and Therefore, or, S.ax. But therefore, V. p (IVBy+mVya+nVaẞ) = o. p | (IV By + mVya + nVaß), p = x (lV By + mVya + nVaß). Sap=xSa (IV By + mVya+nVaß) = xlS. aVẞy + xmS . aVya + xnS. aVaß = xlSaßy. Sap = 1; 1 x= SaBy therefore, p = IV By + mVya + nVaß SaBy (2) This value of p satisfies the three given equations, but no other value of p will satisfy them. For, suppose the three equations to be satisfied by pi and P2. Then Therefore the vector (pip2) is at once perpendicular to a, to ß, and to y. But no real and actual vector can be perpendicular at the same time to three diplanar vectors, which a, B, y are supposed to be. Therefore (pip2) vanishes; therefore Οι = P2. Therefore, the three given equations can be satisfied by cne, and only one, value of p. The principle that no real and actual vector can be at once perpendicular to three diplanar vectors may be put in symbols as follows: Sao = o ; SBr = 0; Syo <= 0; Conversely, if o be an actual and real vector; then SaBy = 0. Had the three given scalar equations been of the form, Sẞyp=p; Syap = 9; Saßp: pa + qẞ + ry=aSẞyp + ẞSyap + ySaßppSaẞy, 81°, qß 132°. A vector, p, cannot be eliminated by fewer than four equations. If we are given only three equations, Saph; Sẞp = 1; Syp = m ; we have, 131° (2), SapVBy+SBpVya + SypVaß - pSaßyo.. (1) an equation into which the vector penters once. Now, suppose we are given a fourth equation, Sop = n. Then, if we multiply (1) by d and take scalars, we get S&V BySap + S&VyaSßp + S&VaẞSyp - SopSaßy = 0, an equation into which the vector p does not enter. SECTION 2 Linear Vector Equations (2) 133°. The general form of a linear vector equation is defined to be φρ = Σβδαρ + V .qp,* (1) where Ρ is an unknown vector, q a known quaternion, and @= (a1 + a2 + .. an), B = (B1 + ß2 + Bn), known vectors. The symbol stands for 'function,' and op is some vector coinitial with p. Similarly, if o be any other vector, σ φσ = Σβδασ + V . σ . *For proof, see Molenbroek, pp. 188-191. (2) If we interchange a and ß, and introduce Kq instead of q, we have o'p ΣaSBp+ V. (Kg) p. 134°. We have now to show that S.o V.qp Soqp S. o (Sq + Vq) p=So (Sq) p + S.σ (Vg) p = = =SqSpo+ S. poVq = S. p (Sq) σ - S. p (Vg) o (1) Functions which, like and p', possess this property are called Conjugate Functions. The function is said to be Self-conjugate when, for any two vectors, p and σ, and Soop = Sppo. 135°. Since ẞSa (p + σ + . .) = ẞSap + ẞSao + V. q(p+o+. .) = V . qp + V . po + ; (2) where and 8 are given. Then it is defined that (1) (2) -1 is a function which possesses properties corresponding to those of 4. As a matter of convenience we write : $(4) = 42, &c. ; -1(μ ̃1) = 4 ̄2, &c. $418=&p== 8; 4-1pp = 4-18 = p. According as mn, (3) 1 ¤ ̄1, 6, 62, &c., are operators which may alter both the length and direction of any vector upon which they operate. 2 is not to be confounded with the square of 4,-(6)2. 137°. We cannot enter here into the general theory of vector equations; suffice it to mention a simple method for their solution suggested by Dr. Molenbroek (" Theorie, &c.," p. 245). Let the given vector equation be Sud; S. po'v Svd.. (3) ρφίν = where λ, μ, v are any three noncoplanar vectors. But, by 134°, we also have P = . . . SV . φ'μφν + Sμδν . φ'νφλ + Sv8V . φ'λφόμ . . (4) As an example, let Then, 78°, 5. φλφ μφ ν Ναρβ = γ. op = y = Vapẞ=aSẞp - pSẞa + ESap; Since a, ẞ, y are any three vectors whatever, we may select them to represent respectively the λ, μ, v of equation (2). SAS Say; Sud Sẞy; Svd = y2. Hence, = = φλ = φα = φα = Va?β = αβ; d'v = d'y = dy = Vayß. γ. V. 'μo'v = p2V. aVayß=ß2Va (ayß — Sayß) V. p'vp'λ = a2V(Vayß. B)=a2V {(ayẞ- Sayẞ)B} =a2B2Vay-Ba2Sayẞ= ẞa2Saẞy — a2ß3Vya; αβεβα V. φ'λφόμ = αβ&Va = = αβαβ. S. (a2 § . φλφ' μφ'ν = S (α? β . β α .Vαγβ) = α2β'S . βαναγβ Therefore, (4), = a2ß2Sẞa (ayẞ - Sayß) = a2ß2SaßSaßy. Sya (aß2Saßy — a2ß3Vẞy) + SBY (Ba2Saßy — a2ß2Vya) — y2. a2ß3Vaß a2B2SaẞSaBy (aß2Sya+Ba2SBy) Saßy — «2ß2 (SayVBy+SByVya+SyyVaß) a B'SaẞSaBy a2ß2(a−1Sya + B−1SBy) Saßy—a2ß2. ySaßy, 82o (1), a-1Sya+ B-'SBY-Y. Saß K |