36 _ REPORT—1857. 
d dZ_dZ.. 
dt dq dq’ 
where the variables g,... correspond to the coordinates of a dynamical 
problem ; if the new variables y,... are any given functions whatever of the 
original variables q, ... and of ¢, this is what may be termed a transformation 
of coordinates. But the proposed system can be expressed, as shown in the 
former part of the memoir, in the generalised Hamiltonian form with the vari- 
ables g,... and the derived variables p, ... (the values of which are given by 
aZ 
dq? .+.): the problem is to transform the last-mentioned system by intro- 
ducing, instead of the original coordinates g,..., the new coordinates 7,..+ 
and instead of the derived variables p,.... the new derived variables a,... 
dZ 
defined by the analogous equations > nia toh in which Z is supposed to 
be expressed as a function of 7,... and 4 The method of transformation is 
given by the theorem 9, which states that the transformation is a normal 
transformation, and that the modulus of transformation (that is, the func- 
tion corresponding to K in theorem 8) is 
K=qp+... 
where g,... are to be expressed in terms of n,.... The latter part of the 
same section contains researches relating to the case where the proposed 
equations are symbolically, but not actually, in the Hamiltonian form, 
viz., where the function H is considered as containing functions of q,... 
P>-++. which are exempt from differentiation in forming the differential 
equations (the author calls this a pseudo-canonical system), and where, in 
like manner, the transformation of variables is a pseudo-normal transforma- 
tion; the theorems 10 and 11 relate to this question, which is treated still 
more generally in Appendix C. The general methods are illustrated by 
applications to the problem of three bodies and the problem of rotation; 
the former problem is specially discussed in section 7; but the results ob- 
tained (and which, as the author remarks, affords an example of the so- 
called ‘elimination of the nodes’) do not come within the plan of the pre- 
sent report. 
66. Bour’s memoir of 1855, ‘On the Integration of the Differential 
Equations of Analytical Mechanics.’—It has been already seen that the 
knowledge of half of the entire system of the integrals of the differential 
equations (these known integrals satisfying certain conditions) leads by 
quadratures only to the knowledge of the remaining integrals; the re- 
searches contained in this most interesting and valuable memoir show that 
this theorem is, in fact, only the last of a series of theorems, here first 
established, relating to the successive reduction which results from the 
knowledge of each new integral. Speaking in general terms, it may be 
stated that the author operates on the linear partial differential equation of 
the first order, which is satisfied by the integrals of the differential equa- 
tions; and that he effectuates upon this equation a reduction of two unities 
in the number of variables for every suitable new integral which is ob- 
tained*. The author shows also that an equal or greater reduction may 
* T have borrowed this and the next sentence from Liouville’s report. It would, I 
think, be more accurate to say, for every suitable new integral after the first one; in the 
case considered in the memoir, the condition of vis viva is satisfied, and there is always one 
integral, the equation of vis viva, which is known; but this alone, and in the general case — 
the first known integral, will not cause a reduction of two unities. 
