Note on the Projection of the General Locus of Space of Four Dimensions into Space of ThreeDimensionsAuthor(s): Thomas CraigSource: American Journal of Mathematics, Vol. 2, No. 3 (Sep., 1879), pp. 252-259Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369241 .

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Note on the Projection of the General Locus of space of four dimensions into space of three dimiensions.

BY THOMAS CRAIG, Fellow of the Johnts Hopkins University.

IN the first number of this Journal, Professor Neveomb has given a proof of a certain property of a closed surface, viz: that suich a surface, il placed in space of four dimensions, could be turned inside out without stretching or tearing it. Of course this is but one out of the many new degrees of freedom which material objects would possess if placed in four-dimensional space; but it is not the object of this brief paper to study the new properties of natural objects placed in space of four dimnensions, but rather to determine the representations in space of tlhree dcimensions of the loci which can only exist in four-diinensional space. The general locus which characterizes space of four dimensions mnay be represented by an equation between the four rec- tangular coordinates x, y, z, t. This locus is, in fact, tle geerccl locuts of three dimiensions. A strface in four-dimnensional space requires for its determination twvo equations between the variables, and, if the surface is one which may exist in Euclidean space, one of these equations must be linear or of the form

oax?+3y+vz +t+ e-O, whiclh denotes the infinite plane Euclidean space. There are of course ani infinite number of these Euclidean spaces in four-dimnensional space which are determined by assigning different values to the constants oa, ~, &c.

The two equations which taken together denote a surface in four-diinen- sional space sustain to this surface the same relation that two equations in Euclidean space sustain to the c urve which is the intersectioni of surfaces corresponding to each of these equations. A surface in four-dimensional space is then given by F(x, y, z, t) O,

PD (x, y, z, t) -0, and F= 0 arid P _- 0 each denotes the general three-dimnensional locUs char- acteristic of space of four dimeensions, just as the surface F (x, y, z) - 0 is the general two-dimensional locus characteristic of space of three climensions.

I obtain in the following paper merely the general differential equations which are necessary to obtain the representation in Etielidean space of the

252

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CRAIG, Note on the Projection of the General Locus of Space. 253

locus F (x, y, z, t) - 0. A transformation to three independent variable para- mleters would probably have the effect of simplifying the results, as in Eucli- dean space the whole theory of surfaces, or more exactly of the curvature of surfaces, is more conveniently and elegantly studied by using two independent varialble parameters instead of the three dependent rectanigular coordinates of any point.

The locus in three-dimensional space F (x, y, z) - 0 is projected into two- dimensional space, say the plane z' = 0, by simply finding the values of xi, Y, the rectangular coordinates of the point in the plane corresponding to the point x, y, z of the surface.

Of course the process of findling, these values is dependent upoln the con- ditions which are to be fulfilled in making the projection. The condition of projection that we shall here emrploy is that of the sinmilarity of the elemenits of the given locus with the representation of these elements in three-dimen- sional space. The conditions to be fulfilled then are, the preservation of angles in their true size, and thle preservation of the ratio between any two elemients of the given locus in the projection of these elements.

The equation of the general four-dimensional locus is F(x, y, z, t) =0.

Let i, , denote the rectangular coordinates in space of three dimnensions; it is required to find i, , < in terms of x, y, z, t, so that the above conditions shall be fulfilled. Since i, v, <are functions of x, y, z, t, we have

= d-x + -` dy + a: dz + -- dt,

dax ay z at d; - a' dxG+at + a' a: + + tdt,

and also aF aF aF aF 0 - dx + -dy + - dz + a dt.

Dx ay Dz at Write for brevity

aF _ aFF a aF aX-a - /- 7, I at

=

as a>* az at __ a _r- h &a - ax ay g az at

a6 a5 - b' a, I

at = a a<t bto It = C, a<_ I." 65

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254 CRAIG, 3ote on the Projection of the General Locus of Space.

then the preceding equations become adlx + 3dy+7dz+Ecdt =- , adx + bdy + cdz + edt - < a'dx + b1dy + c'dz + e'dt =d a"dx + b"/dy + c"dz + e"dt= d;.

We will define here V as the determinant of the above equations when < I dr I d; = 0 that is,

a, b, c, e V a, b', c', e'

a", b", c", e" The minors of V corresponding to ca, Z, y, E are

K, L, iX, N corresponding to a, b, c, e,

k, 1, m, n and similarly those corresponding to a', b', &c., a", b", &c., are

Ic", 1", in", n". If we suppose, now, that in tlhreefold space; is the only quantity which varies, we must have d dry 0, and

adx + Cdy + ydz + Edt = 0, adlx + bdy + cdz + edt O, 0 a'dx + b1dy + c'dz + e'dt_= 0

from these follow dx dy dz d-

Write k2 + 12 + m2+ n2

V=k2+ ? l2 +mIt'2 ?+ 12,

-t ,/Ic"2 + lt 24+ niit 2+ nt 2,

then we have for the "direction cosines" of the line o1 FP- 0 corresponding to d;

k 1,, m n

In like manner if v1 alone vary we have, for the "direction cosines" of the line on F corresponding to dn,

k' It ml1 nm p _ pi _ _ _ pi

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CRAIG, Note on the Projection of the General Locus of Space. 255

and, finally, for $ the onily varying quantity k 1 m n Q p2X

The conditions for orthogonality are now kk' +? i' + nm' + n' -0, k'k" + Il'" -+ m'" +- n'u" - 0, k1k + I"l + rn"m? + n"n =0.

The ratio of the elemnent (d$ to the corresponding element on F is d;

VdX2 + dy2 ? 2 d 2A~ or,

adx + bdy + cdz -+ edt V dX2+ dy2 + dZ2 +dt2'

or finally, ak - bl + cm - en

2 l

Equating this to the ratio of dn to its corresponding element on F,

T (ak/-bl + cmn-en) = (a'k'-bl' + c'm'-e'n').

The quantities in the parentheses are respectively -V and - V, so that this equation becomes simply

1 1=o --0

or, ?1' - -l0

and in like manner .Q"-Q.' =0,

-fl ?- =0.

These equations, written out in full, constitute, with the previously given equations of orthogonality, the entire system of equations of condition from which we mnust endeavor to determine the differential equations affording the solution of the problemn. For convenience I give the entire set of equations here; they are: 1.) kk' + 11' + mi' + nn' - 0 2.) c' - l'l" m'm"+ -nee + n'n" - 0 3.) k"k + I'll + m"'m- + n"n - 0 4.) k +2 +l + M2 +n'2 (2 +12 + m2 +n2) = 0, 5) P2 +1 j"2 + Mr + n"2 (k,2 +. l2 rn'2 - n2) = 0,

6.) k2 + 12 + -m2 4. -2 (kPt2 + ti2 + rn"2 + n"2) = 0.

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256 CRAIG, Note ont the Projection of the General Locus of Space.

Multiply the first, second and third of these equations by 2i and add and substract the results frorn 4, 5 and 6 respectively. 7. (A j ik)2 + (1' ] il)2 + (in' i2un) 2 + in)2 0, 8. (k1 l ik/')2 + (1" ] il')2 (1n in] n')2 + (n" + in')2 -

0

9. (k 4 ik")2 + (I + il")2 + (in ? urn")2 + (n ] in")2 0. The formation of these expressions fromn the determinant V is not difficult though a little tedious, so I merely give the results in which, however, for brevitv, I have written

t61= + in, VI + ? 1 + d

Ub2=g - i, V2=n - i;, W2= ;-t

and also used the symbols U1,2, V1,2, W1,2, when no confusion could arise, to denote that the expressions are the samne in form for u2 as for u1, &c. Equa- tion 7 becomes now

aF aF aF 2 aF aF aF 2

ay' z' at az at' x alU, 2 au,2 aU1, 2 au1, 2 au1, 2 aU1, 2

ay,' az' at + az at' a as- as as as as as ay' az' at az at' ax aF aF aF 2 aF aF aF 2

atX x ay Ux' y' z a1t1, 2 aul,2 aUI, 2 attl, 2 au1, 2 aU1, 2 O + <~r' Ua ' a11 ? ax' a' az as ac as as as a: at' ax' ay ax ay' az

The expanded forms of 8 and 9 are obtained from this by merely changing u into v and w successively. Denoting by 01, 02, 03, 04 any four quantities whatever and writing,

01 , 02 , 03 , 04

aF aF aF aF ax' ay ' az' at

au1, 2 au1, 2 au1, 2 au1,2 = Ul,2 ax' ay1 az' at a: a: a: ag ax a y ' X at

we can place the last equation in the form (auf2) 2 (aQg 28 (auL7) 2 au, 20

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CRAIG, Xote on the Projectionof the General Locuts of Space. 257

or, for convenience, making a2 (2 2 (a 2 a 2 2 0

(observe that (D) is not 2 .

DU1,2-0 -DV1,2 0, DW1,2-0 V and W beinig quantities of the same kind as U.

au, 2

The calculation of the quantities ( 2) is quite simple, since each of

them is a symmetrical determinant. If we expand these expressions so that ) Ul, 2 shall appear in the form

Aa2+ 32+ Cy2+ D2 -2Ea- 2FPy, &c. -0, a, 3, y, being as before the derivatives of F with respect to x, y, z, t, we shall find A, B, C, Z giveni as sums of squares of three determinants of the second order-or, consequently, as one symmetrical determinant of that order; E, F) &c. as sums of products of determinants of the second order. But the most conivenient forin in which to hav-re the equation is that of one symmetri- cal determinant of the third order, and by a known theorem we can write it at once as

a2?+ 2+y2?+2, ala+?blf+?cl/+eiE, a2a+6b2+?c27+e2eI tUI,2= ala+ bl~ + cly + e1c, a'+ bl+c2+ el, a1a2? b1b2?c1c2?e1e2 -0

a2a+ b2f + c2Y + e2E, a1a22+ b1b2 + c1c2+ e1e2, a 2+ b2 + C2 + e2

where a1, bi, c1, e1 are the derivatives of U1,2 2with respect to x, y, z, t respec- tively, a2, b2, c2, e2 are the same derivatives of ;. There are two other equations of this form for v1,2 and W1,2 in which a, 3, y', E have the same significance as in -DU1,2 = 0, at, b,, c1, e1 are the derivatives however of v1, 2 and w1, 2, respec- tively, and a2, b2, c2, e2 denote in DFl 2 = 0 the derivatives of $ and in - Wl, 2 = 0 the derivatives of n with respect to x, y, z and t.

If we had started to project the surface F (x, y, z) = 0 in Euclidean space upon the plane w,e, wewould lhave found as the equations of the problem

a 2 + 2+ 72, ala + bOS + clyl

Iala + bil + clyx alt b + b 12 1-

the a, b, having the same meaning, as before, and a1, b1, c1 to take the values

an, aul au, ax ay az' aU2 aN2 aU2

66 ax - ay az-

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258 CRAIG, Note on the Projection of the General Locus of Space.

suceessively. These are the equations for the orthomorplhic projection of a surface upon a plane.

If we hlave n variables x1, x2, x. , connected by the relation F (xl,,X21. . . Xn) - 0

the general locus of n-dirnensional space, and n I variables 01 42 . ., n -

denoting the coordinates in space of n - 1 dinmensions, then for the projection

of F 0 O into the n - 1-dimlensional space, we shall have ( 1)(n -2) groups of 2

equations of the form

A.S (lih2)2+ (DL2)22 (aU+2) 2

o (V2)+ (a V 2)2+ (aVl2)20

the quantities 0 being anything whatever, aid the derivatives au av ias' ao

being the first minors of the determinants o1 02 XX Oil

aF aF aF ax, ax, ax.

IaDu, 2 aut,2 a?j,2 axr, a-V2 axn a;,3 a3 a-3 =vl

ax,l aX2 axn

. . . . . . . . . . . . .

' an at axn, D1x ai DX1 Dxr1 X

and 69l 7 02 7 * o * 6n

aF aF aF

ax, aX2 ...ax

avI, 2 aV1, 2 aV,, 2

aDX ax2 aDx a3 as3 0s3 =

-i K2 ax1 ax2 ' ax,

..................

ax,: ax.?b - Jt/

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CRAIG, iYOte oni the Projectioni of the General Locus of Sp)ace. 259

where 1b, 2= 1 I2 C, 3 = 2 + iC3, * 7 ,g--1=f + +C

( -, 2 = i3 2 I V), 3 2 Vi, i ]- =i i - i + I

The remaining n (n2 3) groups of equations simiilar to A are of course formed

fronm this group by chang,ing the qt, 2 into u1'2, 3 It,4 , &C., and V1, 2 into v.,., V,,.,,

&c., and also changingw thle : by advancing successively all the subscripts 1 .. .n-1.

By a theorem of determinants (which was unlknown to me until kindly communicated by Mr. Stringham) each of equations A inay be put in the following form:

a(DF)2 aFDaU,2 DaF a,3 xaFi n

Dx, ax, axi axi -a-xi axi axi

aFDu,, 2 a_ 2 au, 2lalfi, 3a,n-;

aDFD,3 ~Du12D%2 3 n (-L

ax, ax Daxi axix xii D* aDx a DaFaD4 XDau1,2D4 a a-34 a, (Dt4V DD__

a,-,, =0

ax, Dx, ax xi aaxDx xxxj '** x aDx

~~~~~~. . . . . . . . . . . . . . . . . . . . . . . ..

X aF-n 1, a I,.n

n- qn- I2

Dx2i Dx i Dxi Dx. ..x,

the summations to be taken from i = 1 to i = n. The equation for V being similar to this, there is no necessity for writing it.

WASHINGTON, Sept. 4, 1879.

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