170 Professor Baker, On the transformation of the equations of 



§ 2. With the symbols e, i, j, k above explained, satisfying the 

 equations 



e^ = i^=^ f =k^=-u, ei=- ie, ...,jk= - kj, ...,eu = ue ^ e, ... , 

 we clearly have 



(er + ii+ jy] + kt) {ecj, + iF + jG + kH) 

 = -u {T(f> + $F + rjG + CH) 



+ jk {r,H - CG) + ki {IF - m) + if HG - rjF) 

 + ei (ri?^ - i<f>) + ej (tG^ - r](l>) + e^^ (r^ - C<^), 

 and if we denote this by 



uW + jkL + kiM + *)W + eiZ + ejY + ekZ, 

 we further have the identity (A) 



(gT- + i^ + j^ + ^•^) {uW + jkL + ^iM + ijN + eiZ + e^T + e^-Z) 

 = e{TW + iX + r]Y+ tZ) + i{^W + rjN - CM - rX) 

 + j {t^W -^N + iL-rY) + k {l,W + iM-7]L- tZ) 

 + ei HL + rjM+ CN) + ii (- r]Z + CY + tL) 

 + J/-1 (- ^Z+ ^Z + tM) + y^i (- £ 7 + ^Z + riV), 

 where e^ == ijk, i^ = ejk, j^ = eki, k^ = eij. dH 



If now we interpret |, rj, t„ r as symbols of differentiation in 

 rjegard respectively to x, y, z, t, the coefficients of e, i, ..., e^, ... 

 arising in these equations are the combinations occurring in the 

 equations of electrodynamics considered by Mr Johnston in the 

 paper referred to, the F, G, H, Z, Y, Z being certain constant 

 multiples of those usually denoted by these symbols, except that W, 

 which vanishes in the electrodynamic case, is here retained, and t 

 is a certain imaginary multiple of the time. 



However, a symbol er + i^ + jr] + kc,, if we use the interpre- 

 tation of e, i, j; k above developed, is the same as 



rfO,-E\+^ fO, I\+7] fO, J\ + I fO, K 



u, 



or 



0, 



\I, 0) \J, 0) \K, 0) 



-rE + il + rjJ + lKv 



.tE + ^I + rjJ+ rji, 0, 



or, taking E, I, J, K as above, is the same as 



0, 0, -T + erj, i^eC 



0, 0, -^-eC, -T-er] 



T + 677, I - e^, 0, 



-i-eC, r-erj, 0, 



