QUATERNION TRANSFORMATIONS. 175 



whatever, and if ada + Mo + ydc+ &c. =D (wliere da de- 

 notes differentiation with regard to a, &c.), then 



D(pcr) =J)o . a-pJ)a + 2{SpD)(r. 

 For 



adaipa) =ctdap .<r + ap. da<T 



=.a.dap . (T — pa. . daO'-\-2{Spada)(T, 



since 



ap-{-pa. = 2Spx, 



If we now form the similar quantities for ^d^, 8cc., and 

 add the respective sides of the equation thus formed, 



.-. D(pc7)=Dp.o— pD(7 + 2(SpD)o-. . . .(II.) 



But T may be written aa. + bl3 + cy, where u, /9, 7 are rect- 

 angular unit vectors. D in this case becomes identical 

 with V, and the preceding form may be deduced. 



The more general form is occasionally useful. 



From I. we may, by taking scalar and vector parts, get 



Sv(po-) = S.vpo— S./5VO-, .... (III.) 



Vv(/5o-)=^Vpo- — VpVo-+2(Spv)o-. • . (IV.) 

 whence we may deduce by putting p = ar 



Vvo-^ = 2V.Vvo-.o- + 2(So-V)o-. . . (V.) 



We may also deduce from (i) expressions for y{Ypa) 

 and V (Spo-) . 



Thus V(Vp<7) = |V(p<^-<rp) 



= S . Vpcr— S 'p'^0'+ (f^pV)cr — (Scry)p + S Vp • o" — Sycr.p, 



and 



V(Spo-) =V . Vv/5 . (7- VpVv<^+ (SpV)cr+ (So-v)p. 



