ON THEORETICAL DYNAMICS. 81 
aay consequently the expression (a, ») will 
dp, dp,’ 
not be identically constant unless 
seen that (a, n)=(a, B,) 
is so: but the integrals, in number 
, 1 
infinite, which result from the combination of a, with B,, 63... Py,_, com- 
bined with the integral a, give identical results. Only the integrals which 
contain G3, lead to results which are not identical. The integrals a and (3, 
connected together in the above special manner, are termed by the author 
conjugate integrals. 
61. Brioschi’s two notes of 1853.—The memoir ‘ Sulla Variazione,’ &c. 
contains reflections and developments in relation to Bertrand’s method of 
integration and to canonical systems of integrals, but I do not perceive that 
any new results are obtained. 
The note, ‘ Intorno ad un Teorema di Meccanica,’ contains a demonstration 
of the theorem in Bertrand’s note of 1852 in the ‘ Cemptes Rendus,’ and an 
extension of the theorem to the case of a combination of any even number 
of the arbitrary constants; the value of the symbol is shown by the theory 
of determinants to be a function of the Poissonian coefficients (a, 3), and as 
these are constants, the value of the symbol considered is also constant. 
62. Liouville’s note of the 29th of June, 1853*, contains the enunciation 
of a theorem which completes the investigations contained in Poisson’s 
memoir of 1837. The equations considered are the Hamiltonian equations 
in their most general form, viz., H is any function whatever of ¢ and the 
other variables: it is assumed that half of the integrals are known, and that 
the given integrals are such that for any two of them a, (3, Poisson’s coeffi- 
cient (a, 3) is equal to zero; this being so, the expression pdq+ ... —Hde, 
where, by means of the known integrals, the variables p,.. are expressed in 
terms of g,...é, is a complete differential in respect to g,...é, viz. it will be 
the differential of Sir W. R. Hamilton’s principal function V, which is thus 
determined by means of the known integrals, and the remaining integrals are 
then given at once by the general theory. 
63. Professor Donkin’s memoir of 1854 and 1855, Part I. (sections 1, 2 
8, articles 1 to 48).—The author refers to the researches of Lagrange, 
Poisson, Sir W. R. Hamilton, and Jacobi, and he remarks that his own 
investigations do not pretend to make any important step in advance. The 
investigations contained in section 1, articles 1 to 14, establish by an inverse 
process (that is, one setting out from the integral equations) the chief con- 
clusions of the theories of Sir W. R. Hamilton and Jacobi, and in particular 
those relating to the canonical system of elements as given by Jacobi’s 
theory. The theorem (3), article 1, which is a very general property of 
functional determinants, is referred to as probably new. The most im- 
portant results of this portion of the memoir are recapitulated in section 4, 
in the form of seven theorems there given without demonstration; some of 
these will be presently again referred to. Articles 17 and 18 contain, I 
believe, the only demonstration which has been given of the equivalence of 
the generalised Lagrangian and Hamiltonian systems. The transformation 
is as follows: the generalised Lagrangian system is 
* The date is that of the communication of the note to the Bureau of Longitudes, but 
_ the note is only published in Liouville’s Journal in the May Number for 1855, which is 
subsequent to the date of the second part of Professor Donkin’s memoir in the ‘ Philosophical 
Transactions,’ which contains the theorem in the question. I have not had the oppor- 
_ tunity of seeing a thesis by M, Adrien Lafon, Paris, 1854, where Liouville’s theorem is 
- Quoted and demonstrated. 
