Conway — Application of Quaternions to Electrical Theory. 7 



is a perfect differential, or that i' is " irrotational " on the surface. If this 

 condition is fulfilled, we have 



- S, Uv = 2nc + S e'Tclv'SUv (Yf - JcH-'X) ; 



and this is of the required form, for we can easily see that g possesses only a 

 logarithmic infinity, and that hence the integral is of the Fredholm type. 

 If we take the coefficient of h in the original equation, we have 



c'r, = 2mUv-iVVfi'Tdv'. 



Now ^VVft'Tdv' = SVi'V'fTdv' (since V/ =- V'/) 



= -jVi' Uv Vdv'A'f - J Ft' WSdv'V.f, 



and J Vi' W VdVV'f = J V. Vdv'Yf. VUv'i = - J Uv'Si'dv'Vf. 



Hence c'n - 2n-£ Uv -r J i' Uv'Sdv'V'f - J Uv'Si'dv'Vf, 



and c-r-nUv = - 2Tn + \ Vi Uv' Uv Sdv'Vf -\VUv Uv'Si'dv'V'.f 



which is a vector integral equation of Fredholm type. 



The solution of M. Poincard is somewhat different, and can be obtained as 

 follows : — 



By Stokes's theorem, 



J V(i' Uv Vdv'V')f = \fV{ Vdv'Vi' Vv' 



= ifUv'Si'dv'V.Uv' + jfUv'Sdv'V'i'. 

 Applying Stokes's theorem* again to the last integral, we find 



jfUv'Sdv'V'i' = - St'dv'V'.fUv'. 

 Making use of this, we get 



c" Ft, Uv = -2Tn+\ Vi Uv UvSdv'Vy - ffVUv'Si'dv'V Uv' Uv 

 + \8idv'V.fVUv'Uv. 



% 4. On the Eelativity Pkinciple. 



Before coming to the principle of relativity let us solve this quaternion 

 problem : — Wliat is the most general form of the linear functions /,( ), 

 /"-{ )> /sC )) so that for any two quaternions p and q 



/.(P)/2(<1) = /3(pq), 

 and in which Sf.i{f)) = where p is a vector ? 



On putting successively p = 1 and q = 1 we get 



/.(P)/.(l) = /3(P) and /,(l)/.(q) = /^(q). 



* The actual equation given by Poincare, Rendicontl del Circolo Mateinatico, ili Palermo, 

 tomo xxix (1910), p. 1S6, equation (3), is (in our notation), 



nliich appeals to be erroneous. 



