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Whence, integrating

dy
dt

2

dr
Y = C,

dt

(3)

c being an arbitrary constant.

Multiply cach term of the first equation by the different sides of this equation ; then

day

d XY

ц M
dt

dt
dt)

dt

da

d!

с

(2

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3

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- +

(5)

Similarly, by means of the second equation,

dy

+b,

dt a and í being arbitrary constants.

Multiply these equations by y and a respectively and add, then we find Mre + ay + bx = (..

(6) d a Multiply (1) by ? (e) by 2

add and integrate; dt

dy

dt'

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++a = x2 (nr + kr“) – 6

(9)

rda whence

9 {2 ???" If we assume x = r cos 0, y = r sin 8, equation (3) becomes

d go?

dt

= C;

nedt

cdr w

(10) r {2 (14r + kro) - ("}! From (10) we know in terms of r, and from (9) r in terms of t + a, so that can be expressed in terms of t+a, and therefore also x and y in terms of the same quantity. There appear to be five arbitrary constants, a, b, c, a, ß, but the equation (6) gives a relation between them which reduces the number of independent constants to four.

(13) Let the equations be

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These are the equations for determining the angular velocities of a rigid body revolving round its centre of gravity and acted on by no forces. Multiply the equations by x, y, x, respectively, and let

XYX =

Then the first equation gives
dt
dx


+(c b)
dt

at

= 0.

dt

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On inverting and integrating we should obtain t in terms of , and therefore o in terms of t, and from the value of o, x, y, z in terms of t.

(11) M. Binet * bas shewn how to integrate the system of simultaneous equation :

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the number of variables u, v, x ... being n, and R being a function of p = (u? + v2 + x + ...), so that

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dR Eliminating between each pair successively we find

dr equations of the form d dau dév

d’u

d'c : 0,

(3) dt? dt?

dt? dt

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0, &c.

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* Journal de Mathématiques, Vol. 11. p. 457.

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d x) 2

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The sum of the squares of these gives
du

du
+

+ &c. = A', (5) dt dt

dt dt where A is the sum of squares of the constants. By adding and subtracting uo

dt
dt,

+ &c., this

dt may be put under the form

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(u' + v* + x? + &c.)

&c.}

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+ &c.)' = A. (6) dt

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du dv On multiplying the proposed equations by 2 2

dt dt' dx 2 - &c. and integrating, we have dt du

+ + &c. = 2(R + B) (7) dt

dt

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dvi 2

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2 B being the arbitrary constant arising in the integration.

Substituting this expression in (6) and putting pol for u' + v + 2* + &c., that equation becomes

dv dic (u

+ &c.)' = 2 r(R + B) A?. dt dt dt

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dt and

{2r” (R + B) - A^}4* By differentiating (8) we find

ďr dR A?

+
dt?
dr

703

(10)

dR Eliminating from the first of equations (2) by means

dr of (10), and multiplying by ”, we have

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Sy2r

' (R + B) – AF}}

rdr t + a =

(14) 2 rk + B) A!! and from (11)

Adr +ß

(15) 2 By means of these we obtain p as a function of r, and go as a function of t+a, and therefore ø as a function of

Then the equations (13) will give u, v, x, &c. in terms of t + a, b, g, h, ,, h2, &c. A and B, the number of

t + a.

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