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or

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do , da 8Q +

+

idx dx dy dy and now eliminating da' dic + dy' dy

άλ αφ , αλ αφ

+ QSM - MOQ=0,

we obtain

which is satisfied by M= Q.

4. [Among Professor Boole's manuscripts I found five pages in German, forming part of a memoir, which was probably intended for Crelle's Mathematical Journal. The memoir was to have discussed two applications of the Calcu

2

2

2n

2n

lus of Variations; one to the Jacobian Theory of the Last Multiplier, and the other to the Solution of Pfaff's equation

X, dx, +Xdx, + ...... + X. dxn=0.

X But there is only a single paragraph relating to the second application.

The manuscript contains the same demonstration of the Jacobian Theory of the Last Multiplier as in Art. 2 of the present Chapter; after this demonstration some remarks occur of which the substance will now be given.]

It is worthy of notice, that Jacobi in the 36th volume of Crelle's Journal, deduced by the aid of the Calculus of Variations the result on which the preceding demonstration of the Theory of the Last Multiplier depends. In fact, he shewed that if V denotes any function of

dz dz X, X, Xne ,

dx,' and V be transformed by the introduction of a new system of independent variables un, Ug, ... Um, then the following relation holds,

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dx,

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dx, du,

dx du,

>

+

Jacobi applies this result to the transformation of the expression

MV IV dV
dx2 + dy
?

dz2 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

are

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m'

dax'
dt

X'ta

dx'

=

th

dic

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Here m is the mass at the point (x, y, z), m' that at (ac', y', z'), X, Y, Z the resolved forces at (x, y, z) tending severally to

, increase those co-ordinates, and so on. Lastly

$=0, x=0,... are the equations of condition each of which may involve all the co-ordinates, and , f... are indeterminate multipliers.

The above is usually termed the first Lagrangean form of the differential equations. In applying it we must either eliminate 1, f... from the giyen equations, and then by the equations of condition just so many of the co-ordinates with their differentials, or we must retain , Ms... as variables so conditioned

da dog that the values of de, die .... in the system shall satisfy iden

dc d'y tically the differential equations involving

de

rived from $=0, p=0,... viz. the equations

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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
d(m*x) x dy

tu dt"

d (m*x) d (m+x)***

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mi

from which we see that if x, y... had been taken to represent

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

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