Chapter III.
Interpretations and Transformations of Quaternion Expressions.

99. If we write, as in §§ 83, 84,

we have, at once, by § 86, p.62 These express in Cartesian cöordinates the propositions we have just proved. In commencing the subject it may perhaps assist the student to see these more familiar forms for the quaternion expressions ; and he will doubtless be induced by their appearance to prosecute the subject, since he cannot fail even at this stage to see how much more simple the quaternion expressions are than those to which he has been accustomed.

100. The expression S . a/3y

because the quaternion a/3y may be broken up into of which the first term is a vector.

But, by § 96,

Here Trj = 1, let L be the angle between 77 and 7, then finally But as rj is perpendicular to a and {3, Ty cos j is the length of the perpendicular from the extremity of 7 upon the plane of a, ft. And as the product of the other three factors is (§ 96) the area of the parallelogram two of whose sides are a, ft, we see that the magnitude of S . a/37, independent of its sign, is the volume of the parallelepiped of which three cöordinate edges are a, ft, y : or six times the volume of the pyramid which has a, ft, y for edges.

101. Hence the equation

if we suppose a, ft, y to be actual vectors, shews either that i.e. two of the three vectors are parallel, or all three are parallel to one plane.

This is consistent with previous results, for if 7 =pft we have

and, if 7 be coplanar with a, ft, we have 7 =poi 4- qft, and

p.63

102. This property of the expression 8 . afiy prepares us to find that it is a determinant. And, in fact, if we take a, ft as in

we have at once The determinant changes sign if we make any two rows change places. This is the proposition we met with before (§ 89) in the

If we take three new vectors

we thus see that they are coplanar if a, /3, y are so. That is, if

103. We have, by § 52,

If q = a/3, we have Kq = /3a, and the formula becomes

In Cartesian cöordinates this is

More generally we have this becomes a formula of algebra due to Euler.

p.64

104. We have, of course, by multiplication,

Translating into the usual notation of plane trigonometry, this the common formula.

Again, F. (a + ft) (a - /3) = - Fa/3 + F/3a = - 2 Fa/3 (§ 86 (2)). Taking tensors of both sides we have the theorem, the parallelogram whose sides are parallel and equal to the diagonals of a given parallelogram, has double its area (§ 96).

and vanishes only when a2 = ft 2 , or Ten = Tft ; that is, the diagonals of a parallelogram are at right angles to one another, when, and only when, it is a rhombus.

Later it will be shewn that this contains a proof that the angle in a semicircle is a right angle.

105. The expression p = afta~ l

obviously denotes a vector whose tensor is equal to that of ft. so that p is in the plane of a, ft. so that ft and p make equal angles with a, evidently on opposite sides of it. Thus if a be the perpendicular to a reflecting surface and ft the path of an incident ray, p will be the path of the reflected ray.

Another mode of obtaining these results is to expand the above expression, thus, § 90 (2),

so that in the figure of § 77 we see that if A = a, and OB = ft, we have OD = p = afta~ l .

Or, again, we may get the result at once by transforming the equation to ^ = K (a" 1 p) - K ? .

p.65

106. For any three coplanar vectors the expression

is (§ 101) a vector. It is interesting to determine what this vector is. The reader will easily see that if a circle be described about the triangle, two of whose sides are (in order) a and /3, and if from the extremity of /3 a line parallel to 7 be drawn, again cutting the circle, the vector joining the point of intersection with the origin of a. is the direction of the vector a/3y. For we may write it in the form which shews that the versor [ -5] which turns yS into a direction parallel to a, turns 7 into a direction parallel to p. And this expresses the long-known property of opposite angles of a quadrilateral inscribed in a circle.

Hence if a, /3, y be the sides of a triangle taken in order, the tangents to the circumscribing circle at the angles of the triangle are parallel respectively to

Suppose two of these to be parallel, i. e. let

since the expression is a vector. Hence which requires either a case not contemplated in the problem ; i. e. the triangle is right-angled. And geometry shews us at once that this is correct.

Again, if the triangle be isosceles, the tangent at the vertex is parallel to the base. Here we have

whence x 7* = a2 , or Ty = Tea, as required.

As an elegant extension of this proposition the reader may T. Q. I. 5 p.66 prove that the vector of the continued product a/rtyS of the vector-sides of any quadrilateral inscribed in a sphere is parallel to the radius drawn to the corner (a, 8). [For, if 6 be the vector from 8, a to /3, 7, a/9e and n8 are (by what precedes) vectors touching the sphere at a, 8. And their product (whose vector part must be parallel to the radius at a, 8) is

107. To exemplify the variety of possible transformations even of simple expressions, we will take cases which are of frequent occurrence in applications to geometry.

[which expresses that if This may be changed to all of which express properties of a plane.

All of these express properties of a sphere. They will be interpreted when we come to geometrical applications.

p.67

108. To find the space relation among five points.

A system of five points, so far as its internal relations are concerned, is fully given by the vectors from one to the other four. If three of these be called a, /?, y, the fourth, S, is necessarily expressible as xa. + yfi + zy. Hence the relation required must be independent of x, y, z.

The elimination of a?, y, z gives a determinant of the fourth order, which may be written Now each term may be put in either of two forms, thus

If the former be taken we have the expression connecting the distances, two and two, of five points in the form given by Muir (Proc. R. S. E. 1889) ; if we use the latter, the tensors divide out (some in rows, some in columns), and we have the relation among the cosines of the sides and diagonals of a spherical quadrilateral.

We may easily shew (as an exercise in quaternion manipulation merely) that this is the only condition, by shewing that from it we can get the condition when any other of the points is taken as origin. Thus, let the origin be at a, the vectors are a, P a, y a, 8 a. But, by changing the signs of the first row, and first column, of the determinant above, and then adding their values term by term to the other rows and columns, it becomes

which, when equated to zero, gives the same relation as before. [See Ex. 10 at the end of this Chapter.] 5-2

p.68 An additional point, with e = x a. -f y ft + 2 % gives six additional equations like (1) ; i. e.

from which corresponding conclusions may be drawn.

Another mode of solving the problem at the head of this section is to write the identity

where the ms are undetermined scalars, and the as are given vectors, while is any vector whatever.

Now, provided that the number of given vectors exceeds four, we do not completely determine the ms by imposing the conditions

Thus we may write the above identity, for each of five vectors successively, as and we have six linear equations from which to eliminate the ms. The resulting determinant is

This is equivalent to the form in which Cayley gave the relation among the mutual distances of five points. (Camb. Math. Journ. 1841.)

p.69 109. We have seen in § 95 that a quaternion may be divided into its scalar and vector parts as follows :—

where is the angle between the directions of a and ft and e UV is the unit-vector perpendicular to the plane of a. and /3 so situated that positive (i.e. left-handed) rotation about it turns a towards ft

Similarly we have (§ 96)

a and e having the same signification as before.

110. Hence, considering the versor parts alone, we have

/ being the positive angle between the directions of 7 and ft and e the same vector as before, if a, ft 7 be coplanar.

Also we have

But we have always

from which we have at once the fundamental formulae for the cosine and sine of the sum of two arcs, by equating separately the scalar and vector parts of these quaternions.

And we see, as an immediate consequence of the expressions above, that

if m be a positive whole number. For the left-hand side is a versor p.70 which turns through the angle m6 at once, while the right-hand side is a versor which effects the same object by m successive turnings each through an angle 6. See 8, 9.

111. To extend this proposition to fractional indices we have only to write - for 0, when we obtain the results as in ordinary trigonometry.

From De Moivre’s Theorem, thus proved, we may of course deduce the rest of Analytical Trigonometry. And as we have already deduced, as interpretations of self-evident quaternion transformations (§§ 97, 104), the fundamental formulae for the solution of plane triangles, we will now pass to the consideration of spherical trigonometry, a subject specially adapted for treatment by quaternions ; but to which we cannot afford more than a very few sections. (More on this subject will be found in Chap. XI. in connexion with the Kinematics of rotation.) The reader is referred to Hamilton’s works for the treatment of this subject by quaternion exponentials.

112. Let a, /3, 7 be unit-vectors drawn from the centre to the corners A, B, C of a triangle on the unit-sphere. Then it is evident that, with the usual notation, we have (§ 96),

Also [7 Fa/3, UV/3y, UVyct are evidently the vectors of the corners of the polar triangle.

Now (§ 90 (1)) we have

Remembering that we have we see that the formula just written is equivalent to

p.71 113. Again, V . Fa/3 F/37 = - /3a/37,

which gives where pa is the arc drawn from A perpendicular to BC, &c.

114. Combining the results of the last two sections, we have

These are therefore versors which turn all vectors perpendicular to OB negatively or positively about OB through the angle B.

[It will be shewn later (§ 119) that, in the combination (cos B+/3smB)( ) (cos B - /3 sin B)t the system operated on is made to rotate, as if rigid, round the vector axis ft through an angle 2B.]

As another instance, we have

The interpretation of each of these forms gives a different theorem in spherical trigonometry.

115. Again, let us square the equal quantities

p.72 supposing a, @, 7 to be any unit-vectors whatever. We have But the left-hand member may be written as whence all of which are well-known formulae.

116. Again, for any quaternion,

so that, if n be a positive integer,

From this at once

If q be a versor we have

so that as we might at once have concluded from § 110.

Such results may be multiplied indefinitely by any one who has mastered the elements of quaternions.

p.73 117. A curious proposition, due to Hamilton, gives us a quaternion expression for the spherical excess in any triangle. The following proof, which is very nearly the same as one of his, though by no means the simplest that can be given, is chosen here because it incidentally gives a good deal of other information. We leave the quaternion proof as an exercise.

Let the unit-vectors drawn from the centre of the sphere to A, B, C, respectively, be a, j3, 7. It is required to express, as an arc and as an angle on the sphere, the quaternion

The figure represents an orthographic projection made on a plane perpendicular to 7. Hence C is the centre of the circle DEe. Let the great circle through A, B meet DEe in E, e, and let DE be a quadrant. Thus DE represents 7 (§ 72). Also make EF = AB = /3a~\ Then, evidently,

which gives the arcual representation required.

Let DF cut Ee in G. Make Ca = EG, and join D, a, and a, F. Obviously, as D is the pole of Ee, Da is a quadrant ; and since EG = Ca, Ga EG, a quadrant also. Hence a is the pole of DG, and therefore the quaternion may be represented by the angle DaF.

Make Cb = Ca, and draw the arcs Pa/3, P6a from P, the pole of p.74 A B. Comparing the triangles Ebi and ea/3, we see that Ecu = e/3. But, since P is the pole of AB, Ffia is a right angle: and therefore as Fa is a quadrant, so is P/3. Thus AB is the complement of Eot or fte, and therefore

Join 6J. and produce it to c so that Ac = bA; join c, P, cutting AB in o. Also join c, 5, and 5, a.

Since P is the pole of AB, the angles at o are right angles ; and therefore, by the equal triangles baA, coA, we have

and therefore the triangles coB and Bafi are equal, and c, 5, a lie on the same great circle.

Produce cA and cB to meet in H (on the opposite side of the sphere). H and c are diametrically opposite, and therefore cP, produced, passes through H.

Now Pa = Pb = P^T, for they differ from quadrants by the equal arcs a/3, ba, oc. Hence these arcs divide the triangle Hab into three isosceles triangles.

[Numerous singular geometrical theorems, easily proved ab initio by quaternions, follow from this : e.g. The arc AB, which bisects two sides of a spherical triangle abc, intersects the base at the distance of a quadrant from its middle point. All spherical triangles, with a common side, and having their other sides bisected by the same great circle (i.e. having their vertices in a p.75 small circle parallel to this great circle) have equal areas, &c. &c.]

118. Let Oa = a. , Ob = /3 , Oc = 7 , and we have

But FG is the complement of DF. Hence the angle of the quaternion

is half the spherical excess of the triangle whose angular points are at the extremities of the unit-vectors a , /3 , 7 .

[In seeking a purely quaternion proof of the preceding propositions, the student may commence by shewing that for any three unit-vectors we have

The angle of the first of these quaternions can be easily assigned ; and the equation shews how to find that of ffa. 1 ^.

Another easy method is to commence afresh by forming from the vectors of the corners of a spherical triangle three new vectors thus :—

Then the angle between the planes of a, @ and 7 , a ; or of ft, 7 and a , j3 ; or of 7, a and /3 , 7 ; is obviously the spherical excess. But a still simpler method of proof is easily derived from the composition of rotations.]

119. It may be well to introduce here, though it belongs rather to Kinematics than to Geometry, the interpretation of the operator

By a rotation, about the axis of q, through double the angle of q) the quaternion r becomes the quaternion qrq~ l . Its tensor and angle remain unchanged, its plane or axis alone varies.

p.76 A glance at the figure is sufficient for the proof, if we note that of course T . qrq~ l = Tr, and therefore that we need consider the versor parts only. Let Q be the pole of q,

Join C A, and make AC=C A. Join CB.

Then CB is qrq 1 , its arc CB is evidently equal in length to that of r, B C ; and its plane (making the same angle with B B that that of B C does) has evidently been made to revolve about Q, the pole of q, through double the angle of q.

It is obvious, from the nature of the above proof, that this operation is distributive ; i. e. that

If r be a vector, = p, then qpq~ l (which is also a vector) is the result of a rotation through double the angle of q about the axis of q. Hence, as Hamilton has expressed it, if B represent a rigid system, or assemblage of vectors,

is its new position after rotating through double the angle of q about the axis of q.

120. To compound such rotations, we have

To cause rotation through an angle -fold the double of the angle of q we write cfBf*.

To reverse the direction of this rotation write q^Bq*.

To translate the body B without rotation, each point of it moving through the vector a, we write a + B.

To produce rotation of the translated body about the same axis, and through the same angle, as before,

Had we rotated first, and then translated, we should have had a + qBq~\

p.77 From the point of view of those who do not believe in the Moon’s rotation, the former of these expressions ought to be

instead of But to such men quaternions are unintelligible.

121. The operator above explained finds, of course, some of its most direct applications in the ordinary questions of Astronomy, connected with the apparent diurnal rotation of the stars. If X be a unit-vector parallel to the polar axis, and h the hour angle from the meridian, the operator is

the inverse going first, because the apparent rotation is negative (clockwise).

If the upward line be i, and the southward j, we have

where I is the latitude of the observer. The meridian equatorial unit vector is and X, /-t, k of course form a rectangular unit system.

The meridian unit-vector of a heavenly body is

where d is its declination.

Hence when its hour-angle is ht its vector is

The vertical plane containing it intersects the horizon in so that

[This may also be obtained directly from the last formula (1) of § 114.]

p.78 To find its Amplitude, i.e. its azimuth at rising or setting, the hour-angle must be obtained from the condition

These relations, with others immediately deducible from them, enable us (at once and for ever) to dispense with the hideous formulae of Spherical Trigonometry.

122. To shew how readily they can be applied, let us translate the expressions above into the ordinary notation. This is effected at once by means of the expressions for X, //,, L, and S above, which give by inspection

and we have from (1) and (2) of last section respectively

In Capt. Weir’s ingenious Azimuth Diagram, these equations are represented graphically by the rectangular cöordinates of a system of confocal conics :—viz.

The ellipses of this system depend upon / alone, the hyperbolas upon h. Since (1) can, by means of (3), be written as

we see that the azimuth can be constructed at once by joining with the point 0, tan d, the intersection of the proper ellipse and hyperbola.

Equation (2) puts these expressions for the cöordinates in the form

The elimination of d gives the ellipse as before, but that of I gives, instead of the hyperbolas, the circles

The radius is

and the cöordinates of the centre are

p.79 123. A scalar equation in p, the vector of an undetermined point, is generally the equation of a surface; since we may use in it the expression p = oca.,

p.79 where x is an unknown scalar, and a any assumed unit-vector. The result is an equation to determine x. Thus one or more points are found on the vector XOL, whose cöordinates satisfy the equation ; and the locus is a surface whose degree is determined by that of the equation which gives the values of x.

But a vector equation in p, as we have seen, generally leads to three scalar equations, from which the three rectangular or other components of the sought vector are to be derived. Such a vector equation, then, usually belongs to a definite number of points in space. But in certain cases these may form a line, and even a surface, the vector equation losing as it were one or two of the three scalar equations to which it is usually equivalent.

Thus while the equation cup ft which is the vector of a definite point (since by making p a vector we have evidently assumed whatever be x. Hence the vector of any point whatever in the line drawn parallel to a from the extremity of a~l (3 satisfies the given equation. [The difference between the results depends upon the fact that Sap is indeterminate in the second form, but definite (= 0) in the first.]

124. Again, Vap .Vp/3 = ( Fa/3)

2 is equivalent to but two scalar equations. For it shews that Vap and V/3p are parallel, i.e. p lies in the same plane as a and /3, and can therefore be written (§ 24) where x and y are scalars as yet undetermined. which (§ 40) is the equation of a hyperbola whose asymptotes are in the directions of a and ft.

125. Again, the equation though apparently equivalent to three scalar equations, is really equivalent to one only. In fact we see by § 91 that it may be written

whence, if a be not zero, we have and thus (§ 101) the only condition is that p is coplanar with a, ft. Hence the equation represents the plane in which a and ft lie.

126. Some very curious results are obtained when we extend these processes of interpretation to functions of a quaternion

instead of functions of a mere vector p.

A scalar equation containing such a quaternion, along with quaternion constants, gives, as in last section, the equation of a surface, if we assign a definite value to w. Hence for successive values of w, we have successive surfaces belonging to a system ; and thus when w is indeterminate the equation represents not a surface, as before, but a volume, in the sense that the vector of any point within that volume satisfies the equation.

represents, for any assigned value of w, not greater than a, a sphere whose radius is Jd2 w\ Hence the equation is satisfied by the vector of any point whatever in the volume of a sphere of radius a, whose centre is origin. p.81 Again, by the same kind of investigation, where q = iu + p, is easily seen to represent the volume of a sphere of radius a described about the extremity of ft as centre. Also S(q*) = a2 is the equation of infinite space less the space contained in a sphere of radius a about the origin. Similar consequences as to the interpretation of vector equations in quaternions may be readily deduced by the reader.

127. The following transformation is enuntiated without proof by Hamilton (Lectures, p. 587, and Elements, p. 299).

which, if we take the positive sign, requires which is the required transformation.

[It is to be noticed that there are other results which might have been arrived at by using the negative sign above ; some involving an arbitrary unit-vector, others involving the imaginary of ordinary algebra.]

128. As a final example, we take a transformation of Hamilton’s, of great importance in the theory of surfaces of the second order.

Transform the expression

T. Q. I. p.82 in which a, /3, 7 arc any three mutually rectangular vectors, into the form which involves only two vector-constants, i, K.

[The student should remark here that t, K, two undetermined vectors, involve six disposable constants :—and that a, ft, 7, being a rectangular system, involve also only six constants.]

But ^2 (Sapf + /T (/) + 7 "2 (^)2 = p 2 (§§ 25, 73).

Multiply by /3 2 and subtract, we get

The left side breaks up into two real factors if /3 2 be intermediate in value to a2 and 7 2 : and that the right side may do so the term in p 2 must vanish. This condition gives Hence we must have where jp is an undetermined scalar.

To determine p, substitute in the expression for /3 2 , and we find

p.83 Thus the transformation succeeds if

Thus we have proved the possibility of the transformation, and determined the transforming vectors i, K.

129. By differentiating the equation

we obtain, as will be seen in Chapter IV, the following, where p also may be any vector whatever.

This is another very important formula of transformation ; and it will be a good exercise for the student to prove its truth by processes analogous to those in last section. We may merely observe, what indeed is obvious, that by putting p = p it becomes the formula of last section. And we see that we may write, with the recent values of i and K in terms of a, /3 y, the identity

6-2

p.84

130. In various quaternion investigations, especially in such as involve imaginary intersections of curves and surfaces, the old imaginary of algebra of course appears. But it is to be particularly noticed that this expression is analogous to a scalar and not to a vector, and that like real scalars it is commutative in multiplication with all other factors. Thus it appears, by the same proof as in algebra, that any quaternion expression which contains this imaginary can always be broken up into the sum of two parts, one real, the other multiplied by the first power of V 1. Such an expression, viz.

where q and q" are real quaternions, is called by Hamilton a biquaternion. [The student should be warned that the term Biquaternion has since been employed by other writers in the sense sometimes of a “set” of 8 elements, analogous to the Quaternion 4 ; sometimes for an expression q + Oq" where 6 is not the algebraic imaginary. By them Hamilton’s Biquaternion is called simply a quaternion with non-real constituents.] Some little care is requisite in the management of these expressions, but there is no new difficulty. The points to be observed are : first, that any biquaternion can be divided into a real and an imaginary part, the latter being the product of V - 1 by a real quaternion ; second, that this V - 1 is commutative with all other quantities in multiplication ; third, that if two biquaternions be equal, as so that an equation between biquaternions involves in general eight equations between scalars. Compare § 80.

131. We have obviously, since \l - 1 is a scalar,

Hence (§ 103) p.85 The only remark which need be made on such formulae is this, that the tensor of a biquaternion may vanish while both of the component quaternions are finite. the above formula gives may be written where a is any vector whatever. and therefore is the general form of a biquaternion whose tensor is zero.

132. More generally we have, q, r, q, r being any four real and non-evanescent quaternions,

That this product may vanish we must have where a is some unit-vector.

And the two equations now agree in giving

so that we have the biquaternion factors in the form and their product is which, of course, vanishes.

p.86 [A somewhat simpler investigation of the same proposition may be obtained by writing the biquaternions as

and shewing that

From this it appears that if the product of two bivectors

is zero, we must have where a may be any vector whatever. But this result is still more easily obtained by means of a direct process.

133. It may be well to observe here (as we intend to avail our selves of them in the succeeding Chapters) that certain abbreviated forms of expression may be used when they are not liable to confuse, or lead to error. Thus we may write

although the true meanings of these expressions are The former is justifiable, as T (Tq) = Tq, and therefore Tq is not required to signify the second tensor (or tensor of the tensor) of q. But the trigonometrical usage is defensible only on the score of convenience, and is habitually violated by the employment of cos"1 x in its natural and proper sense.

Similarly we may write

but it may be advisable not to use as the equivalent of either of those just written; inasmuch as it might be confounded with the (generally) different quantity although this is rarely written without the point or the brackets.

The question of the use of points or brackets is one on which no very definite rules can be laid down. A beginner ought to use p.87 them freely, and he will soon learn by trial which of them are absolutely necessary to prevent ambiguity.

In the present work this course has been adopted :—the earlier examples in each part of the subject being treated with a free use of points and brackets, while in the later examples superfluous marks of the kind are gradually got rid of.

It may be well to indicate some general principles which regulate the omission of these marks. Thus in 8 .a@ or V. a/3 the point is obviously unnecessary :—because So. = 0, and Va. = a, so that the S would annihilate the term if it applied to a alone, while in the same case the V would be superfluous. But in S.qr and V . qr, the point (or an equivalent) is indispensable, for Sq . r, and Vq.r are usually quite different from the first written quantities. In the case of K, and of d (used for scalar differentiation), the omission of the point indicates that the operator acts only on the nearest factor :— thus

while, if its action extend farther, we write

In more complex cases we must be ruled by the general principle of dropping nothing which is essential. Thus, for instance

may be written without ambiguity as but nothing more can be dropped without altering its value.

Another peculiarity of notation, which will occasionally be required, shows which portions of a complex product are affected by an operator. Thus we write

if V operates on a and also on r, but if it operates on r alone. See, in this connection, the last Example at the end of Chap. IV. below.

134. The beginner may expect to be at first a little puzzled with this aspect of the notation ; but, as he learns more of the subject, he will soon see clearly the distinction between such an expression as

p.88 where we may omit at pleasure either the point or the first V without altering the value, and the very different one

which admits of no such changes, without alteration of its value.

All these simplifications of notation are, in fact, merely examples of the transformations of quaternion expressions to which part of this Chapter has been devoted. Thus, to take a very simple ex ample, we easily see that

The above group does not nearly exhaust the list of even the simpler ways of expressing the given quantity. We recommend it to the careful study of the reader. He will find it advisable, at first, to use stops and brackets pretty freely; but will gradually learn to dispense with those which are not absolutely necessary to prevent ambiguity.

There is, however, one additional point of notation to which the reader’s attention should be most carefully directed. A very simple instance will suffice. Take the expressions

The first of these is

and presents no difficulty. But the second, though at first sight it closely resembles the first, is in general totally different in value, being in fact equal to For the denominator must be treated as one quaternion. If, then, we write we have so that, as stated above, We see therefore that