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see in Chap. II. In specifying the motion of a point according to the generalized system of co-ordinates, y, 0, 0 must be considered as varying with the time: 4, 0, , or

dy do do


dt' at di then be the generalized components of velocity: and Ý, , ö, or dy do do

d’h d'd’8 dt' dt' dt'

will be the generalized dt

components of acceleration.


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nates of any

204. On precisely the same principles we may arrange sets Co-ordi, of co-ordinates for specifying the position and motion of a system. material system consisting of any finite number of rigid bodies, or material points, connected together in any way. Thus if y, 0, 0, etc., denote any number of elements, independently variable, which, when all given, fully specify its position and configuration, being of course equal in number to the degrees of freedom to move enjoyed by the system, these elements are its co-ordinates. When it is actually moving, their rates of variation per unit of time, or , $, etc., express what we shall call its generalized component velocities; and the rates at which

, ¢, etc., augment per unit of time, or *, ¢, etc., its component Generalized accelerations. Thus, for example, if the system consists of of velocity. a single rigid body quite free, y, ¢, etc., in number six, may be Examples. three common co-ordinates of one point of the body, and three angular co-ordinates (S 101, above) fixing its position relatively to axes in a given direction through this point. Then 4, , etc., will be the three components of the velocity of this point, and the velocities of the three angular motions explained in § 101, as corresponding to variations in the angular co-ordinates. Or, again, the system may consist of one rigid body supported on a fixed axis; a second, on an axis fixed relatively to the first; a third, on an axis fixed relatively to the second, and so on. There will be in this case only as many co-ordinates as there are of rigid bodies. These co-ordinates might be, for instance, the angle between a plane of the first body and a fixed plane, through the first axis; the angle between planes through the

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Generalized second axis, fixed relatively to the first and second bodies, and components of velocity. so on; and the component velocities, 4, $, etc. would then be Examples. the angular velocity of the first body relatively to directions

fixed in space; the angular velocity of the second body relatively to the first; of the third relatively to the second, and

Or if the system be a set, i in number, of material points perfectly free, one of its 3i co-ordinates may be the sum of the squares of their distances from a certain point, either fixed or moving in any way relatively to the system, and the remaining 3i – 1 may be angles, or may be mere ratios of distances between individual points of the system. But it is needless to multiply examples here. We shall have illustrations enough of the principle of generalized co-ordinates, by actual use of it in Chap. II., and other parts of this book,




(a) In § 491 (c) below is to be found Poisson's extension of Laplace's equation, expressed in rectilineal rectangular co-ordinates; and in § 492 an equivalent in a form quite independent of the particular kind of co-ordinates chosen : all with reference to the theory of attraction according to the Newtonian law. The same analysis is largely applicable through a great range of physical mathematics, including hydro-kinematics (the "equation of continuity" 8 192), the equilibrium of elastic solids (734), the vibrations of elastic solids and fluids (Vol. 11.), Fourier's theory of heat, &c. Hence detaching the analytical subject from particular physical applications, consider the equation do U dot do

47p....... dx

-(1) dy' dza where p is a given function of x, y, z, (arbitrary and discontinuous it may be). Let it be required to express in terms of generalized




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co-ordinates &, Š, É", the property of U which this equation ex- Laplace's presses in terms of rectangular rectilinear co-ordinates. This generalized may be done of course directly ($ (m) below] by analytical trans- dinates. formation, finding the expression in terms of ¢, $', $", for the d dia đ

d operation

But it is done in the form most con

dz venient for physical applications much more easily as follows, by taking advantage of the formula of § 492 which expresses the same property of U independently of any particular system of co-ordinates. This expression is SHOUDS= – 471 SlSpdB.......

... (2), where las denotes integration over the whole of a closed surface S, SSI dB integration throughout the volume B enclosed by it, and 8U the rate of variation of U at any point of S, per unit of length in the direction of the normal outwards.

(6) For B take an infinitely small curvilineal parallelepiped having its centre at (t, &'; $"), and angular points at

($ + } 8É, & + } SE, " + } '). Let R8Ě, R'DE', R''St" be the lengths of the edges of the parallelepiped, and a, a', a" the angles between then in order of symmetry, so that R'R" sin a SESE", &c., are the areas of its faces.

Let DU, D'U, D"U denote the rates of variation of U, per unit of length, perpendicular to the three surfaces & = const., &' =const., É' = const., intersecting in (Ě, Š', $") the centre of the parallelepiped. The value of S&US for a section of the parallelepiped by the surface & = const. through (¢, $,") will be

R'R" sin a DËSE" DU. Hence the values of SS&U ds for the two corresponding sides of the parallelepiped are

d R'R' sin a dE'S" DU = de (RRsin a dě 8t" DU). 186. Hence the value of S&U DS for the pair of sides is


(R'R" sin a d' 8" DU). 8,

(R'R" sin a DU) 8 DE' SE".
Dealing similarly with the two other pairs of sides of the
parallelepiped and adding we find the first member of (2). Its
VOL. 1.



Laplace's equation in generalized Co-ordivates.


DU)}=- 45p.(3).

second member is - 4-P.Q.RR'R" SE DE", if Q denote the ratio
of the bulk of the parallelepiped to a rectangular one of equal
edges. Hence equating and dividing both sides by the bulk of
the parallelepiped we find
1 rd

(R'R" sin a DU) + de )

(R" R sin a' D'U)

(RR sin a" D"U
(c) It remains to express DU, D'U, D'U in terms of the co-
ordinates $, $', ".

Denote by K, L the two points (Ě, Š', &"') and ($ + 8?, &', & ). From L (not shown in the diagram) draw LM perpendicular to

the surface $ =const. through K. Taking an infinitely small portion of this surface for the plane of our diagram, let KE', KE" be the lines in which it is cut respectively by the surfaces É" = const. and š = const. through K. Draw MN parallel to

E"K, and MG perpendicular to KE. Let now P

denote the angle LKM,

We have
ML = KL sin p= R sin p 8,
NM=GM cosec a = ML cosec a cot A' = R sin p cosec a cot A' SF.

Similarly KN = R sin p cosec a cot A" DĚ,
if A" denotes an angle corresponding to A’; so that A' and A"
are respectively the angles at which the surfaces " =const. and
const. cut the plane of the diagram in the lines Kz' and KE".

KN * Now the difference of values of &' for K and N is



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MN and


R" " Hence if U(K), U(M), U(L) denote the values of U respectively at the points K, M, L, we have

dU KN dU NM U(M)=U(K) +

df'R de" · R,

dU and U (L) = U (K) + ६.



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U (L) U (M)

Laplace's But DU

equation in ML

generalized and so using the preceding expressions in the terms involved we dinates. find 1 dU 1 dU

1 dU
DU =
Rsinp de R'sina tan A" dg
R' sina tan A" d& R"sin atan A'

.(4). Using this and the symmetrical expressions for D'U and D'U, in (3), we have the required equation.

(d) It is to be remarked that a, a', a" are the three sides of a spherical triangle of which A, A', A" are the angles, and p the perpendicular from the angle A to the opposite side. Hence by spherical trigonometry

cos a - cos a cos a'


sin a sin a sin A= (1 - cos a - cos' a' - cos' a" + 2 cos a cos a' cos a")

sin a sin a sin p= sin A' sin a' vil - coso a - cos® a' – cos' a" + 2 cos a cos a' cos a")





sin a

To find remark that the volume of the parallelepiped is equal to f sin sin a if f, g, h be its edges : therefore Q = sin p sin &......


whence by (6)



Q=/(1 - cos® a - cos' a' - cosa" + 2 cos a cos a' cos a"')......(8).
Lastly by (5) and (8) we have

tan A
cos a' cos a

(9). (2) Using these in (4) we find

1 sin a dU DU

cos a cos a' dU
Q sin a


- cos a cos a"


cos a"

cos a'

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Using this and the two symmetrical expressions in (3) and adopting a common notation [App. B (g), $ 491 (c), &c. &c.], according to which Poisson's equation is written V'U= - 4P


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