on an application of the theory of scalar and clinant...

lociof scalar and clinant radical On an application of the theory

Alexander J. Ellis, Esq., B. A., F. C. P. S.

January 1860, 393-400, published 1111862 Proc. R. Soc. Lond.

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393

,c On an Application of the Theory of Scalar and Clinant Radical Loci.” By A l e x a n d e r J. E l l is , E sq., B.A., F.C.P.S. Communicated by A. C a y l e y , Esq. Received February 20, 1861. Read March 14, 1861*.

1. The following investigation, which contains a correction and extension of that in Plucker’s ‘ System der Geometric’ (§ 3. art. 64), is a direct application of the theories in the writer’s paper “ On Scalar and Clinant Algebraical Coordinate Geometry” (Proceedings, vol. x. pp. 415-426), to which reference must be made for an explanation af the notation and terminology.

2. 'Letfipe, y) be an algebraical formation (function) o f y and of »+2m dimensions, such that when any scalar value is attributed to x, 0 has n scalar (possible) and ‘Zm. clinant (imaginary) roots, and let A be the coefficient o f ?/w+2m. Then

y ) —Hy ~A x) ... ■ (>—/«*) x ( y - / > ) ( y —/ > ) ...(y —f i {m)x)( y .

3. Now let PQ be a line which, when produced, will cut the curve, whose equations are

O M =a?. O l- j-y . OB, JX.x> y ) ~(where x and y are scalar, and 01 is in the same straight line with OP, and OB is a line equal in length to OI, and parallel to and in the same direction with PQ) in the n points Mp M2, . . . Mre. Then,

will repre-if 0P=a?1.O I, where x1 is scalar, ?M i,OB ’ OB

PM«"OB'

sent the n scalar roots . . f nxv Hence if P Q = y x. OB, thepoint Q not being necessarily a point in the curve, any one of the

factors y x—f rxx will be represented by ^ PMy_MyQMR:'- J OB OB *

^Nowfor the clinant rootsf, put y = r + i . s , where the Roman ^ V and r and s are scalar, then must

f ( x> y) = F 1(«, r, s') + i . F2(a?, r, s),because there are n scalar roots for each of which s= 0 . For the

* An abstract of this Paper has already appeared in the “ Proceedings," page 141. is now printed in full by order of the Council.

flf^PKicker only says that “ no imaginary point of intersection must be neglected" V y«em, p. 45), but he gives no means for taking them into consideration.

T 0 l » - F

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clinant roots, then, we have ^ = 0 , and F2= 0 ; whence we find, byalternately eliminating s and r, F3(a?, r )= 0 , and F4(a?, s)— J.

Now describe the loci of the points R and S, whereO R = j?. OI + r.O B , F3(a?, r) = 0,

and _ _ , .OS = ^ .O I + «.O B, F4(a?,*) = 0.

Then from the nature of the clinant roots, which will be in pairs ol the form rx+ i . sx,if we produce PQ to cut these curves, for every point R' in which it cuts the first there will he two points S'lS S'2,in which it cuts the second. If, then, from the point R' we draw R/M'x= i . PS't and R 'M '2= i . PS '2, two corresponding clinant roots

PM' PM'of the equation will be — - -b so

and

„ _ f v . _ P Q -P M ,w _ M,WQ y1 Jl l~ OB — OB ’

,, /M t - P Q - p m w _ m w q Pi A *1— OB — OB

See the example and figure in (11).Hence we shall have for the geometrical representation of any

algebraical formation, by (2),

M,Q MmQ .M '.Q M'2Q M W Q M W Q OB X OB * OB *’ * OBy i ) = x * OB OB

4. Next suppose that the origin of the coordinates be altered to O', so that 0 'I '= 0 I , and O 'l', when produced, passes through Q. This will be equivalent to putting y + 6 instead of y, and the result will not disturb the coefficient of yn+Also since one of the radical loci for the scalar values remains unaltered, and the others (which are the axes) are parallel to the former, the points Mj, Ma, . . . M» hi which PQ, when produced, will cut the curve, remain the same, and consequently there are the same number of scalar roots as before.

5. But as the terms themselves of the equation f{x , y )—® are altered, the radical loci of R and S in (3) will be altered, and as the distances are now measured from Q and not from P, the lines Q R ',+ 1 . QS'3, QR', + i . QS'4 will now be different and = Q m \, Qm'„. I f then Q'Q=a?2. O 'l', and Q P = y 2 • B 0 = —y2. OB, then

/ (* _ y ) - ' M,P m'(m)P m' m)OB

MKP V m\P m'2POB X OB ' OB



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5r, replacing OB in each case by BO, we have

N M.P MmP P m ^ Pf ( * 2> yd — • gQ • • • g o x b o ’ ’ ' boT 5

vhere, since there are n + 2m factors,

\ ' = ( — l ) w+2w.X.

6. But if, instead of simply changing the origin, we change the nclination of PQ and make it parallel to OC, then if OB and OC nake the angles (3 and y with 01, and OJ be at right angles to 01, he lengths of 01, OJ, OB, OC being the same, and M be any point, we have

OM=sx . 014- y • OB= ( x + cos (3) . 01 + sin . 0 J,rad

OM=aj,.O I+ y '. O C = (^ f + yf cosy). 0 1 + / sin y .O J,

vhich give two simple equations to determine y in terms of / , so that on substitution f (x , y) becomes / ) , a formation of thesame number of dimensions. But the coefficient of y ,n+‘Zm will be different from that of y n+2m. Let it = p . The curve for the scalar ralues of x', y' in

O M = y .O I + y .O C , </>(V>/)= 0will be precisely the same as that in (3) ; but as PQ, having a diffei - ent inclination, will cut it in very different places or not at all, the numbers of scalar and clinant roots of 0 may be individually different from, although their combined number will be the same as, those o f /= 0. If then there be hscalar and 2k clinant roots of 0 = 0 , where h + 2k=n + 2m,and we determine Nx, N2. N f2. . . inthe same way as we previously determined Mx, Ma, . . . M \, M^, . . . we find

Vi) =»*(*!» y d —i1N,Q NaQ N \Q N WQ OC ' " OC oc oc

7. This investigation shows that if there be any number of lines drawn from a point, cutting those radical loci of an equation for scalar and clinant roots which correspond to the inclination of these lines to the axis and to the position of the origin, coefficients X, p, &c. can always be assigned such as to make the product of these ratios each equal to the other; and that these coefficients have only to be multiplied by ( _ i ) M+2m} jn order to give the coefficients corresponding to

2 f 2



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such a change of origin as would make the axis pass through th other extremities of the lines.

8. Now let 0 X0 2 . . . 0 , ^ be a completely enclosed polygon, th sides of which, when produced, cut the radical loci of a known algt hraical equation of n dimensions in known points. Let the produc of the series of ratios formed as above (3), corresponding to line drawn from the point Or in the direction Or Or+i (on which las) named line the unit line Or 0 'r+ i, corresponding to 01 in (3), is mee sured), be represented by

[Or, r+lM -r Or OV+i].

And let Xr>r+i be the coefficient necessary to make this product correct representation of the formation on the left-hand side of th given equation for the values of x and y, corresponding to the poii; O,. and the direction Or Or+i-

Then for the point Or+i and the direction Or+J Or (for wMch tli corresponding unit line i3 Or-fiO'r), we shall have the product of th corresponding series of ratios represented by

X»—f-1, r [ O j- f l, r M ~ O r + i O r]>

where

X r + l, r == ( —’ 1 ) M • Xr, r+1*

Proceeding then round the polygon, and forming the products fc each of the two sides terminating at each point, we have by (7),

Xi, w[Oi; re M -r OjOfre] =Xi, 2 [Oi, 2 M -r O jO y (■-1)*X,, , [ 0 2>, M -T- 0 , 0 ' , ] = X,, , [ 0 2, ,M -T 0 ,0 ',] ,

( l ) WXre—2, m—l [ 0 M—i, re—2 M-7-Ore—1 O^—2] —XM—i,«[Ore—i,«M . On_ i /> ( l ) ”Xre—1,« [Ore, n—l M + O n 0 'w] =X„,, [Ore,! M -r0„0'

Hence multiplying all these n equations together, and rememberin that Xljre= (— l ) re-Xre, 1, and that ( —l) ww= ( —1)M, since and are both odd or both even, we have

[Oj.reM-rO 're] . [02, , M-r 0*0',] . . . [Ow,re-l M-r0„0'„_I = (—l)w.[Oif*M-r 0,0'*]. [O*. 3 M-r 0 20'3] . .. [Ore, 1M-r OreO'i

Now since Or>r+iM and Or+ijrM are on the same straight lint their ratio is scalar; and since OiO're=— 1 . OreO'j, we have, by a



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ibvious extension of the notation, and by altering the order o f the actors,

[0, ,aM-f 0 2 , i M] . [ 0 2,3 M-f-O3, 2 M] . . . [Ow,iM-rOI)#M] = ( -1 )* . . [ o 2 o '3 ^ o 3o y . . . [OwO'x- -OjOw]= (—-l)w. [ (—l) w. ( — 1)M. . . . (nfactors) . . . ( —1)M]= ( —1)«. ( —l)w» = ( — 1)». (— l)»= (_ l )2»— 1 *.

9. Hence if all the roots are scalar, that is, i f each side o f the jolygon cuts the scalar radical locus in as many points as its abstract •quation has dimensions, we have the following fundamental proposition of the theory of transversals :—

“ If we take the points at the extremities o f a side o f a polygon, md find the ratios of the distances o f these points from the intersec- iions of that side with a curve, and proceed in this manner regularly round the polygon in one direction, the product of all these ratios will be unity.”

10. Thus if the triangle O ^ O g cut a straight line in A, B, C so that these points lie on 0 2Oa, C^Og, OjOj respectively, then

OjC OgA OgB OaC 0 3A * O.B T*

If the same triangle C^OjOg cut a conic in the points A v A2 by 0 20 3 by 0 10 3 ; Cp C2 by 0 X0 2 respectively, then

9 f i . 2 f i v O A . o A O A o ,bO .c, 0 3Ca x 0 SA1 O„Aa x 0 A O . B , - 1*

Plucker says, = ± 1 , and adds, that “ it is immediately evident that when the curve is of an even order, the upper sign must be taken ; and when it is of an meven order, either the upper or the lower, according as the number of the angles t the polygon is even or odd” (System, p. 44). Accordingly most of his exam- * ®hown m (10)’ *ive an erroneous result. It must be observed that

dn* uTu "0t giVC the equati0n in the form above b itten , but places the pro- con pn a x e aatecedents of the ratios on one side, and the product of all the direJJv * 0n f ° ther* As’ however’ these antecedents and consequents are those , Pr°dUf 8 haVC no Pr°Per meaning. He has in fact consideredwith resneet + f s°alar8’ but bas neglected to assign the directed unit lines

espect to which they are to be considered as scalars. Hence his error.

e r r o n e o u fV f f n he *** °f *his equation ^ - l . which is manifestly0 ,0 o oBUo n ° ia’ °**’ ° sC bC draW“ fr°m ° lf ° 2’ °3 through any point M

angles o n A A * resPe<*lve,y ' then by considering the intersections of the tri- the equation^ the text^ ° 3° ^ respectively» we immediately deduce from

Oj£ O a 03g_Osc * 08a ‘ O J~



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I f the triangle touches the conic in A, B, C, then Ap A2 coincide with A, &c.j and the above equation becomes

0,C 0„A OsB :1 or — 1.0 2C 0 3A OxB

The first case shows that the three points lie on a straight line; the second, that the three lines C^A, 0 2B, 0 3C meet in a point*.

I f the point C were to lie at an infinite distance from Ox and 0 2,

then since O = 0 ^ + 0 ^ , and 5 £ = 1+ where the lastU2'~'

o cterm is infinitesimal, we have merely to consider --i—- = 1, or omit itU2v

from the product f . Thus if we take as two sides of the triangle the asymptotes of an hyperbola meeting at Ox, and suppose the third side 0 20 3 to cut the curve in Ax and A2, then since both 0 j0 2 and 0 j0 3 cut the curve at an infinite distance, the equation reduces to

^ 2 ^ 1 ^2^2 .o A ' o 3a 2 ’

whence we readily find

^3^2 ~ Qj A2, or 0 2Ai.

The first value is impossible, because A2 lies between 0 2 and 0 3; the second gives the well-known property of the hyperbola £.

11. As regards the clinant roots, or the intersections of the sides of the polygon with the clinant radical loci, the wording of the proposition (9) requires to be changed to adapt it to the more complicated form of the product; and perhaps it may he considered suffir cient to refer to the forms of the products in the final equation of (8),

* Pliicker interchanges the two cases, and makes the positive result show the intersection of the three lines in a point, which is shown to be wrong in the last note. He rejects the first case, “ because a conic can only be cut in two points by a straight line ” (p. 45). But in this case both the conic and the triangle reduce to a straight line (that is, the points Oj, 0 2, 0 3, A, B, C all lie in the same straight line), and the points A, B, C do not indicate contacts, but double-intersections, which have the same analytical expression, owing to the rejection of infinitesimals in the above calculation of contact.

t Pliicker says that “ in every such case there must be a fresh change of sign (p. 46). This apparently arises from his having neglected to particularize the directed unit, or to note the primary relation 0 1C = 0 10 2-j-02C.

+ Pliicker makes 0 2A1= 0 3A.2 (p. 46); that is, he neglects the relation of direction, which is all-important in such investigations.



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and append a single simple example in which the product admits of being easily formed.

Let the triangle 0 10 a0 3 touch a circle BC in the points B and C, and let the side 0 20 3 be perpendicular to the line 0 L0 drawn through the centre of the circle, O, but be wholly exterior to the circle. Take 0 as the origin of coordinates, and suppose that the equations to the circle are

O P=a?. O I-py • OJ,Now change the origin to Of, where 0 0 f= ^ 0 30 2= —5 . OJ. Then the equations to the circle become

O 'Psstf'.O I + .O J ,

And for the radical loci, when y{ is clinant, we have (3)0 ^ = ^ ' . OI + rf . OJ, r1—5—0,

that is, the line 0 10 , andO 'S ^ ' .O I + s ' .O J , *'2- * ' 2= e 2,

that is, the rectangular hyperbola Sfx S'2. The roots will then be

OaR + i . 0 2S'X _ 0 2R + R M \ _ 0 2M \OJ ~ OJ OJ

andOaR + i . 0 2S'2 _ 0 2R + R M '3 _ 0 2M'2

OJ “ OJ OJ Jand the factors corresponding to them in the final product in (3) will be

0 20 3- 0 2M \ 0 20 8 - 0 2M's _ M^Os M f20 3 OJ ‘ OJ OJ OJ



400

Similarly, by transferring the origin to O", where 0 0 " = 6 .0 J , the equations to the circle become

0 " P = * " .0 I + y " . OJ, x"2 + (y" + b y = c\The radical loci, when y" is clinant, will have for their equations

0 " R = > ' .0 I + r " .0 J , > " + 6 = 0 , that is, the line OjO, and

0"S=*"> 0 1 +s".OJ, *"2- * " 2= c2, that is, the rectangular hyperbola S"x, S"a. Hence the roots will be

Q3R + i . Q3S"X _ O aR +R m ', 0 3m'xOJ o J “ "OJ"

and. 0 3R + i • OaS"2 __ 0 3R + R 0 3m',

o j ■ — o t — = - o r ;and the factors corresponding to them in the product (5) will be, after replacing OJ by JO, as there shown,

Q3Q2~ Q 3mfi 4 Q3Q2- Q 3m f2 _ m\ o 2 m\ 0 2JO JO JO * JO '

Hence the final equation in (8) gives

( 9 £ \ i . ° 3m '2\ A WVO,C/ VO 2m'2 0 2m ' J ’ { o p ) - L-

hiow the second factor of the product on the left-hand side of this equation, is itself a product of which each factor = — 1, and it may consequently be omitted from the equation, so that

0 XC _ , CXB OsC ± OaB*

The under sign cannot be taken, because it would necessitate one of the points B or C lying without, and the other within the corresponding side of the triangle. Hence we must have

0 i C _ 0 1B .OaC 0 3B ’

that is, BC will be parallel to 0 3Oa, a result which readily follows from other considerations.

12. The great complexity of the process, and the necessity of knowing the algebraical form of the equation to the curve, will probably prevent this employment of clinant roots from having any practical value beyond the completion of the theoretical harmony between the abstract and the concrete, the algebraical formula and the geometrical figure.



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