124 Mr Brill, On the Application of Quaternions [Nov. 24, 



5. In conclusion, we may notice that Laplace's Equation is 

 a particular case of a more general equation to which the qua- 

 ternion method is applicable, viz. the equation 



d\i d 2 u d\i ^ u _o 

 dtf + dy i+ d7 + d?~ 



In this case, as in the former, we have four related solutions, 

 but the equations connecting them are 



8S_<57_8/3 da 

 dx dy dz dt ' 



dy dz dx dt ' 

 dS_d/3_da dy 

 dz dec dy dt ' 



dx dy dz dt 



These four equations are equivalent to the single quaternion 

 equation 



{st + i l + ik +k s) < ~ S + h+j0 + h) = ° (4> 



We however require three special solutions of this equation, for 

 which we may take the following : 



u =t — ix+ jy + hz, 



v = t + ix — jy + hz, 



w = t + ix + jy — hz. 



And, proceeding as in Article 2, we obtain a relation of the 

 form 



ds = du . U+ dv . V + dw . W, 



where s is any solution of (4), and IT, V, W are quaternions whose 

 values depend upon x, y, z, t and not upon dx, dy, dz, dt* From 

 this we can deduce by proceeding as in the former case that if 



We have 



^-(»+*5>--('B + >S)- 



