unchanged. Hence the product of two perpendicular vectors must be a vector, and a simple extension of the same reasoning shews that it must be perpendicular to each of the factors. It is easy to carry this farther, but enough has been said to shew the character of the reasoning. EXAMPLES TO CHAPTER II. 1. It is obvious from the properties of polar triangles that any mode of representing versors by the sides of a triangle must have an equivalent statement in which they are represented by angles in the polar triangle. Shew directly that the product of two versors represented by two angles of a spherical triangle is a third versor represented by the supplement of the remaining angle of the triangle; and determine the rule which connects the directions in which these angles are to be measured. 2. Hence derive another proof that we have not generally pq = qp. 3. Hence shew that the proof of the associative principle, § 57, may be made to depend upon the fact that if from any point of the sphere tangent arcs be drawn to a spherical conic, and also ares to the foci, the inclination of either tangent arc to one of the focal • arcs is equal to that of the other tangent arc to the other focal arc. 4. Prove the formulae 28.αβγ = αβγ-γβα, 2 Γ.αβγ = αβγ + γβα. 5. Shew that, whatever odd number of vectors be represented by a, ẞ, y, &c., we have always 6. Shew that and Γ.αβγδε = Γ. εδγβα, Γ. αβγδεζη = Γ. ηζεδγβα, &c. S.VaẞVẞy Vya—(S.aßy)2, = V.VaẞVẞy Vya Vaß (y2Saß-SpySya)+....... 7. If a, ẞ, y be any vectors at right angles to each other, shew that (a3 +ß3+y3) S.aßy = a1Vßy +ß1Vya+y1Vaß. 8. If a, ß, y be non-coplanar vectors, find the relations among the six scalars, x, y, z and §, n, s, which are implied in the equation xa+yẞ+zy = {Vßy+nVya+¿Vaß. 9. If a, ß, y be any three non-coplanar vectors, express any fourth vector, d, as a linear function of each of the following sets of three derived vectors, and Γ. γαβ, Γ.αβγ, Γ.βγα, V.VaẞVẞy Vya, V.VBуVyaVaß, V.VyaVaẞVẞy. 10. Eliminate p from the equations Sapa, S3pb, Syp = c, Sop = d, where a, ẞ, y, d are vectors, and a, b, c, d scalars. 11. In any quadrilateral, plane or gauche, the sum of the squares of the diagonals is double the sum of the squares of the lines joining the middle points of opposite sides. CHAPTER III. INTERPRETATIONS AND TRANSFORMATIONS OF QUATERNION EXPRESSIONS. 94.] AMONG the most useful characteristics of the Calculus of Quaternions, the ease of interpreting its formulae geometrically, and the extraordinary variety of transformations of which the simplest expressions are susceptible, deserve a prominent place. We devote this Chapter to some of the more simple of these, together with a few of somewhat more complex character but of constant occurrence in geometrical and physical investigations. Others will appear in every succeeding Chapter. It is here, perhaps, that the student is likely to feel most strongly the peculiar difficulties of the new Calculus. But on that very account he should endeavour to master them, for the variety of forms which any one formula may assume, though puzzling to the beginner, is of the most extraordinary advantage to the advanced student, not alone as aiding him in the solution of complex questions, but as affording an invaluable mental discipline. 95.] If we refer again to the figure of § 77 we see that Hence, if e be a unit-vector perpendicular to a and ẞ, or Thus the scalar of the product of two vectors is the continued product of their tensors and of the cosine of the supplement of the contained angle. The tensor of the vector of the product of two vectors is the continued product of their tensors and the sine of the contained angle; and the versor of the same is a unit-vector perpendicular to both, and such that the rotation about it from the first vector (i. e. the multiplier) to the second is left-handed or positive. Hence TVaß is double the area of the triangle two of whose sides are a, ß. 97.1 (a.) In any triangle ABC we have Hence AC = AB+BC. AC2 = SAC AC = S.AC (AB+BC). With the usual notation for a plane triangle the interpretation of this formula is These are truths, but not truisms, as we might have been led to fancy from the excessive simplicity of the process employed. E 98.] From § 96 it follows that, if a and ẞ be both actual (i. e. real and non-evanescent) vectors, the equation Saß = 0 shews that cos 0 = 0, or that a is perpendicular to ß. And, in fact, we know already that the product of two perpendicular vectors is a vector. Again, if Vaß = 0, we must have sin = 0, or a is parallel to B. We know already that the product of two parallel vectors is a scalar. where Also r = √√x2 + y2+z2, r′ = √x2 + y2+2′2. Vaß = rr { yd2 = 2y i + 2x2=ad j + my—your b} - Γαβ xy-yx k}. These express in Cartesian coordinates 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 may be written δ.αβγ because the quaternion aßy may be broken up into (Saß) y+(Vaẞ) y of which the first term is a vector. |