28 REPORT—1857. 
of the theorem relating to Lagrange’s coefficients (a, 6), and of the corre- 
sponding theorem for the coefficients of Poisson. 
55. Bertrand’s memoir of 1851, ‘On the Integrals common to several 
Mechanical Problems,’ is one of great importance, but it is not very easy 
to explain its relation to other investigations. The author remarks that, 
given the integral of a mechanical problem, it is in general a question ad- 
mitting of determinate solution to find the expression for the forces; in 
other words, to determine the problem which has given rise to the integral ; 
at least, this is the case when it is assumed that the forces are functions of 
the coordinates, without the time or the velocities ; and he points out how 
the solution of the question is to be obtained. But, in certain cases, the 
method fails, that is, it leads to expressions which are not sufficient for the 
determination of the forces; these are the only cases in which the given 
integral can belong to several different problems; and the method shows 
the conditions necessary, in order that these cases may present themselves. 
It is to be remarked that the given integral must be understood to be one 
of an absolutely definite form, such for instance as the equations of the 
conservation of the motion of the centre of gravity or of areas, but not such 
as the equation of wis viva, which is a property common indeed to a 
variety of mechanical problems, but which involves the forces, and is there- 
fore not the same equation for different problems. The author studies in 
particular the case where the system consists of a single particle; he shows, 
that when the motion is in a plane, the integrals capable of belonging to 
two or more different problems are two in number, each of them involving 
as a particular case the equation of areas. When the point moves on a 
surface, he arrives at the remarkable theorem—“In order that the equa- 
tions of motion of a point moving on a surface may have an integral inde- 
pendent of the time, and common to two or more problems, it is necessary 
that the surface should be a surface of revolution, or one which is develop- 
able upon a surface of revolution.” When the condition is satisfied, he gives 
the form of the integral, and the general expression of the forces in the 
problems for which such integral exists. He examines, lastly, the general 
case of a point moving freely in space. The number of integrals common to 
several problems is here infinite. After giving a general form which com- 
prehends them all, the author shows how to obtain as many particular 
forms as may be desired: it is, in fact, only necessary to resolve any pro- 
blem relative to motion in a plane, and to effect a certain simple trans- 
formation on the integrals; one thus obtains a new equation which is the 
integral of an infinite number of different problems relating to the motion 
in space.” 
As an instance of the analytical forms on which these remarkable results 
depend, I quote the following, which is one of the most simple :—“If an 
integral of the equations of motion of a point in a plane belongs to two dif- 
ferent problems, it is of the form 
a=F(¢', x, y, ¢), 
where ¢! is the derivative with respect to ¢, of a function of 2, y, which 
equated to zero gives the equation of a system of right lines.” 
56. Bertrand’s memoir of 1851, ‘On the Integration of the Differential 
Equations of Dynamics.’—The author refers to Jacobi’s note of 1840, in 
relation to Poisson’s theorem; and after remarking that there are very few 
problems of which two integrals are not known, and which therefore might 
not be solved by the method if it never failed; he observes that unfor- 
tunately there are (as was known) cases of exception, and that, as his me- 
