102 
PROFESSOR DONKIN ON THE 
last article are merely transformations of the two first of Mr. Cayley’s (18.) ; and (iii.), 
though not identical with the third, is of the same degree ; so that the algebraical 
difficulty is precisely the same in both methods.) 
33. If we suppose A = B, the algebraical difficulty disappears, and the solution of 
the problem can be explicitly completed. But on account of the importance and 
interest of this case I shall make it the subject of a separate section, in which it will 
also be shown that the solution of the general case may be made to depend upon it, 
by means of the variation of elements. (See Section III.) 
34. Suppose any complete normal solution of the system of differential equations (I.), 
art. 14, be known, i. e. a solution involving the 2n elements 
^15 ^2J ••• ^15 ^25 
which satisfy the conditions (23.), art. 9; then an infinite number of other sets of 
normal elements can always be found. 
For if we determine the 2 n quantities ... a„,j8 15 ... (3 n , as functions of a 1} &c., b. L , &c. 
by the 2 n equations 
d A 7 d A n 
dH~ 
where A is any arbitrary function of 
^27 ••• a i5 a 25 ••• 
it is obvious that the whole of the reasoning by which the formulae (19.), art. 7, were 
established may be repeated, merely putting A in place of X, and a, [3 instead of x,y. 
And repeating in like manner the reasoning of art. 9, mutatis mutandis, it will follow 
that if f, g represent any two of the 2 n quantities «„ &c., (3 1} See., the expression 
T d(f, 9 ) 
d(bi, (ii) 
will be equal to unity if f, g be a pair of the form a,, (3 j; and will vanish in every other 
case. But it was also shown ((25.) art. 9) that the above expression is equivalent to 
— [/’ g~\ ; it follows then that 
L u j> ft] = ~h [«;> «/] = O ft] = [ft> ft] = ' 0 * ; 
or, in other words, that a,, ... a n , (3 1} ... (3 n , fire a new set of normal elements. 
This method however can hardly be of much use in practice, because we cannot 
(at least without the solution of partial differential equations) determine what form 
* I shall have occasion to refer afterwards to M. Desboves’ Memoir in Liotjville’s Journal, vol. xiii., 
“ Demonstration de deux theoremes de M. Jacobi.” But it may be observed here that the proposition in the 
text is not the same as that expressed by the same notation in the memoir alluded to, p. 400. For M. Des- 
boves uses the symbols [a,-, xj] in a different sense. His theorem, in the notation of the present paper, is 
d(ai, bi) 
= 1, or =0, 
according as /, g are of the form ctj, fa or not, which is easily established without the help of relations analogous 
