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220 THE DIFFERENTIAL EQUATIONS OF DYNAMICS. [CH. XXXII.

d'.

do dels

dy
= X +λ

dx
dạy

do

drf
Υ +λ

tu dx

Y+ dy

te dy

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 trans-' formation 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 g, n, ... 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, @=0,... with respect to any one of the new variables & we have

do do do dy

+
dx az dy dě

: 0,

df dx dy dy

+

dy whence if we multiply the equations of the given system by dx dy

and add, we have
' de
dx d’x, dy d'y dx dy

+ Y
"dt* de

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= 0;

dx dě

+

= X

CHAPTER XXXIII,

ON THE PROJECTION OF A SURFACE ON A PLANE.

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[The following memoir was found among Professor Boole's manuscripts; a T'itle 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'ail 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; a', y' the co-ordinates of the corresponding point on the plane of projection. Let the equation of the given surface be

F(x, y, z) = 0; or, for simplicity,

F=0.

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The condition of projection upon which Sir John Herschel's investigations are founded, and which we shall adopt here, is that of the similarity of corresponding infinitesimal areas on the surface and on the plane. The object of the problem then in general is the discovery of the mode in which x, y' depend upon x, y, and z in accordance with the above condition; its object in any particular case is the determination of ac', ý as functions of x, y, z.

Regarding then ac', y' as ultimately functions of x, y, z we have

dx dr dac' dx'

dy+ dac dy

dxt

dz,

dz

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in which dx, dy, dz are not independent, but are connected by the condition

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Now the condition of the similarity of infinitesimal corresponding areas may be resolved into the two following conditions, viz.:

1st. The equality of their corresponding angles.

2ndly. The proportionality of their corresponding sides. And these conditions we shall introduce separately.

1st. Assuming any point x', y' on the plane of projection, let ac

' alone vary, and the infinitesimal line generated is doc', while (since dy' = 0) (2) and (3) become

a'dx + b'dy+ c'dz=0,
Adx + Bdy+Cdz =0,

=

whence, if we write
L=Bc' Cb', M= Ca' – Ac, N=AB' Ba',

dx _dy_dz
L MN

we have

(4),

so that the direction cosines of the infinitesimal line on the surface F corresponding to the line dac' on the plane (oc', y') will be L M

N

(5). (L' +Mo+N2)+' (L’ + M° +)' ([’+Mo+N2) &

N2' L! In like manner, if y' alone vary, we shall find for the direction cosines of the infinitesimal line on the surface F which corresponds to dy' on the plane L' М'

N'

(6), (L*+M+N)” (L+M? +N) ” (L? +Mo+N?) where I'=Bc Cb, M' = Ca – Ac, N' = Ab Ba.

By the first of the conditions of similarity the angle between these lines on the surface must be a right angle since d' and dy are at right angles. Hence we have, from (5) and (6), LL' + MM' + NN' = 0

...(7). .

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2ndly. The ratio of the length of the element dac' to the corresponding element on the surface is

dx

s dx2 + dyř + d22 or, by (1),

adx:+ bdy + cd2

Nda+ dy? + dz?' and therefore by (4)

aL +6M +cN NL + M" + N'

equating which to the corresponding expression for the ratio of the length of dy to that of its projection on the surface,

we have

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Now if we substitute for L, M, N, L', M', N' their values, we shall find

aL+6M +cN= A (b'c bc') + B (c'a-ca') + C (a'b ab'), a' L'+b'M'+c'N'= A (bc'— b'c) + B (ca' c'a) + C (ab' -- a'b), and the second members of these equations differ only in sign.

Thus (8) may be expressed in the form
{a{bc—bo") +B (c'a— ca) + C (ató – a'])}

1
12+

-2007) = 0... (9). (L' + M° + N2) (L" + M" + N')!

? M?

1

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