292 



Sir W. Rowan Hamilton on Quaternions. 



consequences, and which presents itself under the form of a 

 quaternion : 



, ^ /dt du i\v\ 



/dv du\ . (dt dv\ , (du 

 \d~v ~ Tz) +"^ Vd^^ ~ dI7 ■^'^ Vd^ 



+ 1 



di\ 

 d3// 



(e.) 



In fact the equations (a.) give generally (see art. 21 of the 

 present series), 



( ix -{-jy + kz) {it +ju + kv)=:— {xt + yu-{- s:v) 

 + i[yv—zu) +j{zt — xv) + 1i{xu —yt), 



}• 



(f.) 



\{ xyztuv denote any six real numbers ; and the calculations by 

 which this is proved, show, still more generally, that the same 

 transformation must hold good, if each of the three symbols 

 iyj^ k, subject still to the equations (a.), be commutative in 

 arrangement, as a symbolic factor, with each of the three 

 other symbols .v, _?/, z ; even though the latter symbols, like 

 the former, should not be commutative in that way among 

 themselves ; and even if they should denote symbolical instead 

 of numerical multipliers, possessing still the distributive cha- 

 racter. We may therefore change the three symbols x,y, z, 

 respectively, to the three characteristics of partial differentia- 

 tion, -r-i -r-) -,- ; and thus the formula (e.) is seen to be in- 

 dx dy dz 



eluded in the formula (f.). And if we then, in like manner, 

 change the three symbols /, ?/, v, regarded as factors, to 



-T-i» -T-n -r-.i that is, to the characteristics of three partial dif- 

 dx' dy' d.s' 



ferentiations performed with respect to three new and inde- 

 pendent variables x', y\ z\ we shall thereby change -p to 



-T--r-;, and so obtain the formula: 

 dd^da? 



^ d .d-d 

 ^dx^^dy-^^'dz 



)0- 



d_ ._d_ _d^\ 



d^'"^'^d7/''^ d^7 



\dxdotI ^ 



+ 



dy dj/' 

 dzdy 



.(±± _i.A^ 



'^ \dxdy' dydx'J' 



.{d d 



'^'KdydF' 



z)+J' 



dy' 



dzdz'J 

 ./d_ d_ 

 Vd^d^' 



dx d^' 



) 



(g.) 



