‘Oi eae ui 
‘st ee 
: ; 
ON THEORETICAL DYNAMICS. 39 
tion is an elegant and valuable one, but it is not in anywise to be inferred 
that there is any advantage in considering the particular case (which is thus 
shown to be capable of including the general one), rather than the general 
one itself: such inference does not seem to be intended, and would, I think, 
be a wrong one. 
69. Brioschi’s note of 1855 contains an elegant demonstration (founded 
on the theory of skew determinants) of a property which appears to be a 
new one, of the canonical integrals of a dynamical problem, viz. if g,p stand 
for a corresponding pair of the variables g,...p,... then 
3(a, B) 
23(G, p) 
where the summation refers to all the different pairs of conjugate integrals 
a, 3 of the canonical system, the pair g,p in the denominator being the same 
in each term ; but if the variables in the denominator are a non-corresponding 
pair out of the two series g,.. and p,..., or else a pair out of one series only 
(that is, both g’s or both p’s), then the expression on the left-hand side is 
equal to zero. This is in fact a sort of reciprocal theorem to the theorem 
which defines the canonical system of integrals. ‘There are two or three 
memoirs of Brioschi in Crelle’s Journal connected with this note and the 
note of 1853; but as they relate professedly to skew determinants and not to 
the equations of dynamics, it is not necessary here to refer to them more 
particularly. 
70. Bertrand’s memoir of 1857 forms a sequel to the memoir of 1851, on 
the integrals common to several problems of mechanics. The author calls 
to mind that he has shown in the first memoir, that, given an integral of a 
mechanical problem, and assuming only that the forces are functions of the 
coordinates, it is possible to determine the problem and find the forces which 
act upon each point; and (he proceeds) it is important to remark, that the 
solution leads often to contradictory results,—that, in fact, an equation as- 
sumed at hazard is not in general an integral of any problem whatever of 
the class under consideration: and he thereupon proposes to himself in the 
present memoir to develope some of the consequences of this remark, and to 
seek among the most simple forms, the equations which can present them- 
selves as integrals, and the problems to which such integrals belong. The 
various special results obtained in the memoir are interesting and valuable. 
71. In what precedes I have traced as well as I have been able the series 
of investigations of geometers in relation to the subject of analytical dyna- 
mics. The various theorems obtained have been in general stated with 
_ sufficient fulness to render them intelligible to mathematicians; the attempt 
_ to state them in a uniform notation and systematic order would be out of the 
_ province of the present report. The leading steps are,—first, the establish- 
ment of the Lagrangian form of the equations of motion; secondly, La- 
grange’s theory of the variation of the arbitrary constants, a theory perfectly 
complete in itself; and it would not have been easy to see @ priori that it 
_ would be less fruitful in results than the theory of Poisson; thirdly, Poisson’s 
theory of the variation of the arbitrary constants, and the method of inte- 
gration thereby afforded; fourthly, Sir W. R. Hamilton’s representation of 
_ the integral equations by means of a single characteristic function deter- 
“minable @ posteriori by means of the integral equations assumed to be known, 
_or by the condition of its simultaneous satisfaction of two partial differential 
_ equations ; fifthly, Sir W. R. Hamilton’s form of the equations of motion ; 
‘sixthly, Jacobi’s reduction of the integration of the differential equations to 
_ the problem of finding a complete integral of a single partial differential 
=1 
