THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES "his b ;he SOUTHERN BRANCH, UNIVERSITY OF CALIFORNIA, LIBRARY, LOS ANGELES, GALJF. THE DIRECTIONAL CALCULUS, BASED UPON THE METHODS OF HERMANN GRASSMANN. BY E. W. HYDE, PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CINCINNATI, CINCINNATI, OHIO. 48807 BOSTON, U.S.A.: PUBLISHED BY GINN & COMPANY. 1890, COPYRIGHT, 1890, BY E. W. HYDE. ALL RIGHTS RESERVED. TYPOGRAPHY BY J. 8. CUSHING & Co., BOSTON, U.S.A. PHESSWORK BY GINN & Co., BOSTON, U.S.A. Engineering & Mathematical Sciences Library QA a THE 4, Mechanics, should be generally introduced to the knowledge of the coming generation of English-speaking mathematicians, PREFACE. wonderful and comprehensive system of Multiple Algebra invented by Hermann Grassmann, and called ^ by him the Ausdehnungslehre or Theory of Extension, though long neglected by the mathematicians even of Germany, is at the present time coming to be more and more appreciated and studied. In order that this system, with its intrinsic natural- ness, and adaptability to all the purposes of Geometry and 1 V it is very necessary that a text-book should be provided, suitable for use in colleges and universities, through which students may become acquainted with the principles of the ^V subject and its applications. The following pages present, in part, the results of eight or nine years of study and experience in lecturing to university classes upon this subject, and the Author hopes that they may aid in some measure to bring about that general study and use of directional methods, which, in his opinion, will ultimately cause the comparatively awkward and roundabout methods of Cartesian coordinates to be superseded by them, for many of the purposes of analysis. As the great generality of Grassrnann's processes all results being obtained for n-dimensional space has been one of the main hindrances to the general cultivation of his iv PREFACE. system, it has been thought best to restrict the discussion to space of two and three dimensions. The Author, though formerly an enthusiastic admirer of Hamilton's Quaternions, has been brought, by study and expe- rience in teaching both, to a firm belief in the great practical, as well as theoretical, superiority of Grassmann's system. This superiority consists, according to the judgment of the writer, first, and largely, in the fact that Grassmann's system is founded upon, and absolutely consistent with, the idea of geometric dimensions. Second, in the fact that all geometric quantities appear as independent units, viz. : the point ; the point at oo , or line direction ; the definite line ; the line at oo , or plane direction ; the definite plane ; and, finally, the plane at oo, equivalent to a volume, which is scalar. The same holds for space of any number of dimensions ; in fact, it seems scarcely possible that any method can ever be devised, compar- able with this, for investigating w-dimensional space. Now Quaternions deals only with the vector, or line direc- tion, and the scalar for a quaternion is only the sum of these two; it knows nothing of a vector having a definite position, which is the complete representative of the space qualities of a force. Further, Hamilton's vector is not a vec- tor pure and simple, but a versor-vector a fact which gives its peculiar character to his system, as being really a calculus of directed imaginaries. This, which Tait regards as " one of the main elements of the singular simplicity of the Quaternion Calculus," appears to the writer in a very different light. It gives rise to such equations as * = ji, i.e. multiplying two quantities together is the same as dividing one by the other, which the author has found a great stumbling-block to the student. It can hardly be regarded as a natural geometrical PREFACE. V conception, that the product of two vectors should be another vector perpendicular to each of them ; still less, that it should be, as in the general case, a scalar plus such a vector. In fact, the Quaternion system practically throws overboard the idea of geometric dimensions. It has been deemed advantageous, however, to make use of certain terms and symbols introduced by Hamilton, such as scalar, tensor, with its symbol T, etc. Although this work is based upon the principles and meth- ods of Grassmann, yet much matter will be found in it that is believed to be original with the author. Thus the idea of the complement in a point system, with its geometric interpretation, as developed in Arts. 40-44 and 63, does not occur in Grassuianu's works, either of 1844 or 1862. This idea is of great value in geometric applications, and cor- responds to that of duality in Modern Geometry. In Arts. 68 and 69 are treated some properties of what I have called screivs, which, as such, are not discussed by Grass- mann. One of the most beautiful and valuable theories developed by Hamilton in his great works is that of linear and vector functions. This theory I have found to be equally capable of application to point functions in. n-dimensional space. In Chapters IV. and VI. this linear, point function has been used in the discussion of the conic and quadric, giving rise to a treatment of these loci bearing the same analogy to trilinear and quadriplanar methods that the vector treatment of these curves and surfaces bears to ordinary Cartesian methods, and possessing likewise the same superiority in clearness, concise- ness, and intelligibility which the vector methods have over the Cartesian. VI PREFACE. It appears to the writer that Hamilton's method of dealing with linear functions much excels that of Grassmann, as devel- oped in the second part of his work of 1862, for practical util- ity and convenience. It is hoped and believed that the exposition of the funda- mental ideas and principles of the system in the first two chapters will be found sufficiently full and explicit to give the student a working knowledge and grasp of the subject, such as will enable him to take up without difficulty the work of the succeeding chapters, or to read, with comparative ease, the original treatises of Grassmann. A large number of exercises have been inserted, in the belief that only by the repeated application of the principles which he has learned to the solu- tion of actual problems can the student acquire any real com- mand of any branch of Mathematics. Eight or nine blank pages will be found at the end of each chapter, for the recep- tion of notes, solutions, etc., the Author's experience having convinced him of the advantages of such an arrangement. E. W. HYDE. TABLE OF CONTENTS. CHAPTER I. ADDITION AND SUBTRACTION. PAGE Definition of vector, plane- vector, etc 1 Equality 1 The point, equality of points 2 Difference of unit points 2 Sum and difference of vectors 3 Addition of points 4 Case in which 2m = 5 Conditions that two vectors may be parallel, and that three may be parallel to one plane 7 Dependence and independence 8 Tensor, unit, reference systems 9 Exercises 10 CHAPTER II. MULTIPLICATION. Fundamental notion of a geometric product 23 Posited quantities 23 Definitions of combinatory products 24 Laws of combinatory multiplication 25 Product of two points 25 Product of two vectors 20 Product of three points 27 Product of three vectors 28 Product of four, or more, points, or vectors 29 Equations of condition, planimetric products 30 Product of points expressed in terms of reference points 31 Planimetric product of two lines 32 Product of a point and two lines 33 Viii TABLE OF CONTENTS. PAGE Conditions L^ = and L^L.^ = 34 Product of two || lines and of two vectors 34 Sum of point- vectors 35 Complement 36 Co-product and co-square 37 Complement in a plane point system 39 Geometric interpretation of the same 40 Distance between two points 41 Anti-polar property of complement 41 Projections 43 Multiplication table for a point system 44 Division indeterminate 49 Exercises 50 Stereometric products 56 Products of points expressed in terms of reference points 56 Product of a line and a plane 57 Product of two planes; of three planes 58 Product of four planes; products of plane-vectors 59 Equations of condition 60 Addition of planes and plane- vectors 60 Addition of point-vectors, or lines 62 The complement in three-dimensional space 62 Complement in a point system in three-dimensional space 64 Condition that a linear function of the edges of the reference tetrac- dron shall be a point-vector 65 Point system in solid space a fifteen- fold algebra 66 Projections in a vector system 67 Projections in a point system 68 Normal form of the screw 71 Product of two screws 72 Products Sfi Sp 2 and Sp l Sp t Sp 3 73 Quaternion expressions corresponding to some expressions of this calculus 73 Exercises . . 74 CHAPTER III. APPLICATIONS TO PLANE GEOMETRY. Preliminary statements 87 Equations of plane space 88 Equations of right lines 89 TABLE OF CONTENTS. IX PAGE Transformation from a point to a vector system, and vice verscl 90 Exercises 91 Equations L 1 p L. 2 p = C and p 1 L p t L = C 92 Conies through three points, and touching three lines 93 Differentiation 94 Examples of differentiation 96 Tangent and normal. The circle 97 Exercises on the circle 98 The parabola 99 Exercises on the parabola 101 The ellipse and hyperbola 102 Tangent and normal. Diameter 104 The
function 114
The invariants m a,nd m, 116
Exercises on inversion 117
The general equation of the second degree in plane space 117
Center of the locus 118
Case when tj>\ i to
p 2 , we write
P-Pi = e, ......... (1)
and the development of the subject will show this to be in
accordance with the above proviso.
CHAP. I.]
ADDITION AND SUBTRACTION.
3
If p 2 pi = e = 0, or p 2 = PI, there is no difference of position
between the points ; i.e. they coincide, as previously stated.
Adding p l to both sides of (1), we have
in which, of course, the meaning of the sign + is extended in
a manner similar to that of the minus sign.
Eq. (2) shows that the sum of a point and a vector is a point,
distant from the first point by the length of the vector, and in
the direction of the same.
6. Let e,, e 2 , e s be any three unit points, and e 1? e,, c 3 vectors
|| and equal in length to the sides of the triangle e-f,^ taken in
order, so that cj = e 3 e 2 , c 2 = &i e s , e 3 = e 2 e t : then
that is, the sum of the vector sides of a triangle, taken posi-
tively in the same sense around the triangle, is zero.
We have also
so that the sum of two vectors is a
vector which represents the differ-
ence of the initial and final posi-
tions of a point which moves along
the two vectors successively.
Write e/ = e 2 - e s = - Cl ;
then / 2 = e 2 e. A Ci-\-e 3 = e 2 ei = e s j . . . (4 a)
that is, the difference of two vectors is the vector joining their
extremities when they are drawn outwards from a common
point.
Similarly, if we take n unit points e 1} e 2 , e n , we have
- 3 -i 1- e n - e n _ = e n - e 1 ;
that is, the sum of any number of vectors is the vector repre-
4 DIIIECTIONAL CALCULUS. [ART. 7.
senting the difference between the initial and final positions of
a point which moves along all the vectors in succession. The
result is evidently independent of the order in which the
vectors are taken.
7. Addition of points. In order to interpret the meaning
of such an expression as m^p-i + m 2 p 2 + etc. = 2,mp, we will
assume,
1st. That the sum of any number of points will be itself a
point, which we shall call the mean point of the system.
2d. That the weight of the sum will be equal to the algebraic
sum of the weights of the points.
These assumptions are allowable if they lead to no contra-
dictory or inconsistent results. In accordance with them we
write for two points,
Transposing, we have
*(Ps-p) = mi(p-Pi), ........ (6)
which shows that p is on the straight line
joining p l and p^ at distances from these
points inversely proportional to their weights. p
Let e be any point whatever, and subtract from both sides of
(5) (mt + mde;
-' m>i(pi e) + m 2 (p 2 -e) = (m,i + m 2 ) (p e),
1 1 vi / \ , 7/fco / \ / ~ \
or p- e = - -( Pl - e ) + - _(#,_), . . (7)
m l + m 2 mi + ^2
by which p is easily found.
For three points we have
m 1 pi + m 2 p 2 + m s p s = (m 1 -\-m 2 -\-m 3 )(p)', . . (8)
or, by (5),
(?n,! -f- m 2 )p + m s p 3 = (w^ + m. t + m z } (p) ' ;
CIIAF. I.] ADDITION AND SUBTRACTION. 5
so that (j>) ' is on the line joining p and p s at distances from
them inversely proportional to their weights.
Subtracting from both sides of (8) (m 1 + m 2 + m 3 )e, we
have, on dividing by 2w,
>-v, 1l - 2 -e) + m s (p s - e)
(P) 6 = - - -
by which (p) ' is easily constructed.
Similarly, for any number of points we have
p"2m, ..... ........ (10)
and p e = 2[m(p e)] ......... (11)
2m
The reader will notice the analogy to "center of parallel
forces."
8. Case in which 2m = 0. In (5) and (6) let
m l + ni 2 = 0,
id these equations become
aid jp Pa=.p Pi ............. (13)
If m 2 be not zero, and jp a not coincident with p 2) these equations
can only be satisfied when p is at an infinite distance on the
line pip 2 - Eq. (6) shows at once that, when m a is negative,
p is not between pi and p 2 , and that, as the numerical value of
m 2 approaches that of m^ p recedes farther and farther, until,
when mi + m 2 = 0, j? is at GO. Thus the meaning of eq. (12)
is, that a point of zero weight at an infinite distance is equivalent
to a vector of definite length directed towards this point. It will
often be convenient to regard vectors as points at QO.
Consider next the general case of n points. Eqs. (10) and
(11) become, when 2^ = 0,
and 2"[m(p e)] =0 (p e).
6 DIRECTIONAL CALCULUS. [ART. 8.
Let
(pY^in = m'(p)', say:
then, since
mj + -m' = 2"?n = 0,
+m'(p)'= m 1 ((p)' -p v ) = 0-p,
and 2m-e=m 1 p 1 -e)+
Hence the direction of the point at co is found by constructing
the mean point of all the points except one, when the vector
from the excepted point to this mean point has the required
direction. As p is a unique point for any given system of
weighted points, and p t may be taken as any point of the
system, it follows that, in any system of points whose total
weight is zero, a line drawn from any point of the system to
the mean point of all the rest is parallel to any other such
line.
Finally, suppose 2mp=p-2m = 0, ...... (16)
which gives as a necessary consequence 2m = 0. Then, by
(15), if m' is not zero, we must have
that is, P! is the mean point of the system consisting of the
remaining points. But, as p l may be any point of the system,
it appears that any one of the points is, in this case, the mean
of all the rest. For example, let n = 3 ; then (16) becomes
m iPi + W 2 p 2 + m s p s 0, which requires also m t + m 2 -f- m 3 = 0;
whence
~ m s p s = (?rcj
or p s is the mean point of m^ and m 2 p 2 . Now we have seen
in Art. 7 that the mean of two points is collinear with them ;
hence
Q ...... (17)
is the condition that the three points shall be collinear.
CHAP. L] ADDITION AND SUBTRACTION. 7
If three points p l} p z , p 3 are not collinear, and yet are con-
nected by a linear relation such as (17), this equation can only
be satisfied when each iveight is separately zero, that is
mi = m 2 = m 3 = ; ........ (17 a)
for, as we have just seen, for all values of the weights different
from zero the three points are collinear, but when each weight
is zero, (17) is satisfied independently of the positions of the
points. %
Similarly, m\p\ + m 2 p 2 + m s p 3 + m 4 ^4 = 0, ... (18)
which requires also m l + m z + m z + m^ = 0, is the condition
that four points shall be coplanar, and if the points are not
in one plane, then we must have
M! = m 2 = m 3 = m 4 = ....... (19)
A similar condition between Jive points would imply that
the points were all in one tri-dimensional space, which leads to
the consideration of space of higher dimensions than three.
9. Let cj and e 2 be two vectors, and 7^ and n 2 scalars ; then
e 2 =0 ......... (20)
is the condition that the vectors shall be parallel, as appears at
once by Art. 2, if we write the equation n^ =
Similarly, n^ + w 2 e 2 -f n s e s = ....... (21)
requires the three vectors to be parallel to one plane; for, by
Art. 6, the vectors n^ n 2 c 2 , ri 3 e 3 , may be represented by the
three sides of a triangle. The same appears from (17) if we
take pu p 2 , p s as points at GO.
If we have the additional condition
(22)
the extremities of the vectors, if drawn outwards from a point
will be in one right line. For let e , e^ e 2 , e s be four points
8 DIRECTIONAL CALCULUS. [Aux. 10.
so taken that e 1 e e 1 , ee =c.,, e 3 e = e 3 ; then (21)
becomes
rti(gj e ) + n 2 (e, e fl ) + n s (e 8 e ) = 0,
or, by (22), n^ -+- n 2 e 2 + n 3 e 3 = ;
which, by (17), requires e { , e 2 , e s to be collinear.
Similarly, it may be shown that the eqs.
=
sn 4 4 = -j
?i 4 = I
are the conditions that the extremities of the vectors tj, e 2 , etc.,
drawn outwards from one point shall be coplanar.
10. If the equation Swip = can be satisfied by finite values
of the ra's different from zero, then, as we have seen, any one
of the points can be expressed in terms of the others, and is,
therefore, dependent on them ; thus the points are mutually
dependent. If, however, the equation can only be satisfied
when all the m's are zero, no relation of this kind exists be-
tween the points ; i.e. they are independent. This is evidently
as true of vectors as of points.
Thus, by Art. 4, two different points are always independent.
Three non-collinear points are independent, while three col-
linear points are mutually dependent.
Four non-coplanar points are independent, while four coplanar
points are mutually dependent.
Any five, or more, points are always mutually dependent.
Similarly, two non-parallel vectors are independent, but two
parallel vectors are dependent.
Three vectors not parallel to one plane are independent, but
when all are parallel to one plane they are mutually dependent.
Four, or more, vectors are always mutually dependent.
In a system of independent points or vectors no one can be
expressed linearly in terms of the others.
CHAT. I.] ADDITION AND SUBTRACTION. 9
11. The tensor of a directed quantity is its numerical magni-
tude, taken always as a positive quantity. It will be denoted,
as in Quaternions, by T ; as, Te. = tensor of e.
That portion of a directed quantity whose magnitude is
unity will be called its unit, and will be denoted by U; as,
Ue = unit of e.
Hence we have
Te Ue = e . (24)
12. Reference systems. Let p be any point, and e , e 1} e 2
three fixed reference points ; then writing
p = xe + ye! + ze 2 , (25)
we must have, if p is to be a unit point,
x + y + z=l, (2G)
and p is the mean point of xe , ye and ze 2 . Eliminating x
between (25) and (26), we have
P = e + y(ei-e l) )-\-z(e 2 -e ) ) .... (27)
from which it appears that, by varying y and z, p may be made
to occupy any position whatever in the plane of e , e,, e 2 .
Hence any 1 three unit points e , e 1? e 2 may be taken as a refer-
ence system for plane space, in terms of which all poifTts of
this plane space may be expressed.
Writing p e fl = p, e, e = e l5 e 2 e = e 2 , eq. (27) becomes
P = yi + e (28)
and any vector in plane space may be expressed in terms of
two reference vectors c^ e 2 in that plane. When ej and e 2 are
of unit length and at right angles, the system will be called a
unit normal reference system, and ^ and i 2 will be substituted
in this case for cj and e 2 .
Similarly, in solid space any point p may be expressed in
terms of four non-coplanar points e , e 1} e 2 > ^ by the equation
p = we + xe l + ye 2 + ze 3 , (29)
10 DIRECTIONAL CALCULUS. [ART. 13.
Avhich requires, if p is a unit point,
w + + y + z = l, ......... (30)
whence p = e -\-x(e l - ) + y(e s - e ) + z(e s e ). (31)
Thus any four unit points may be taken as a reference sys-
tem for solid space.
Putting, as before, p =_p e , cj = ej e n , etc., (31) becomes
P = a*i + yg + za, ......... (32)
so that any vector may be expressed in terms of three given
vectors e^ e 2 > 3- When Tcj = TV, = 2Vs = 1> and the three vec-
tors are at right angles to each other, we have a unit normal
system, and, in this case, substitute t 1? t 2 , i 3 for e l9 e 2 , c 3 .
13. A number of exercises will now be given illustrative of
the application of the preceding principles.
(1) The mean point of unit points at the vertices of a tri-
angle coincides with the center of
gravity of its area. The center of
gravity of the area will be at the
common point of lines through the
vertices and the middle points of
the opposite sides. Thus, if p be
the point,
2 (e 2 +e 3 ) =x'e. 2
.-. (x - iy'
by eqs. (17) and (17 a) this gives
* - iy' = %y x< = y y' = Q.
But, since p is a unit point, we have
x + y = 1 = x' + y'.
From these equations we find
x = x' = i, 2/ = y = |;
whence p = $ e l + |p, = 1 (e l + e 2 -f e 3 ),
CHAP. I.] ADDITION AND SUBTRACTION. 11
which shows that p is the mean point of e\, e 2 , e s , and trisects
the distance from e l to p^.
(2) To find the common point of the perpendiculars from
the vertices of a triangle on the opposite sides.
Let Z, m, n be the ratios of the
sides of the triangle to the cosines
of the opposite angles ; i.e.
m/ _ ~ \
-, etc. ;
cos(Z at
m0 -I- 77.1
then
m + 1
_ ne 3 + fcj _ Ze t + me.,
n-\-l I -+- 7)1
Proceeding now precisely as in the first problem, with the
above values of p^ p 2 , p s instead of those there used, we find
, _le L + me, 4- ne x
l + m + n
(3) Find the common point of perpendiculars to the sides
of a triangle through the middle points of the respective sides.
By the figure for the last problem we have
i" Ifo _1_
P" = S(e l +
l + m + n
m + n
Hence, equating to zero coefficients of e l and e 2 ,
y (m + 71) _ __ 1 a
l + m + n I + m + n 2
|
2
.: x=y=-, and p" =
12 DIRECTION, AL CALCULUS. [ART. 13.
(4) Show that p, p', p" of examples 1, 2, 3 are collinear.
It is easily seen that 3p p' 2p" = 0, which, by (17),
proves the collinearity of the points. It is evident also that
p, p', and p" are collinear whatever values be assigned to I, ra, n ;
for nothing in the demonstration depends on I, m, n having
the values assigned in ex. 2.
(5) The center of gravity of the volume of a tetraedron
coincides with
(a) The mean of unit points at the vertices ;
(6) The center of gravity of the tetraedron whose vertices
are the centers of gravity of its faces ;
(c) The mean of the middle points of its edges.
(G) Find a point such that the sum of the vectors drawn
from it to n given points shall be zero.
Ans. p = -2te = mean of points.
n
(7) Construct by eq. (11) the mean point of points at the
four corners of a square, whose weights are respectively
1, 2, 3, and 4.
(8) Let e 1} e 2 , e 3 , e 4 , be the corners of a parallelogram, and
e & = %(d + e 2 ) : show that e^ and e 4 e 5 will trisect each other.
(9) Let 6j, e 2 , e s , e 4 be the corners of a parallelogram, and let
p l} p 2 be points on a line || to e^ : if q is the common point of
P& and p 2 2, and q 2 the common point of p^ and p 2 e s , show
that g,g 2 is || to e^.
(10) Let 1, e 2 , e s , e 4 be four non-coplanar points, and let
Pi, Pv Ps> P* be points taken on e^, e 2 e 3 , e^ and e i e l respec-
tively: find the condition to be fulfilled -in order that p^p 2 ,
6163, and p s p 4 may have a common point.
Write p l ra^ + w,e 2 , p 2 = m z e 2 -\- n 2 e s , etc. ; then the p's
must be coplanar, which leads to the condition
CHAP. I.] ADDITION AND SUBTRACTION. 13
(11) If through any point within the triangle e^e-fa lines be
drawn || to the sides and terminated by them, and if I, m, n, be
the respective ratios of these lines to the sides to which they
are || ; then I + m + n = 2.
(12) The center of gravity of the sides of a triangle coin-
cides with the center of the circle inscribed in the triangle
formed by joining the middle points of the sides.
(13) Find the center of gravity of the faces of a tetraedron ;
also of the edges.
CHAP. II.] MULTIPLICATION. 23
CHAPTER II.
MULTIPLICATION.
14. Grassmann's first conception of a geometrical product is
that it is what is produced or generated by the first factor, as
it moves over a distance determined by the second.
Thus ejea, if ci and t% are two vectors, signifies the directed
plane area bounded by the parallelogram
whose sides are \\ and equal in length to
cj and 2 ; that is, the plane area gener-
ated by EJ as it moves, || to itself, along
e 2 from its initial to its final point. EI
This is a plane-vector as defined in Art. 1. The product will
evidently have the same value whether the initial point of c x
moves in the direction e 2 or in any other path, provided that tj
itself moves in the plane of cj and e 2 from the initial to the
terminal point of e 2 .
Evidently e 2 j, interpreted in the same way, gives a genera-
tion of the same parallelogram in the reverse sense, and should
therefore be the negative of e^
Similarly, if p l and p 2 are two unit points, p t p 2 is that which
is generated by pi in moving from its position to that of p 2 in
a right line ; thus prf 2 is a line of definite magnitude, direc-
tion, and position ; i.e. a point-vector, according to Art. 1.
Evidently p 2 pi = Pip*
Such products as the above are called combinatory, because
the factors combine to form a new geometric quantity different
from either of the component factors.
15. The term posited will be applied to such geometric
quantities as have definite positions ; as, for instance, a right
line or plane passing through a definite point not at oc.
24 DIRECTIONAL CALCULUS. [ART. 16.
Any two geometric quantities which differ only in magnitude
are said to be congruent.
16. Another conception of combiuatory products of posited
quantities will now be given, which will be found to be con-
sistent with that given in Art. 14.
(a) The product of two posited quantities which have no com-
mon figure is some multiple of the connecting figure.
(b) TJie product of two posited quantities ichich MUST have a
common figure in the space under consideration, is the common
figure, multiplied by a scalar quantity.
EXAMPLES. Under (a), the product of two points is the
connecting straight line as in Art. 14.
Under (6), the product of two point-plane-vectors, which
must necessarily have a common line, is that common line
(point-vector) multiplied by a scalar to be determined here-
after.
(c) A continued product of several posited quantities is to be
interpreted by taking together first the two factors on the right,
then the result of this multiplication with the next factor towards
the left, then this result ivith the next, etc. ; but if at any stage of
the process the product of the factors treated up to this point
becomes a scalar quantity, then this scalar will not form a COM-
BINATORY product with the next factor to the left, but is to be
treated like any other merely numerical factor, obeying as a
WHOLE the laws of ordinary algebraic multiplication.
This statement cannot well be illustrated here, but its mean-
ing will appear in the sequel.
Grassmann calls a product of the kind (a) progressive, be-
cause it is of a geometric order higher than that of either
factor ; while one of the kind (6) is regressive, because it is
of lower order than either factor. If in a continued product
of factors, as in (c) , some of the successive products are pro-
gressive and some regressive, then the product as a whole is
said to be mixed.
CHAP. II.] MULTIPLICATION. 25
17. Lairs of combinatory multiplication of any number of
points or vectors not exceeding the number of INDEPENDENT
points or vectors possible in the space under consideration.
These are :
The associative law ;
PlP*Ps=P\'P*P* = PlP*'P* (33)
The distributive law ;
Pi(pt+Ps)=PiPi+PiPs (34)
The alternative law ;
PiP 2 = P 2 Pi (35)
If p. 2 = p 1 in (35), we have pip\ = Q, which agrees with the
meaning of the product of two points as given in Art. 14.
Also pMPi = Pi p 2 pi = -Pi- PiP 2 = - PiPi P 2 = 0,
\)j (33) and (35) ; hence, if there occur tico identical factors in
a product of the kind stated at the head of this article, the product
is zero.
The meaning of certain products will be different according
as they are interpreted in two- or three-dimensional space ;
hence we shall apply the term planimetric to such a product
when it is to be taken in two-dimensional space, and the term
stereometric when it is to be taken in three-dimensional space.
We proceed now to a detailed discussion of all the geometric
products possible in two- and three-dimensional space.
18. Product of tico points. Let p l and p 2 be two unit points.
Their product, Pip^ has already, in Art. 14,
been stated to be the portion of the right line Pl 'ft
fixed by p l and p 2 extending from PI to p 2 , and
it will be called a point-vector, or simply a line for brevity.
Let f=p. 2 p 1 ; then
Pi(P*-Pi)=Pi* = PiPs-PiPi=PiP - (36)
by the last article ; so that the product of two points is equiva-
lent to that of the first point into the vector from the first to
26 DIRECTIONAL CALCULUS. [ART. 19.
the second. Hence multiplying a vector by a point changes
it into a point- vector ; i.e. fixes its position. Let e' be any
vector whatever; then, by eq. (2), p l + xe' may be any point
in space by suitably choosing x and e : multiply into e ;
therefore
Hence the point-vector obtained by multiplying the vector e
by the point p t + xe' is not in general equal to pje, but differs
from it by the quantity xe'e. If, however, we have e' = e, then
p l ' i ..... (38)
so that, when c is multiplied by any point on the line through
Pi and p 2 , the resulting point-vector is equal to p^.
Hence, in order that two point-vectors may be equal, they
must have the same length, the same direction, and must be
situated upon the same straight line, while their position on
this line is indifferent.
19. Product of two vectors. Let ^ and e 2 be any two vec-
tors : their product e^ has already, in Art. 14, been stated to
be a plane area || to ej and e 2 and equal to the area of the par-
allelogram whose sides are parallel and equal in length to c t
and e 2 . Write
and c' =
then c and e' may be any two vectors || to the plane-vector e^
by giving suitable values to 1} x 2 , y lf y 2 . Multiplying, we have
* (39)
Hence the product of any two vectors || to Cjc 2 only differs
from cjCjj by a scalar factor. If x$ 2 xfli = 1, then te' = e^.
Thus in order that two plane-vectors should be equal, it is
only necessary that their plane directions and areas should be
the same, without regard to the directions of the component
vectors. Of course direction as applied to a plane-vector must
CHAP. IT.] MULTIPLICATION. 27
include the sense in which the generation takes place. See
Art. 14.
Regarding ej and c 2 as points at oc, we see that, just as a
zero point at oc is a vector, so the product of two such, that is
a point-vector at oo, is a plane-vector ; or, in other words, just
as a point at oo gives a line direction, so a line at oo gives a
plane direction.
20. Product of three points. Let p t , p% p 3 be three unit
points. By Art. 16, (a) and (c), the product should be a
multiple of the connecting tri-
angle whose vertices are p^ p^ p s .
By Art. 17 the product obeys the
associative law, so that
Hence, by Art. 14, the product is what is generated by the
point-vector p t p 2 in moving, || to itself, in the plane of the
three points, from its original position, till it passes through
p s ; that is, & parallelogram whose area is twice that of the con-
necting triangle. Let p. 2 p\ = c, and p 3 p l = c' ; then
PiP*Ps=Pil>*'=Pia' (39)
The product is thus a posited and directed plane area of given
magnitude; that is, a point-plane-vector, or simply a plane for
brevity, especially when the magnitude is a matter of indiffer-
ence. Eq. (39) shows that multiplying a plane-vector by a
point fixes its position by making it pass through the point,
since by Art. 14, ee' is a plane- vector. Let e" be a vector in
any direction ; then, as in Art. 18, p^ + are" may be any point
in space. Xow
(?! + *")' =;?,' +o*"c f ; (40)
so that the point-plane-vector obtained by multiplying ' by
the point p l + a*" is, in general, different from p^t'. If, how-
ever, we have e" = ye. + 2e', so that e" is || to ct', and p l + a*" is
a point of the plane p\p. 2 p^ then
ee' = ( Pl + x(y e + '))'=l>i'- (41)
28
DIRECTIONAL CALCULUS.
[ART. 21.
Hence two point-plane-vectors are equal when they have the
same area and direction, or sense, and lie in the same plane,
without regard to position in that plane.
21. Product of three vectors. Let
vectors not || to the same plane ; then
Art. 14, the product is the parallele-
piped generated by the plane-vector
e^a as it moves, || to itself, from the
initial to the terminal point of e 3 . p
Let e =
c 2 , c 3 be any three
= (.& c 3 , and, by
then e, e', e" may be any three vectors whatever with suitable
values of the scalar coefficients, and
1 2 3
2] 2 2 #3
(42)
Hence two triple products of vectors can only differ in magni-
tude and sign, and two such products will be equal when their
magnitudes and order of generation, or sense, are the same.
It follows therefore that the combinatory product of three
vectors is always a scalar quantity, by the definition given in
Art. 1. If the three vectors are parallel to one plane, the
volume of the parallelepiped becomes zero, and the product
therefore vanishes. Hence the planimetric combiwxtory product
of three vectors is always zero. If we regard e 1? c 2 , c 3 as points
at oo, we see that a point-plane-vector at GO is equivalent to a
solid, which carries out the analogy mentioned at the end of
Art. 19.
* For the benefit of any reader who may not be familiar with deter-
minants, it may be stated that the coefficient of eje.^ in (42) is an ab-
breviated way of writing
which expression will be obtained by multiplying out the values of
and remembering that terms containing repeated factors vanish.
CHAP. II.] MULTIPLICATION. 29
22. Product of four points. Let p^ p& p^, p^ be four unit
points ; then, by Arts. 17 and 14, we have
and the product is the volume generated by the point-plane-
vector 2hP*P3 when it is moved || to itself from its initial posi-
tion till it passes through p 4 ; that is, & parallelepiped, of which
Pip?, p\Pv and pp+ are three conterminous edges. This volume
is six times the connecting tetrahedron of the four points,
which accords with Art. 16. (See figure of last Art.) If
Pa Pi = <> Pa Pi= '> and p 4 p l = e",
then PiP*P^=PiP*P&" =PiP*'*" = PI'C". . . (43)
The point p l + a* -f- yt' + zt" may be any point whatever,
with suitable values of x, y, z, and we have
( Pl + a j e + y e ' + c")'e"= S p 1 ee'c\ ... (44)
so that any point whatever may be substituted for p^ in (43)
without changing the value. By the above, and by the pre-
ceding article, it appears that a product of four points is equal
to the product of any other four points having the same mag-
nitude and intrinsic sign, or order of generation : thus such a
product is a scalar; and, in fact, differs in no manner what-
ever from a product of three vectors which has the same mag-
nitude and sign. We may therefore write
#,'" = ee'e" ........... (45)
If the four points are in one plane, the product is zero,
because the volume of the connecting tetrahedron is zero.
Thus the planimetric progressive product of four points is
always zero.
23. The stereometric, progressive product of four or more
vectors is always zero. But we may have e l e^e^ meaning
the vector e l times the scalar c^s 6 ^ etc.
The stereometric, progressive product of five or more points is
always zero. But we may have such products as p t p a
30 DIRECTIONAL CALCULUS. [ART. 24.
meaning the ordinary algebraic product of the point-vector
p\p into the scalar
24. From the preceding articles we have the following
conditions :
is the condition that two points shall coincide.
i> l p.,p.. = Q ......... (47)
i
is the condition that three points shall be in one right line.
y,y,/':;y'4 = . .... (48)
is the condition that four points shall be in one plane.
e,t, = .......... (49)
is the condition that two vectors shall be parallel.
1 o 3 =0 .......... (50)
is the condition that three vectors shall be parallel to one
plane.
In the further development of the subject it will be con-
venient to treat separately two- and three-dimensional space,
considering the former first.
PLAXIMETRIC PRODUCTS.
25. In two-dimensional, or plane, space two plane- vectors,
or two point-plane-vectors, cannot differ from each other except
in magnitude and sign, since both are restricted to one plane.
Hence they become scalar quantities. Furthermore, by
Arts. 19 and 20, the product of two vectors is now identical
in meaning with the product of three points. Thus, if Pipi=f
and p 3 p l = c', we have, in plane space,
*,' = ' .......... (51)
Plane space is the locus of all points dependent on three
fixed reference points. We shall call these e , e,, e 2 , and shall
always take the area <& as the unit of measure of area, when
CHAP. IT.]
MULTIPLICATION.
31
dealing with a point system. That is, we write always in plane
space,
W,>=1 ............ (52)
This is a great practical convenience, and in no way affects the
generality of results.
The sides of the reference triangle taken around in order are
called the complements of the opposite vertices; thus
^e.2 = (complement of e ) = \e
^o = (complement of e t ) = !!
e v = (complement of e 2 ) = \e 2
(53)
the vertical line before the point being called the sign of the
complement.
In dealing with a vector system we shall usually refer to a
unit normal system of vectors tj, i,, as stated in Art. 12. We
shall then have in plane space
1*=! ............. (54)
26. Let
Pi = ? u<
then pip 2 =
m. 2
TOo
TOO
[e , e,, e.,] =
|e,
(55)
The third and fourth members of (55) are simply different
ways of expressing the second. The fourth member is espe-
cially noticeable for its symmetry. Eq. (55) shows that any
point-vector is expressible in terms of the sides of the reference
triangle.
'ii ( ] (>
Again, p^p^Ste
TOO TO! ni.,
n ??i n
(56)
as will be found on multiplying out and putting e^e, = 1, by
(52).
32 DIRECTIONAL CALCULUS. [ART. 27.
27. EXERCISES. 1. Find by eq. (47) the condition that
the three points p+^ p + e 2 > P -f f shall be collinear, and
illustrate geometrically. .4/1,9. e^ + c^g + e^ = 0.
2. Show that if a line be expressed in terms of the sides of
the reference triangle, and the sum of the coefficients be zero,
the line passes through the mean point of the reference points.
28. Since the product of three points obeys the associative
law, it can be regarded as the product of a point into a point-
vector, or of a point-vector into a point. Thus, if L = p->p 3 ,
PlPsPsPl'PiPs=PlL=P3PsPi = Lpl} ' ' (57)
so that this product is commutative.
29. Product of two lines, or point-vectors. The products
hitherto considered have all been progressive ; we now come to
one which is regressive. Since two lines in plane space must
intersect, they come under Art. 16, (6). Let
the lines be L^ and L 2 , let p be their com-
mon point, and let p 1 and p., be so taken that
Li=p pi, and L 2 =p pt.
We may also write,
A = J>o ( Pi -Po), Lo = /) (p a -po) ;
now the product of the vectors (Pip ) (Pz Po) is the area of
the parallelogram on these vectors, and is scalar ; the product
of the pomi-vectors should certainly give this result, and, in
addition, the point fixed by them, viz. their intersection. This
is in accordance with Art. 16, (6). The product
(Pi-Po)(P2-Po)
is equivalent to p pip 2 ; hence we may write
L l L 2 =p p 1 -p p 2 =p p l p. 2 -p (58)
It is to be carefully noted that the third member of eq. (58) is
not derived from the second by interchanges, according to the asso-
ciative and alternative laws, but is an independent expression,
CHAP. II.] MULTIPLICATION. 33
which gives the meaning of the product of two lines. It may be
regarded as a model form for the treatment of regressive prod-
ucts. Thus, if AB and AC are any two quantities whose
product is regressive, and if A is their common figure, we
shall always have
AB-AC=ABC-A ......... (59)
We have accordingly
LLi = p p 2 pop! = p pspi -P = - PoPiPz -Po = - LiLz, (60)
or the product of two lines is non-commutative.
30. Product of a point and two lines. Let Z/j and L 2 be as
in the last article, and p be some point ; then
pL t L 2 = p p pi - p p 2 = p poptfs po = popiPa PPo- (61)
PoPiPs can be placed first because it is a scalar. This is a
mixed product, that of the two lines being regressive, and that
of p into their common point, p , being progressive. Also,
pLiL 2 = -2)L,L i = L,L l -p ...... (62)
The period is necessary in the last member of this equation
to preserve the meaning ; that is, the product of the points
.LjZ/a an( l P' Without the period the expression would mean
the line L 2 multiplied into the scalar L t p.
31. Product of three lines. Let L 1} L. 2 , L s loe three lines, and
PD Pi, Ps their common points, and take scalar factors n 1} n 2 , n s
so that Li^tiipzps, L 2 = n 2 p s pi, J^s = n sPiP-2'i then
= - njn 2 n 3 p 2 p 3 - prfspz p l = W 1 w 2 w 3 (|) 1 |> 2 jp 3 ) 2 . . (63)
It thus appears that in plane space lines obey the same laws
of multiplication as points.
32. Let two points be given each as the common point of
two lines, viz. p^ = Z/ ^i an( i P = L L 2) then
L L 1 L. i 'L , ........ (64)
34 DIRECTIONAL CALCULUS. [ART. 33.
which is a reciprocal equation to (58). Similarly, if
we have
P-2 =
2 . (65)
33. The condition
............ (66)
requires that LI and L 2 shall be congruent, that is, be situated
on the same right line; for, by (58), this gives
The condition
. .' ......... (67)
requires that the three lines shall jxtss through a common
point; for L 2 L 3 is some point, say p, and L^p = makes L^
pass through p, by eq. (47).
34. Product of parallel lines. Let the lines be L = p^ and
L-2 = np.f. ; then
L^ z = np^ - p.f mp! ep. 2 = ntpipz c = np^-f. e. . . (68)
Thus the product is the common vector, or point at GO, mul-
tiplied by the scalar
35. Product of two vectors. Let
ti = m^i +
then, since i^o = 1,
(69,
But, since e^ is a parallelogram whose sides are || to ^ and
we have m
,.,= Te 1 7
Also, Wj = Tej cos < EI = Tt! cos a^ say ; m 2 = Tf l sin aj ;
ri! = TV, cos a 2 ; H 2 = Ze 2 sin u 2 .
CHAV. II. ] MULTIPLICATION. 35
Therefore
which affords a proof of the trigonometrical formula for the
sine of the difference of two angles.
36. Sum of fjoint-vectors. Using L L , L^p^p^pz as in Art.
29, we have
LI + L, = PM + p Q p. z =p (P! + p 2 ) = 2p p, . . ( 71 )
if p is the mean point of 2h an( i Ps- Thus, L l -+- L 2 passes
through the common point of the two lines and is equal in
length to that diagonal of the parallelogram on L l and L 2
which passes through p (> . Similarly,
so that the difference of L v and L 2 passes through p n and is
equal in length to the other diagonal of the parallelogram.
Suppose L! and L to be parallel, and equal respectively to
npif and j>-f 5 then
A + A = (^i + l> 2 ) = ('t + l)7'e; .... (73)
that is, the sum is a || point-vector passing through the mean
point of npi and p 2 . Finally, let n = 1, so that
and the sum is, in this case, the product of two vectors ; that
is, in plane space, a scalar.
The reader will notice the exact correspondence between
the results of this article and the resultant of forces in plane
space.
37. Sum of sides of a polygon. Let 1, 2, 3 be any three
points ; then
12 + 23 + 31 = 23 - 21 + 11 - 13 = 2(3 - 1) - 1(3-1)
36 DIKECTIONAL CALCULUS. [ART. 38.
Hence the sum of the three point-vector sides of a triangle
taken around in order is equal to twice the area of the triangle.
Similarly, let 1, 2, 3, 4, n, be n points, the vertices of a poly-
gon of n sides ; then
12 + 23 + 31 = (2 - 1) (3 - 1)
13 + 34 + 41 = (3 - 1) (4 - 1)
-l)n + nl= [(n - 1)-
and, adding,
12 + 23 + 34 + - -f n 1 = twice the area of the polygon.
38. COMPLEMENT. (a) The complement of a reference unit
is the product of the other reference units, so taken that thex>roduct
of the unit into its complement shall be positive unity.
This definition is perfectly general, and applies to either a
point or vector system in space of any number of dimensions.
We have already had examples in eq. (53,) as e = &, whence
== ^^^ = *
(6) The complement of a scalar quantity is the quantity itself.
Thus, \n = n ; {i^ = i^ = 1.
(c) The complement of the product of several factors is equal
to the product of the complements of the factors.
Thus, [ (rip) = |nj p = nip ; jtjig = Itjjig.
(d) The complement of the sum of several quantities is the sum
of the complements of the quantities.
39. Complement in a plane vector system. Taking a unit
normal system t,, 4, we have, according to the previous article,
Complement of i 1? written ]i : = i?, for ij|ii = 1^2 = 1 ;
Complement of i% written J^ = i 1? for ijij = I 2 t 1 =t 1 i 2 =l ;
II*i = 1*2 = .*i ; 11*3 = |*i = - *.
CHAP. II.] MULTIPLICATION. 37
Let EI = 7H.ii! -f-
and 2 = i h + n-ih. j then
By the figure it is evident that J X is 5*j
a vector of the same length as e^ and
perpendicular to it, or, in other words,
taking the complement of a vector in plane space rotates it posi-
tively through 90.
The product e^ is the parallelogram whose sides are cj and
[c 2 ; if tj is parallel to je^ the area of the parallelogram van-
ishes, or cjje, = ; but, since |c 2 is _L to c a q must, in this case,
be _L to c 2 ; hence the equation
i!*-0 (76)
is the condition that the two vectors ^ and e, shall be perpendic-
ular.
The product e^ is the area of a square each side of which
is of the length T^ ; hence
-I- 7 12 * e 2 CO-IT (T7\
e ii e i * i r5 sa y i. * * )
The form e^ is merely another way of writing e^ which is
often convenient. If n is a scalar, we have
nn = nn =
whence the analogy is apparent.
Grassmann calls e^ the " inner product " of ei and c 2 , regard-
ing the complement sign as a species of multiplication sign, and
accordingly calls ^ the " inner square " of e. It seems prefer-
able to the author not to introduce a new species of multipli-
cation, but to regard 1 'e 2 as simply the combinatory product
of cj into Icj, a way of looking at it which is practically far
more simple, and renders interpretation easier. Somewhat
after the analogy of the word cosine for sine of the complement,
we may call ^ the co-square of e, and, just as we read a 2 , a
square, we may read c?, e co-square. Similarly, c a ]e 2 may be
called the co-product of ej and 3.
48607
38 DIRECTIONAL CALCULUS. [Aux. 39.
We have also,
e,fc 2 = (m^! -J- ?n 2 i 2 )| ( n i l i + 7| i; l :>) = wVi 4- wi 2 M 2 = f-2 'i- (^8)
We obtain the third member because
*ll l l = 1 1 1 2 = 1> I l! l 2 = l l'l = 0, I 2 'li = 1 2 1 2 = 0,
and iz\i 2 = toti = iii 2 = 1 ;
the fourth member is apparent from the symmetry of the third.
It will be found that we have always, when A and Z> are quan-
tities of the same order in the reference units,
A\B = B\A ........... (79)
Since Tie = T e, as shown above, we have, as in Art. 35,
ejcs = T I Te 2 x sin (ang. bet. e, and ],) = 7^ Te., cos < c l (80)
Also, taking the values of m 1} m 2 , i], w 2 as i n -A- 1 ^- 35, we have
tjjc, = m^i + wi 2 n a = Tcj Te 2 (cos aj cos a., + sin a! sin a.,)
= r I 3 r7 2 cos< e -, ....... (81)
which affords a proof of the trigonometrical formula for the
cosine of the difference of two angles.
If c 2 = EU we have
*
'
Square and add eqs. (70) and (80) ; therefore
r 2 i r 2 2 = 1 i 2 i = ( 1 2 ) 2 +( 1 | 2 ) 2 . . . . (83)
If i be any unit vector, and p any other vector, we have
t p i = i Tp- cos<^ = projection of p on direction of t,
and ip \L =\t Tp sin<^ = projection of p on direction X "to t;
hence we may write
p = i p|i + ip - i ......... (84)
This equation may be verified by multiplying successively
by t and |i.
CHAP. II.]
MULTIPLICATION.
39
40. Complement in a plane point system. Taking as reference
points e , e l5 e. 2 , with the condition e^e^ = 1, we have, in accord-
ance with Art. 38,
=
(85)
= K-K. e,, by Art. 38, (c),
- O]^O9 * OQOQ ' ^Q^l ( ^0^1^2/ *""~ '
Note that the complement of the complement of a reference
point is here the point itself, while in the vector system the
complement of the complement was negative. The general law
is that when the number of reference units is even, the com-
plement of the complement of one of them is negative, and
when the number is odd, it is positive.
two points ; then
_ _ n (, i \ n g o i >e \ . (86)
so that the complement of any point is a point-vector. The
fourth member of (86) expresses this point-vector as the prod-
uct of the points in which it cuts two of the sides of the refer-
ence triangle, so that it may be easily constructed. We have
also
so that eq. (79) is here verified. Note that the product is
scalar. We have, however,
.,-... (88)
and ei|eo e i = e^, = |e )
so that, when the quantities on each side of the complement
sign are of different order, the product is not scalar, nor commu-
tative about the sign.
40
DIRECTIONAL CALCULUS.
[ART. 41.
If LI=\PI and L 2 = \p 2 , we have pi=\L l and ^ 2 = \L 2) and
L,\L, = |l>i \ Ps = \ Pl p 2 =p.^p l = Pl \ Pa = L 2 \L,, (89)
which also agrees with eq. (79). Again,
L,L,=\ih'\P>> = \lhIh\ (90)
so that the common point of the lines L and L., is the comple-
ment of the line PlPi . Also,
L ] \p 2 = ^(L ] \p 2 ) = \(\L l -p 2 ) = \(-p 2 \L l ) = -\(p 2 \L 1 ), . (91)
of which eqs. (88) are a special case.
41. Geometric interpretation. For the sake of simplicity,
suppose e , e 1} e 2 to be the cor-
ners of an equilateral triangle.
With e, the mean point of P
the triangle, as a center, draw a
circle of radius = , a being the
VG
side of the triangle; then, p
being any point whatever, \p is
its anti-polar with reference to
this circle ; that is, a line par-
allel to its polar, and equidistant
Avith it from the center of the
circle, but on the opposite side.
It is evident that with reference to the above circle each
vertex is the anti-pole of the opposite side of the reference
triangle ; for the respective distances of a vertex and its oppo-
site side from the center are
aV3
and -
V3 3 2 2V3
and the radius of the circle is a mean proportional between
these. The figure shows the construction for the radius of the
circle, and also for the anti-polar of any point p.
If the reference triangle is not equilateral, it can be obtained
by projection from an equilateral one, and the circle corre-
CHAP. II.] MULTIPLICATION. 41
spending to that will be projected into an ellipse such that,
with reference to it, \p is always the anti-polar of p. Hence it
will only be necessary to prove the property in the case of an
equilateral reference triangle. Before proceeding to the proof,
however, it will be necessary to find an expression for the dis-
tance between two points in terms of their coefficients.
42. Distance between two points. Let the points be p^ and
p 2 as in Art. 40. Since they are unit points, we have
2^ = 1 = 2iWi ; .'. In = 1 /i Z 2 , m n = 1 m l m 2 .
Hence
and p. 2 = 2we = e + tn>\(e\ e o) + m. 2 (e. 2 e ) = e {] -f- m^ -}- m&.
Thus, p. 2 pi = (wj ?0 fi + ( w 2 Z 2 ) c 2 ,
and the required distance
by eq. (82),
(92)
In the case of an equilateral reference triangle, whose side
is a, we have e/ = e 2 - = <* 2 > and ! c 2 = a 2 cos 60 = a 2 ; so that
eq. (92) becomes
2 -Z 2 ). (93)
43. Proof of the anti-polar property of the complement in a
point system. Referring to the figure of Art. 41, we propose
to show that
which will establish the proposition. We have, since p' is the
common point of \p and pe, p' congruent with
pe ' \p = xp + ye.
42 DIRECTIONAL CALCULUS. [ART. 43.
Multiply both members by pp^ ; then,
ppi ' p& ' \P = ppfi ' P\P = ypp^ '>
whence y = p--
Multiply the same equation by p,e ; therefore
p^e -pe-\p ep,p e\p =
whence x = e\p.
Substituting, we have
p- p e
Hence we have
r e e\p p e\p-(p *
ep' = e J ff - = |J ^ _
p- e\p p- e\p
and
pt e\p
Now, by eq. (93), taking Z , l>i, k as the coefficients forp, and
m = m l = m 2 = ^ as those for e, we have
Also,
p*=p\p = tf
and e|p = K^o + i i + ^) = i'
so that
& - e\p = 2(1? + I? -I,- I, + 1,1, + 1).
Hence (95) becomes
- e) T(e -p>) = * a ^ + *** + ?1/2 ~ Z ' " ?2 + i> =^- Q.E.D.
CHAP. II.] MULTIPLICATION. 43
44. The conception of the complement in a point system,
as developed Arts. 40-43, is not found in Grassmann's works.
He deals exclusively with the complement in a unit normal
vector system. See Die Ausdehnungslehre, 1862, Art. 330. In
a remark at the end of Art. 337 he shows how the idea of the
complement might be extended to a point system, but in a
way entirely different from mine, and one which he himself
evidently considered of no practical value, since he has made
no application of it. On the contrary, the method above de-
veloped is of great utility, giving at once reciprocal properties,
according to the principle of duality.
In a subsequent chapter the anti-polar property of the com-
plement will be established in a different manner, directly, for
a reference triangle of any shape.
45. A multiplication table for a point system in plane space
is given on page 44. The product of any quantity at the left
into any at the top is found at the intersection of the corre-
sponding row and column. Thus, e.^e efa = e& e 2 \e = 0, etc.
Algebraically considered this system forms a seven-fold alge-
bra, seven reference quantities being required to express all
quantities of the system, including scalars.
46. Projections. If we write the equation
P = Xfy + x.fy ........ . (96)
Xfa is evidently the projection of p on 1 \\ to e 2 , and x.f. 2 is the
projection of p on c 2 || to c^ Multiply /
(96) into c 2 ; therefore pc 2 = x^e^ since /
pe,
e-jCa = ; thus, x l = -. Similarly, mul-
1 C 2
P e i
tiplying into e l5 we have x 2 = - ; whence
'-o e , c i
p = li-^- 2 + ^l 1 . ....... (97)
Hence the projections of p on e l || to c 2 and on e. 2 \\ to cj are
44
DIRECTIONAL CALCULUS. [ART. 46.
A MULTIPLICATION TABLE FOR A POINT SYSTEM IN
PLANE SPACE.
II II
CHAP. II.] MULTIPLICATION. 45
respectively - - and - - -. These are particular cases of
iz si
a general proposition which may be stated as follows.
Let B be a quantity of the nth order in the reference units,
and C one of the mth order, and let the number of reference
units be m + n, so that BC is scalar : then the projection on B
of any quantity A, directed by C, is
B-AC
BC '
Similarly, the projection of A on "(7, directed by B, is
C-AB
CB
We shall have also
A _B.AC . C.AB
~~
CB
Since we have restricted ourselves to space of two and three
dimensions, we shall not give a general proof, but shall verify
and explain the proposition in such cases as arise under this
restriction. See Die Ausdehnungslehre, 1844, Chap. 5, and the
same, 1862, 127-129.
If in (97) EU c 2 are a unit normal system, replace them by ^
and i 2 ; then, since i^ = 1,
p = ti pt-2 2 P l \ = i P! I I + IIP 4 u .... (99)
which agrees with eq. (84).
47. Consider next the point equation
P = XnPo + x\p\ + x?Pz 5 ........ (100)
multiply successively by pip 2J p 2 po, and p tt pi, and we find
pPiPs PPzPo PPoPi
whence p = - - [ p - pp&t + Pi pp 2 po + Ps PPoPi] -
46 DIRECTIONAL CALCULUS. [ART. 47.
Each, term in the brackets taken with the outside factor is
of the typical form (98), and is the projection of p on one of
the points on which it depends. Write again,
P =
and multiply into \p^ \p 1} \p 2 successively, and we find
PoPiP?
' ( 102 )
We might also have obtained (102) from (101) by putting
\pip 2 for PQ, etc. The terms of the right-hand member of (102)
are again of the typical form, and are the projections of p on
the anti-poles of Pip 2 , P*PO, and p pi. Note that
\PlPa ' \Po=\(PlPaP*) = PlPaPo = PoPlP2-
If in (101) we take the last two points together, their sum
is some point on p t p 2 and also on pp^ since p is expressed in
terms of p and this point. Hence this point is congruent with
PPo> an( l we ma y write
or
Miiltiply into p\p& and we have
PPiPa = vpoPiP*
Multiply into p pi, and we have
PPoPi yPiPz 'PPo 'PoPi = VPiPaPo -PPoPi,
Substituting values of x and y, we obtain
PiPa'PPo
_ ,
an equation of the same form as (98). The second term of
the right-hand member of (103) is the projection of p
directed by p&
CHAP. II.] MULTIPLICATION. 47
The second members of (101) and (103) must be identically
equal ; hence we have
PiPa PPo = Pi PPaPo + Ps ' PPoPi,
or, writing p s instead of p, and p^ instead of p , for symmetry,
PiP2 PsP* = -Pi- P 2 PsP4 + P 2 PsP*Pi I 1()4 ,
= Ps PtPiPz P* PiPzPa f '
The last expression is obtained by interchanging the suf-
fixes 1 and 2 with 3 and 4. If in (101) and (102) we put the
reference points e , e ly e 2 for the p's, the equations become iden-
tical, viz.,
j> = o- J pe<> + i-.pK4-j-p|s ....... (105)
Similarly, (103) becomes
p = e 'p\e +\e -pe n = e 'p\e -{-e p-\e , . . . (106)
a form analogous to eq. (99), expressing p in terms of its pro-
jections on e and e^.
48. The operations of the last article would have been pre-
cisely the same if lines (point-vectors) had been used through-
out instead of points. Hence, substituting L's for p's in eqs.
(101) to (106), we have
L = 1 [7,o LL,L, + L, LL 2 L + L, LL L,l . (107)
JjQ^L/2
L = - [|AA L L +\L 2 L n -LL,-}- \L L 1 - L\L.^ (108)
\LZ jjp.Li ^ LI ' L^L^L^ -f- L. 2
L = e^., - Le Q + e. 2 e Le + e e i Le 2 , ...... (HI)
L = e^ Le u -j- e Le v e 2 = e^ L\e^ -{-(e^ L) e-^e.,. (112)
48
DIRECTIONAL CALCULUS.
[ART. 49.
In equations (107), (108), and (111) L appears as equal to
the sum of its projections on three given lines ; in (109) and
(112) it appears as the sum of its projections on a line and
point. The projection on the point in (109), viz. 1 2 ' - ? ,
LiL 2 L
is a certain portion of the line joining the common point of L t
and L 2 with the common point of L and L , the reciprocal idea
to that of the projection of p on pip 2 in (103).
49. In eq. (104) put p 3 p 4 = \q 1} and we have
PiPt'\Qi = -Pi-Pa9i+Pa'PiQi ...... (H3)
In (110) put L 3 L 4 = MI, and we have
L t L 2 M l = - L, L 2 M l + L, -L.M^ . . . (114)
Note that q l is a point, the anti-pole of p s p 4 , and M l is a line,
the anti-polar of the point L 3 L 4 .
Again, in (110) put L 1 L 2 = p 2 , L 3 = \QI, L+ = \q 2 ; then, since
! . 2 = , we have
and similarly from (104),
l ..... (116)
Multiply (115) and (116) respectively by p^ and LI, or (113)
and (114) respectively into* |0 2 and \M 2 , and we have
(118)
If 0i =PI and Q 2 = p 2 , Mi = A and M 2 = L% (117) and (118)
become
P\Pz\P\Pz = (P\P2)~ = Pi~P2- (Pi\P2) 2 ) /im\
2 I ' ' (11")
* ^1 multiplied &/ .B means BA ; A multiplied into B means AB.
CHAP. II.]
MULTIPLICATION.
49
Put q 2 for p in (102), and multiply by p pip 2 2 .
Then, PjP 2 = Lp 1 Lp. 2 = Lp,p 2 -L >
P t P 1 = l4> s -l4> 1 = Lp a p 1 .L = -P l P a ) 5
so that the product of two planes, like that of two points, is
non-commutative.
If P! and P 2 are parallel, L is at oc and becomes a plane-
vector ; call it t]j and substitute in (130) ; then we have for
the product of two planes, having a common line t] at oc,
. . (131)
56. Product of three planes. Let p be the common point of
PI, Pa, and P 3 , and take p lf p 2 , p s on the common lines of these
planes, so that
then,
PiPsPa = Pop a p 9 poPsPi PoPiPa = 023 013 . 012
= -023.0132.01 = 0123.023.01 (132)
= 0123 . 0231 = (jpoPiM.) 2 -l>o
In this equation we have used for p , 1 for p^ etc., for con-
venience. This we may frequently do when no ambiguity
will result. In eq. (132) we have worked according to Art.
16, (c), by which P 1 P 2 P S = P l P 2 P 3 ; but if we had combined
P! and P 2 first, and the result with P 3 , we should have obtained
the same result. Hence planes obey the same laws * of multi-
plication as points, in solid space.
* We have here assumed the distributive law to hold, as, in fact, it
does, for all products, progressive, regressive, or mixed ; but it is easy to
prove the law for planes or lines, assuming it to be true for points. Thus,
taking the planes as above,
P,P, + PjP 3 = 023 031 + 023 . 012 = 0123 (03 - 02), because 0123 is scalar,
= 0123 0(3 - 2) = 01 (2 - 3)3 0(3 - 2)
= 0(3 - 2)13 0(3 - 2) = 0(3 - 2) 1 0(3 - 2)3
= 023 01(2 - 3) = 023 . (031 + 012) = P,(P, + P 3 ).
CHAP. II.] MULTIPLICATION. 59
If the three planes are parallel to one right line, the common
point is at GO, and e may be substituted for p Q in (132).
57. Product of four planes. Let p^ p. 2 , p s , p be the four
common points of four planes P^ P a P 3 , P 4 taken three by
three, and take four coefficients n 1} n^ so that P 1 = ihp a p 3 p^
etc. ; then
P!P 2 P 3 P 4 = W i 3 tt 4 . 234 . 341 . 412 - 123 |
= n l n 2 n 3 n 4 (2hp-2p 3 p4Y > >
Mixed products are to be interpreted according to Art. 16,
(c). Thus,
has this meaning. P?L S is a point; this, multiplied by L&
gives a plane ; this, by P l} a line ; and this, by LI, a scalar
quantity.
58. Products of plane-vectors. Let 7/1 and 7/ 2 be two plane-
vectors (lines at oo ), and let c be parallel to each of them,
while Ci and e 2 are so taken that 7/1 = eti and i) 2 = ee 2 ; then,
This result may be obtained directly from eq. (58) by
regarding the points and lines of that equation as all at oo,
and therefore necessarily in the plane at oc.
The product of two plane-vectors appears as a vector parallel
to each of them, multiplied by a scalar quantity. We have
at once
Next take a third plane-vector j/ 3 , and let e t be || to rj 2 and
rj s , e 2 || to 773 and 7/1, e 3 || to 7/1 and 7/ 2 , while the tensors of l} etc.,
are such that 7/1 = e>e 3 , rj.> = e 3 j, 7/ 3 = e^o > then
As an exercise let the student discuss yP, the product of a
plane-vector and a point-plane-vector.
60 DIRECTIONAL CALCULUS. [ART. 50.
59. Equations of condition. By eq. (48),
pP=0 (137)
makes p lie on P, or P pass through p ;
L 1 L 2 = Q (138)
makes the two lines intersect ;
LP=0 (139)
makes L lie in P, or P pass through L ;
AP^O (140)
makes the two planes coincide ;
P!P 2 P 3 = (141)
makes the three planes pass through a common line ; for P 2 P 3
is a line, say L, and, by (139), P,Z,=0 makes P! pass through L;
PaP 2 P 3 P 4 = (142)
makes the four planes pass through one point, for PjPgPg is
some point, say p, and pP 4 , by (137), makes P 4 pass through p ;
PjLP^O (143)
makes Pj and P 2 cut L at the same point ; for, writing
L = PgP,, the result follows from (142) ;
W = (144)
makes the two plane-vectors parallel ;
W72>?3 = (145)
makes the three plane-vectors all parallel to one straight line.
Equations (138), (140), (141), (142), (144), (145), should
be compared with equations (66), (46), (50), respectively.
60. Addition of planes and plane-vectors. Let P^ and P 2 be
two planes intersecting in L, and let p t and p 2 be so taken that
P! = Lp l and P 2 = Lp 2 ; then
P! +P 2 = L Pl + Lp, = L( Pl +p,) = 2Lp, (146)
in which p is the mean of p l and p 2 . Thus the sum is that
CHAP. II.] MULTIPLICATION. 61
diagonal plane of the parallelepiped, of which two adjacent
faces are Pj and P 2 , which passes through L; the parallelo-
grams P! and P 2 being so placed as to have a common side L.
If the t' vo planes are parallel, let rj be a plane-vector parallel
to each of them, i.e. their common line at oo, and let p l and p 2
be points of the respective planes ; then we may write
P 1 = n^r] and P., = n 2 p 2 rj ;
whence Pj + P 2 = n l r 1 p l + n 2 r)p 2 = rj (n^ + n 2 p 2 )
= i 277
If HI + n 2 = 0, then
(148)
so that the sum, in this case, becomes a volume, and is scalar.
Cf. eq. (74).
Take P u P 2 , P 3 , p > Pi> Pv Ps as in Art. 56 ; then
Thus the sum is a plane through the common point parallel
to the plane p\p 2 p z -
If p is at oo, call it c ; then each plane is || to c, and the sum
becomes the product of three vectors, and therefore scalar.
If P 1 + P 2 = 0, or P 1 = -P 2 , ..... (150)
the two planes are coincident.
If P 1 + P 2 + P 3 = 0, ........ (151)
the three planes pass through one right line, as appears by
comparison with eq. (146) .
Similarly, P l + P 2 + P 3 +P 4 = ....... (152)
causes the four planes to pass through a common point, as
appears from eq. (149).
Take rji, r/ 2 , e, cj, e 2 as in Art. 58 ; then
>7i + >?2 = i + 2 = e( e i + c 2 ), .... (153)
so that the sum is a plane-vector parallel to c.
62 DIRECTIONAL CALCULUS. [Aur. 01.
61. Addition of point-vectors, or lines. Take n point-vectors
Pi f D P2 f 2> ' 'Pn f n) an( l ca ll their sum S ; then
'
It appears that /S is, in general, composed of two parts, of
which one is a point-vector, and the other a pZome-vector. If
this plane-vector is parallel to the point-vector, i.e. capable of
expression as the product of some vector a into 2, then their
sum can be expressed as a point-vector only ; for we have, in
this case,
S=e 2e + a2c = 0o-f a)2e,
a point-vector of the same length as e^e, \\ to it, and distant
^
from it by the amount To. sin < e -
S being composed of two parts which cannot be equal to
each other, if we have the equation S = 0, it can only be satis-
fied by making each part separately zero, so that S = implies
2e = and 2(p e )e = 0. The quantity S maybe called a
screw,* and we shall hereafter consider some of its properties.
62. TJie complement in three-dimensional space. Following
the definitions of Art. 38, we have for a unit normal vector
system ij, t 2 , i 3 ,
Let cj = Z^ + l. 2 L 2 + Z
and c 2 = m i l i +
then |i=^i 2 t 3+Wi+^ t i'2 = 7 (li<*t k l i)(li<-3 ^1)5
* See "The Theory of Screws," by R. S. Ball, Dublin, Hodges, Foster
& Co. ; and also a paper by the author on "The Directional Theory of
Screws," Annals of Mathematics, Vol. IV., No. 5.
CHAP. II.]
MULTIPLIC ATION.
63
so that e l is a. plane-vector. The third member of (156) is the
product of two vectors ; the first, ^t 2 l-*i, is easily seen, by
the figure, to be _L to l^ -f- ^ 2 > the projection of q on the plane
-i
iji,, and hence _L also to ej, because _L to the plane that projects
Ci on iji 2 ; similarly, l^ Z 3 i x is _L to Z x i x + Z 3 t 3 , hence to the plane
that projects e x on i 3 i x , and therefore to e x itself.
Hence |e x is a plane- vector perpendicular to e x . Since ||e x = q,
it follows that the converse is true ; that is, the complement of
a plane-vector is a line-vector perpendicular to it.
It is evident from the figure that c x is a diagonal of the
rectangular parallelepiped whose edges are Zj, Z 2 , l s in length ;
hence,
T I = VV + tf + tf. (157)
Multiply (156) by tj ; therefore
e iei = l * = ^ + ^ + Z 3 3=T* ei , .... (158)
so that, as in plane space, the co-square of a vector is equal to
the square of its tensor. The product e^q is that of the vector
E! into a _L plane-vector, as has just been shown ; it is therefore
64
DIRECTIONAL CALCULUS.
[ART. 63.
a volume which is equivalent to Ttj times the area of ej ; hence,
by (158), the area of |c t is numerically equal to Te 1} or
T\^ = Tfy' ........... (159)
Thus the complement of a vector in solid space is a perpendic-
ular plane-vector having the same tensor.
We have j
/
. (Io9)
I 3 m 3 = e 2 |e 1 )
Now e 1 |e 2) being the product of the vector e,
into the plane-vector |e 2 , is equivalent to
T|e 2 sin < Cz = Tej Te 2 cos <
that is,
(160)
If cj and c 2 were unit vectors, 1 1} 1 2 , 1 3 , m^ m 2 , m 3 would be
direction cosines, and thus (160) gives a proof of the formula
for the cosine of the angle between two lines in terms of the
direction cosines of the lines.
By (160) the condition that e 1 and c 2 shall be at right angles
Cl c 2 = (161)
Let r)i = \*i and rj 2 = |c 2 ; then
7?! 77 2 ^ Ci C 2 ^^ C 2 Cj zzz Ci C 2 ^= J. ^JL 2 COS <^ ^^ J. 7?j J. 71 2 COS <^ . ( 1 () )
e l ^l
and -qi\r]z = (163)
is the condition of perpendicularity of two plane-vectors.
63. Complement in a point system in three-dimensional space.
Let e , e 1} e% e 3 be four unit reference points, so taken that the
product 606,6263 = 1 ; then
I it j i 1
OQ 1 2>? I! 0""~ 123 "~~" ~~* 0? 1 ^~ ^2 3? ^01"^ 2 3^~ 0^1?
|^1 == "~~ ^2^3^0? 11^1^ "~ ^2^3^0 == ^1? ^0^2 == ^3^1? 1 1 ^0^2 ^ | ^3^1 == ^0^2?
|^2^ 6360^], ||^2 == I^S^O^l 213 ^2? |^0^3 == ^1^2J jl^O^S" &1&2^^&$3'
10 > O > \\O ^~
tf 3 o"l^2? 1 1 "3
CHAP. II.] MULTIPLICATION. 65
Note that the complement of the complement of a reference
point is the point with negative sign, but that the complement
of the complement of a reference line, or edge of the reference
tetraedron, is the line with positive sign. We have
\e fa -\e. 2 = e^Cg e 2 e 3 e e & e^
which agrees with Art. 38, (c).
Let Pi = 2,lke, and p 2 = 2o?e ; then
+ etc.,
so that the complement of any point is a point-plane-vector,
or plane, and any plane may be expressed in terms of the four
faces of the reference tetraedron.
From eq. (123) it follows that the complement of any point-
vector or line is another point-vector. Again,
Pi\Ps = 2fce| 2Ze = Z m -f- l^ -f I 2 m 2 + I 3 m s = p a \pi. . (165)
Let P! = \PH P 2 = \p 2 5 therefore,
Pl\P 2 =\Pl'\\P 2 =-\Pl'P 2 =P-2\Pl=Pl\P2=P2\Pl- (166)
Let LI = fc^i + &26o6 2 + ^36063 -f- fc/e 2 e 3 + ^'636! -f A: 3 'e 1 e 3
and L 2 = l^e^ + l^fa + etc. ;
then L 1 \L 2 =Jc 1 l 1 -\-k 2 l 2 +k 3 l s +Jc 1 'l 1 '+k 2 'l 2 '+k 3 'l 3 '=L 2 \L l . (167)
Also L 1 L 2 = k l l 1 ' + k 2 l. i l + kJ s ' + k l 'l l + k 2 'l 2 + k 3 'l 3 . . (168)
If L 2 = L l} we have
L l 2 = 2(k l k 1 ' + k 2 k 2 ' + k 3 k 3 ') ........ (169)
But if LI is a point-vector, its square must be zero, and as
the second member of (169) is not necessarily zero, it follows
that LI and L 2 are not, in general, point-vectors ; in fact, they
are screws, as shown in Art. 61. We have then for the condi-
tion that LI shall be a point-vector,
kM + kJcJ + kJc^Q .......... (170)
From (165), (166), and (167) it appears that a co-product
in which the factors on opposite sides of the sign are of the
66 DIRECTIONAL CALCULUS. [ART. 64.
same order is commutative about that sign, and always scalar.
If the factors are not of the same order, this is not the case ;
for example,
P\L = -\\(P\L) = -\(\P-L) = -\(L\P). . . (171)
Proceeding in a similar manner to that of Art. 43, it may be
easily shown that \p is the anti-polar plane of p, with reference
to an ellipsoid so situated that each vertex of the reference
tetraedroii is the anti-pole of the opposite face. If the refer-
ence tetraedron is regular, and a be one of its equal edges, the
ellipsoid becomes a sphere whose radius is easily found to be
2V2
With this geometric interpretation
Pi/> 2 = (172)
causes p l to be in the anti-polar plane of p 2 with reference to
the reciprocating ellipsoid, and vice versd;
PjjP^O (173)
causes P 1 to pass through the anti-pole of P 2 , and vice versd;
14^ = (174)
causes LI to intersect the anti-polar line of L 2 , and vice versd;
P\P=V (175)
makes p the anti-pole of P.
64. All the quantities we have to deal with in three-dimen-
sional space viz. scalars, points, lines, screws, and planes
are expressible in terms of fifteen quantities, which are all
either the reference points or products of them of different
orders ; they are the four reference points ; their six products,
two by two, i.e. the edges of the reference tetraedron ; their
four products, three by three, or the faces of the reference
tetraedron ; and the product of the four, which is numerical
unity. A multiplication table can be easily constructed sim-
ilar to that in Art. 45. Considered as an algebra it appears
that this system is fifteen-fold.
CHAP. II.] MULTIPLICATION. 67
65. Projections. We have the same fundamental formula
for projection as in Art. 46, viz. :
R
(Projection on B of A, directed by C7) = ' ^ , . (176)
.LiC
in which BC is scalar, while B and C separately are not. If
we substitute in the equations of Art. 47 vectors for points,
and plane-vectors for point-vectors, we shall obtain a set of
corresponding formulae for a vector system in solid space, as
follows :
p s i + 3-p i 2), (177)
s i p| 2 + |ij ' P\es)> - (178)
1 2 3
_ 1 ' P 2 3 _[_ 2 3 ' P l
1 2 3 2 3 1
These are derived from eqs. (101), (102), and (103), and
the last one gives p in terms of its projections on ^ parallel to
2*3, and on 2 e 3 parallel to c^ Also, from (104), (105), and
(106), we have
1 2 ' 3 4 = e l ' 23 4 -f 2 ' 3 4 1 = 3 ' 4 1 2 ~ 4 ' ^3*3, (180)
P = !*2 + 1 3 ' Pi's.' ........ (181)
= ' '/W?! + Vs ' if'Mz)) (183)
\>wii ylyz), - - (184)
T?2*?3 ' Tmi^ (185)
(186)
77 = ^ . 77^ + igij ,7713 + tit., 7713, . ....... (187)
= ws wi + i i y^ ............ (188)
68
DIRECTIONAL CALCULUS.
[ART. 66.
Finally, from the equations of Art. 49, we have
1 2 ' l' = 1 ' 2| l' + 2 ' l| l',
1 2 1 l =
i| i
' 1721V
2| 2
1 2 1 2 ( 1 2)" 1~ 2 ( li 2) ,
l! l' l 2 f l| 3
1 2| 2 2 3
(189)
(190)
(191)
(192)
(193)
(194)
(195)
(196)
3| 1 C 3| 2 3| 3
P ' ( c l 2)~ == 1 " P f 2\ f l f 2 2 ' P l| l 2-
From eq. (192) on, the plane-vector equations have not been
written; to obtain them we have only to substitute plane-
vectors for vectors in eqs. (193)-(196).
66. Projections in a point system. Write
and multiply successively by p\p-2pz, PzPzPn> e tc., and we find
PPiPzPs PPzlhPo
whence
P =
x, =
-, etc.,
Oo 'PPiPsPs Pi 'PPsPsPo +P-> 'IWaPnPi
Ps'PPoPiPa)) .... (197)
which gives p in terms of its projections on any four points.
Write next
2 ~~ OJL o ~i ^1? i ~i **^*>?-'*> ?-^3 * X^/A) y i?
and multiply successively by Pip 2 p 3 , l^PsPot an d PsPoPi ', the
values of x and ^ will be the same as before, but that of x 2
1
will be x =
; hence
PoPiP 2 Ps
(P'PPiP*P*Pi
CHAP. II.] MULTIPLICATION. 69
which gives p as the sum of its projections on p , p^, and the
line p*,p s . Similarly, we have for the expression of a point in
terms of its projections on any point p t and any plane P 15
......... (199)
and for the expression in terms of its projections on any two
lines Z/j and L.,,
p== ^'P^ + ^2-p^ ^ (20Q)
-/Vji>2 X/jX/o
In equations (197) -(200) the projected point p may be replaced
by a screw S, or a plane P. We may also write in (197) and
(198) planes for points throughout.
Let P\=PzPzP^ P-2=PsPopi, etc.;
the projection of p on P, directed by \p^ is ' '-Plff^ and we
Pipi
may write
_ \Po-ppo \Pi'P\Pi , P 2 'PP 2 , \Ps-P\P*
I
P 3 p
or, taking the complement of both sides,
\P*-P*'pM- (202)
Let there be two planes, P = prf^Ps and Q = q^^ ; then
PQ = PiPsPa ' p 4- p\$dp = 2dp\ p = 0,
or, by (282),
o-i^ = l, ........... (284)
as the equation of the tangent to the curve.
For the normal we have the equation
(o-- P ) n+m p = 9"<-i i-i\ 'p' = afi^p 1 = p', and p'^p = 0, so that (p 1 -|-p2) = 0, and that
these diagonals are conjugate in direction.
(5) Find the condition that the line cr|e = C shall be tangent
to the curve p| p = 1 ; comparing this
with the given equation, we have c p. p. ;
also, writing p in terms of its projections on \cf> c \ and |< C /A, we
have
P = TTT
or, substituting from the preceding equations,
'
terfi.Xic), . (308)
the proposed formula. If < be self-conjugate, then <^> c = <, and
the suffix may be dropped in (307) and (308).
To invert such a function as fa + k p + p\ + p = Ap + we p e ; suppose that Tt = 1, and put
A = e and fjt. = e ; then 00 A. = (<^> -4)e =7ie is _L to the a axis, so that the axes
are now completely determined.
CHAP. III.] APPLICATIONS TO PLANE GEOMETJRY. 123
(2) Discuss the following equations :
(a) (ap)'(#,) 2 = (a0) S .
(6) p*-a|p.0|,o--ap.j3p-(a+/8)[p=l, when Ta=T(3=l.
(C) (pj(a-/3)) 2 -(pj(a + /?)) 2 = 4(a/?) 2 , When Ta=T(3=l.
(d) t ? p ? = e 2 [cj(p e)] 2 , e being a scalar constant.
In the last equation consider the cases when e>l, e p 3 will all be on the line q^o, so that their product
will be zero. Conversely, if we have
fcPi<&>*&>3 = ......... (348)
for any three points whatever, p l9 p 2 , p 3 , then the function <
must have the above form, and p\(f>p is factorable. The ex-
pression ~ l e, and let > c be the function conjugate to !(e + e 1 + e 2 + e 3 ) = 4p;e = l ..... (377)
The equations of curves and surfaces in solid space may
appear under the following forms :
Expressed in terms of points
Non-scalar equations.
Scalar equations.
Expressed in terms of planes
Expressed in terms of vectors
Expressed in terms of points
Expressed in terms of planes
Expressed in terms of vectors
132. The non-scalar equation
p = 2o(ze) = CQ + 2
(378)
164 DIRECTIONAL CALCULUS. [ART. 133.
cu ca, e 3 having the values given in the last article, and x being
eliminated from the third member of the equation by the rela-
tion x -f- x l + x 2 + Xg = 1, may be called the equation of solid
space, since, by giving proper values to XQ, x^ etc., p may
become any point whatever in space.
If a relation be given between the scalar quantities, such as
fi(x<>, a* ** a*) = 0, or f^x,, x,, z 3 ) = 0,
then p will lose one degree of freedom of motion, and will lie
on some surface. If another relation be given, such as
ft(*a> *i, * ZB) = 0, or / 2 (Zi, x,, x s ) = 0,
then p will be compelled to move on a second surface simul-
taneously with the first, and hence along the curve of inter-
section of the two surfaces. It follows that the non-scalar
equation of a surface has two independent scalar variables,
while that of a curve has only one. Eq. (378) may be written
in the form
p = e 9 'p\e t + e l 'p\e l + e,-p\ei + e s -p\e m . . (379)
from which it appears at once that the scalar coefficients are
proportional to volumes of the tetraedra formed by joining p
with the reference points. Since p is a unit point, the truth
of (377) appears also from (379). It is easily seen likewise
from (379) that when p passes through any face of the refer-
ence tetraddron, the coefficient of the opposite point changes
sign. Thus when the coefficients are all positive, p is inside
the tetraedron ; when one is negative, it has passed through one
face ; when two are negative, it has passed through two faces,
i.e. through one edge; and when three are negative, it has
passed through three faces, i.e. through one vertex.
133. If P \p, then the complementary equation to (378)
or (379) is
P=$l(x> e)=2o(|e-eP), ...... (380)
and P may be any plane whatever in space. But if a relation
exists, as in the last article, between the scalar variables, then
CHAP. V.] SOLID GEOMETRY. 165
P moves according to some definite law, and envelops a sur-
face.
If a second relation exists between them, then P touches
two surfaces simultaneously, or rolls on them, and therefore
envelops a developable surface, which is reciprocal to a curve.
134. Writing in (378) p e = p, we have
/>-=3!(0, ........... (381)
a vector equation which, regarding p as always drawn outwards
from a fixed origin, represents a surface or a curve under the
same conditions as previously given.
135. Equations of planes, lines, and points. If, together with
eq. (378), we have the linear relation
S2(maj) = 0, .......... (382)
then (378) represents a plane; for, since x =p\e , etc., we
have
e) = 0, ...... (383)
which is the condition that p shall be always on the plane
| (m e + mfr + m^e 2 + m s e 3 ) .
If we have also the relation
Sj(na;) = 0, ............ (384)
p must also lie on the plane whose equation is
^(ne) = 0, .......... (385)
and hence must lie on their common line.
The equation of a plane through any three points p^ p 2 , p s
may be written
P'*p l + x(p-pi) + y(ps-pi), . (386)
of which the scalar form is
Q, .......... (387)
166
DIRECTIONAL CALCULUS.
[ART. 136.
which may be derived from the non-scalar form by multiplying
it by pipzPs-
The equation of a plane through p l and || to P l is
(p-p 1 )P 1 = 0; (388)
for it is satisfied when p=p\, and is the condition that the
vector p pi shall be || to Pj.
The complementary equations to (383), (385), and (387), viz. :
P2,(nie) = Q P20ie) = PPP>P =0
are the conditions that the variable plane P shall always pass
through the fixed points 2(wie), 2(ne), and P^PoP^ and hence
are the plane equations of these points.
Eqs. (233), (234), and (235) are the equations of lines in
solid space as well as in plane space, though in the former
they are all non-scalar.
136. Vector equations of planes and lines. The equation
%l(Ax) = <7, (389)
taken in connection with (381), represents a plane, if p be
always drawn outward from a fixed point ; for, eliminating x 3
between the two equations, we have
+ A* I -""I i>
*i e i - -r-
(390)
or
the non-scalar and scalar forms of the equation of a plane
through the end of Ce s -*-A 3 and parallel to the vectors
A& A& and A 3 f 2 A.f%. If a third equation,
%l(Bx) = C', (301)
be given, we have, eliminating o^ and x.,,
Cj , 2 ,
\, A 2 ,
x 3 A 3 C
xB-C'
= 0, or
(392)
an equation which evidently represents a right line.
CHAP. V.] SOLID GEOMETRY. 167
Eqs. (390) and (392) are of the respective forms
p = e + a*' + ye" I
(p _ e ) V' = L ........ (393)
which are those practically used.
The equation of a plane through the ends of ej, e 2 , c 3 drawn
out from the origin may be written at once in either of the
forms
or ( p _ l ) (2 _ i ) (3 _ ei) =
the last being equivalent to
P( 2 e 3 + c 3i + eie2) = ei23> ...... (395)
as will be found on multiplying out.
137. EXERCISES. (1) What is the meaning of the two
equations Pp^ = and Pp 2 = taken simultaneously ?
(2) Interpret the following equations written in comple-
mentary pairs:
f PPl^2 = ) | PPiP& = ) f > 2 AA = )
^O)' \PP 1 P 2 q l q 2 e = 0} ) I PL, L lPl = j '
P 3 P 4 P 5 = ) ( p^P! + Vi) = )
= o r ( Pfc 1 i> 1 + te -or
j P(> 2 -Pi) PiP* = 1 ip 2 p 3 = C
and ppip 2 pi = C' is
(Pa~ Pi) (PiPzPsPi 'Pi +C(Pil ) i) C'(P 3 pi))-
(6) Show that the common point of the three planes,
*n T> n C 11 f>n n n C 111 i
2 2 s2 } 2 s/ \L
e + C'(e p\) + C"(e p 2 ) + C'"(ep 5 }.
(7) Show that Pip 2 p s = is the condition that the three
planes p\pi = p\p 2 =p\p 3 = shall have a common line.
(8) Find the condition that the plane p\p 4 = 0, together
with the three planes of the last exercise, shall have a common
point.
(9) Show that if the equations of three planes, on being
multiplied by any constants and added, vanish identically, i.e.
for all values of p, then the three planes pass through a com-
mon line. Also that if the same holds for the equations of
four planes, then the planes have a common point.
(10) Show that when Pip 2 p s p t = C l C 2 + C s C i} the four
planes pp\p 2 p 3 = C^ ppzPsP* = C 1} pp^P\P\ @2> PP*P\Pz @3
have a common point.
(11) Interpret the equations /a|e = (7, pee' =(7, (p )|e = 0,
(12) Find the vector perpendiculars from the origin on the
planes of Ex. (11).
Ans - "T> 7~rS' c > Cl T/ ' T*!^ 3
* (ee')= ((2 ejcs)-
(13) Find the conditions of perpendicularity and parallelism
of the two planes p|e = C and p|c' = C'.
CHAP. V.] SOLID GEOMETRY. 169
(14) Find the vector perpendicular from the end of the
vector 8 upon the line (p e)c' = 0.
Ans. e'(e 8) ['.
(15) Find the conditions that the three lines (p e)e' = 0,
P l P. 2 p = 0, Pipzp = shall lie respectively in the three planes
Ans. e'|e" = and e\t" = C,
(P* Pi) \Pz and p 2 \p s = C.
(16) Show that the shortest distance between the two right
lines (p-e,)e 2 = and (p- ei ')e 2 ' = is ( g i~ c i')W. (Use
Art. 46.) T ** 2
(17) Show that the equation of the common line of the two
planes (p e)e'c" = and (p eiXV = is
(18) Show that the equation of a plane through the line
(p c i) e a = parallel to the line (p fi)e 2 l may be writ-
ten (p ej)e2 W = 0, and hence, by Ex. (17), that the line
,f f > -I- V\ f c '
"(W) 2 2 H
cuts these two lines at right angles.
(19) What are the conditions that the pairs of right lines
JfcfcC*-*,) =0| and J p.
(8) Show that, when T p, p\(f>p = 1, and