32 REPORT—1857. 
a @t_dZ 
dt dq dq’ 
where Z is any function of ¢ and of g,...q',... And writing aan ssa 
then if H=—Z+q'p+ ... where, on the right-hand side, g',.. are ex- 
pressed in terms of ¢,g,.. p,--. so that H is a function of ¢, g,... p,.++3 
then the theorems in the preceding articles show that 
dq_ du dp_ dH 
dt dp’ dt dq’ 
which is the generalised Hamiltonian system. 
In section 2, articles 21 and 22, there is an elegant demonstration, by 
means of the Hamiltonian equations, of the theorem in relation to Poisson’s 
coefficients (a,b), viz., that these coefficients are functions of the elements 
only. And there are contained various developments as to the consequences 
of this theorem; and as to systems of canonical, or, as the author calls 
them, normal elements. The latter part of the section and section 3, 
relate principally to the special problems of the motion of a body under the 
action of a central force, and of the motion of rotation of a solid body. 
64. Part II. (sections 4, 5, 6 and 7, articles 49-93, appendices ).—Section 
4 contains the seven theorems before referred to. Although not given as 
new theorems, yet, to a considerable extent, and in form and point of view, 
they are new theorems. 
Theorem 1 is a theorem standing apart from the others, and which is used 
in the demonstration of the transformation from the Lagrangian to the 
Hamiltonian system. It is as follows: viz., if X be a function of the  vari- 
ables z,..., and if y,... be ” other variables connected with these by the 
m equations 
dX 
were eee 
then will the values of ,..., expressed by means of these equations in 
terms of y,..., be of the form 
dY 
ee 
and if p be any other quantity explicitly contained in X, then also 
dX dY 
Gohdp 
the differentiation with respect to p being in each case performed only so 
far as p appears explicitly in the function. 
The value of Y is given by the equation 
Y=—X+ay+... 
where, on the right-hand side, z,... are expressed in terms of y,... 
Theorems 2, 3 and 4, and a supplemental theorem in article 50, relate to 
the deduction of the generalised Hamiltonian system of differential equa- 
tions from the integral equations assumed to be known. In fact (writing 
V5 'Gs-- + Dy --- O; ene Gye's-, instead of the authors &, 2.-- 2s Masia 
@,+++G,, 6,...6,), it is assumed that V is a given function of ¢, of the 2 va- 
