the equation of the right cone of which ẞ is the axis, and a a side. ADDITIONAL EXAMPLES TO CHAP. IX. Prove that S. (a + B) (B + y) (y + a) = 2S′. aßy. 1. 2. S. VaẞVByVya=-(Saẞy). 3. 4. S(VByVya) = y3Saß - SẞySyo. S. V (VaßVßy) V (VßyVya) V (VyaVaß) =— (S. aßy)*. 9. S(Vaßy Vẞya Vyaß) = 4SaßSBySyaS. aßy. 10. The expression VaẞVyd+ Vay V§ß + Vad Vßy denotes a vector. What vector? (Tait's Quaternions. Miscellaneous Ex. 1.) 11, SapS. Byd-SBpS. yda + SypS. daß - SspS. aßy=0. 12. (aßy)2 = 2a2ß3y3 + a3 (By)2+ ß2 (ay)2+ y2 (aß)3 — 4aySaßSBy. (Hamilton, Elements, p. 346.) 13. With the notation of the Note, Art. 69. 5, we shall have DABC = OABC – OBCD +OCDA–ODAB. 14. When A, B, C, D are in the same plane, a. BCD-B. CDA + 7. DAB-8. ABC = 0, where BCD, &c. are the areas of the triangles. 15. δ. αβγ+αν .βγδ +βΓ . γδα + γ V . δαβ = 45'. αβγδ. 16. VaẞVyd+ Vßy Vda + VydVaß + VdaVẞy is a scalar. What is its geometrical meaning? 17. Find the equation of the sphere circumscribing a given tetrahedron. 18. A straight line intersects a fixed line at right angles, and turns uniformly about it while it slides uniformly along it. Find the equation of the surface described (1) when the fixed line is straight, (2) when it is circular. CHAPTER X. VECTOR EQUATIONS OF THE FIRST DEGREE. WITH the object of giving the student an idea of one of the physical applications of Quaternions, we will treat the solution of linear and vector equations from an elementary kinematical point of view. For this purpose we choose the problem of the deformation of a solid or fluid body, when all its parts are similarly and equally deformed. DEF. Homogeneous Strain is such that portions of a body, originally equal, similar, and similarly placed, remain after the strain equal, similar, and similarly placed. Thus straight lines remain straight lines, parallel lines remain parallel, equal parallel lines remain equal, planes remain planes, parallel planes remain parallel, and equal areas on parallel planes remain equal. Also the volumes of all portions of the body are increased or diminished in the same proportion, as is easily seen by supposing the body originally divided into small equal cubes by series of planes perpendicular to each other. After the strain, these cubes are all changed into similar, similarly placed, and equal parallelepipeds. It is thus obvious that a homogeneous strain is entirely determined if we know into what vectors three given (non-coplanar) vectors are changed by it. Thus if a, ß, y become a', B', y' respectively: any other vector, which may of course be expressed as No needful generality is lost, while much simplification is gained, by taking a, ß, y as unit vectors at right angles to one another. This is, in fact, the method already spoken of, i. e. the imaginary division of the body into small equal cubes, by three mutually perpendicular series of equidistant planes. We thus have Comparing these expressions we see that Homogeneous Strain alters a vector into a definite linear and vector function of its original value. In abbreviated notation, we may write (as in Art. 63, though our symbol, as will soon be seen, is more general than that there employed) where op=-(a'Sup + P'Sßp + y'Syp), itself depends upon nine independent constants involved in the three equations фа φβ = β' φγ = γ For a', B', y' may of course be expressed in terms of a, ß, y: and, as they are quite independent of one another, the nine coefficients in the following equations may have absolutely any values whatever ; = Aa + cẞ+b'y pa = a' by=y' = ba+a'ß + Cy) .(a). In discussing the particular form of which occurs in the treatment of central surfaces of the second order we found, Art. 44, that it possessed the property whatever vectors are represented by p and σ. Remembering that a, ß, y form a rectangular unit system, we find from (a) δ. βφα with other similar pairs; so that our new value of satisfies (b) if, and only if, we have in (a) as will be seen im The physical meaning of this condition, mediately, is that the distortion expressed by takes place without rotation. In this case the nine constants are reduced to six. But, although (b) is not generally true, we have S.opp=-(SaoSap + Sẞ'oSßp + Sy'σSyp) = − S. p (aSa'o + BSB'o + ySy'o), where the expression in brackets is a linear and vector function of σ, depending upon the same nine scalars as those in ; and which we may therefore express by ', so that po=-(aSa'o + BSB'o+ySyo)....(d). And with this we have obviously which is the general relation, of which (b) is a mere particular case. By putting a, ẞ, y in succession for σ in (d) and referring to (a) we have |