Wang — The Differentiation of Quaternion Functions. 77 



tlierefore 



fZ.e«=Limf^(l + 7^r7)"=LimL(l^/^^)" f/?+ )(! + «?)" Vq-V.[Unq)"\ r(Vdq:Vq)^ 



= e^dq + {el Vq - V. e<>) F{ Vdq : Vq). 



(5.) Differentiate any function of q that can be expressed in a series of 

 powers of q (including the limit form loge q) with scalar indices and coefficients. 



fq = 2fl.r. 

 dfq = %a^d.(f- = 'S.a^xq'^-'' dq + {S.a^xq'-' Vq - Vta.^q") V( Vdq : Vq] 



= f'l-dq + (fq . Vq - Vfq) V( Vdq : Vq). 



(6.) Differentiate Fq=ff2q, where/, and /j are any functions which have 

 previously been proved that satisfy the general expression 



dFq = dfM =f,f,q . dfq + (/, /,? . Vfq - Vffq, F( Fdfq: Vfq) 

 =.Unq . dfq + (f'fq . Vfq - Vffq) V(Vdq: Vq) 

 = /./// { /"32 • ^? + (f'^i ■ Vq - Vfq) V{Vdq:Vq)\ 



+ (f\fq ■ Vf.q - Vffq) V( Vdq : Vq) 

 = F'q . dq + (F'q . Vq - VFq) V{ Vdq : Vq) 

 {■.• rj,q.f\q-F'q). 

 The successive operation by either q or q- y on fq has also general expres- 

 sions. However, for simplicity, we shall first solve some questions connecting 

 to them. 



(1.) Find the values of al^iQ^i)' USViQ^i), aS^iQVdq), a\V[QVdq), 

 (J I Q Vdq, and (jj V{Vdq : Vq), where dffdq denote -/I + ^'/^ +jfj + kfk. 



Let q = vj + i.r + jy + kz and Q = W + iX + jY + kZ. 

 Hence, - SQ = - W, iS .Qi = - iX, jS .Qj=-jY, and kS .Qk = - kZ ; 

 thus, aSS.Qdq = -Q. [1] 



Again, -VQ = -iX-jY- kZ, iV . Qi = - W + JY + kZ, 



jV.Qj=-W+iX+kZ, and kV . Qk = - W + iX + jY; 

 thus, a\V.Qdq = -3SQ+VQ. [2] 



Also, (i\S.QVdq = q\S.Qdq without the first term = - VQ. [3] 

 Similarly ai^.Q Vdq = - 3SQ + 2 VQ. [4] 



a i QVdq ^ a Is.QVdq + a i V. QVdq ^ - ssQ + vq. [5] 



And iV(i: Vq) = iV(i. Vq) : Vq' = - {jy + kz) : Vq\ 



similarly 



j V{j : F?) = - {ix + hz) : Vq\ and kV{k:Vq) = - (ix +jy) : Vq' ; 



therefore a\ F( Vdq : Vq) = - 2Vq : Vq' = - 2 Vq-\ [6] 



