Jacobi applies this result to the transformation of the expression

But neither Jacobi himself, nor any other person, so far as I know, has drawn attention to the application of the result which I have given here.

[The substance of the single paragraph relating to the second application of the Calculus of Variations will now be given.]

Clebsch has earned the thanks of all who are interested in the higher parts of the Theory of Differential Equations, since he has performed the same service for Pfaff's problem as Jacobi did for the Theory of Partial Differential Equations of the first order, and thereby for the equations of Dynamics. But while I recognise the great importance of the results, I consider it desirable to give a simpler deduction of the system of partial differential equations therein involved, and on which the other results depend.

CHAPTER XXXII..

THE DIFFERENTIAL EQUATIONS OF DYNAMICS.

[IT will be seen that this is only a fragment of the Chapter which was to have appeared under this title.]

I do not propose in this Chapter to discuss the origin and interpretation of the differential equations of motion or to enter into those details of their application which are found in all ordinary treatises on Dynamics. But they constitute a system analytically so remarkable from the forms in which it is capable of being expressed, and from the general methods of integration which emerge out of those forms, that they are well deserving of a special attention.

Referred to rectangular co-ordinates the differential equations for the motion of a system of points free or connected

Here m is the mass at the point (x, y, z), m' that at (x', y', z'), X, Y, Z the resolved forces at (x, y, z) tending severally to increase those co-ordinates, and so on. Lastly

= 0, y =0,...

are the equations of condition each of which may involve all the co-ordinates, and λ, μ... are indeterminate multipliers.

The above is usually termed the first Lagrangean form of the differential equations. In applying it we must either eliminate X, ... from the given equations, and then by the equations of condition just so many of the co-ordinates with their differentials, or we must retain X,,... as variables so conditioned in the system shall satisfy iden

that the values of

dx d'y
dt2 dť2

dx dy

tically the differential equations involving de de

rived from =0, y=0,... viz. the equations

The first Lagrangean system may by a slight transformation be reduced to a form in which all the equations are of one type, viz. of the type which they would have if all the masses were equal to unity.

For taking the first equation of the system and dividing by m we may express the result in the form

from which we see that if x, y ... had been taken to represent the entire system of co-ordinates taken in any order and multiplied each by the square root of the corresponding mass, and X, Y... the corresponding resolved forces taken in the same order and divided each by the square root of the corresponding mass, the system of equations would have been

220 THE DIFFERENTIAL EQUATIONS OF DYN MICS. [CH. XXXII.

all being of one type. In general investigations this form is to be preferred.

From the first Lagrangean form another known as the second Lagrangean, and from this again a third known as the Hamiltonian are derived. The second Lagrangean form is properly speaking an expression for the effect of a transformation of co-ordinates in the most general sense upon the original system, i.e. of a transformation which in place of x,y,... the entire system of given co-ordinates substitutes a new system of variables §, 7, ... the expressions of which as functions of x, y, are known. It is not necessary that this new system of variables should be co-ordinates in the proper sense of that term, determining three by three the positions of the several masses; it suffices that they should in their entirety determine and be determined by the co-ordinates given.

...

The second Lagrangean form may be established as follows:

Differentiating the equations = 0, y=0,... with respect to any one of the new variables § we have

whence if we multiply the equations of the given system by

dx dy

and add, we have

αξ' αξ

CHAPTER XXXIII.

ON THE PROJECTION OF A SURFACE ON A PLANE.

[THE following memoir was found among Professor Boole's manuscripts; a Title and Introductory Remarks were to have been prefixed, but with this exception the memoir appears to be finished for publication. It is sufficiently connected with the subject of Differential Equations to find a place in the present volume.

The memoir by Sir John Herschel to which allusion is made is entitled, On a new Projection of the Sphere; this was read before the Royal Geographical Society of London on the 11th of April, 1859, and was printed as part of the Journal of the Society, Vol. xxx. 1860, pages 100...106. A chart of the World on Sir John Herschel's projection has been published by A. and C. Black of Edinburgh.

The history of the subject will be found in Chapter XXIII. of the Coup d'œil historique sur la Projection des Cartes de Géographie... Par M. D'Avezac, Paris, 1863.

For the materials of this introductory notice I am indebted to Sir John Herschel.]

1. Let x, y, z be the rectangular co-ordinates of any point on the given surface; x', y' the co-ordinates of the corresponding point on the plane of projection. Let the equation of the given surface be

or, for simplicity,

F(x, y, z) = 0;