it would probably be easy to do so. 
Note on art. 2, Section I. 
The theorem established in this article may be more shortly demonstrated as fol- 
lows : — 
Since d^l^yj = X^dy,) + XiVidxj 
* The conditions [aj, 6 ;] = 1, [<q, fy] = 0, [&,fy]=0, 
[a;, cij]= 0 will give, as is easily seen. 
n(n— 1) 
2 
+ » : 
equations; and the analogous conditions (21.), art. 9, in which the summation refers to the numerators of the 
differential coefficients, will give the same number, so that upon the whole we shall apparently have 3 w 2 — k 
equations, to determine the 2 n‘ partial coefficients required. 
It is not difficult to make mistakes in this subject. I was for some time under the impression that the pro- 
blem could be solved when any n independent integrals were given. Even the illustrious Jacobi himself 
appears to have been misled, at first sight, as to the consequences of Poisson’s theorem (art. 22.). See the 
beginning of M. Bertband’s Memoir mentioned above ; I do not know the fact from any other source. 
