342 REPORT — 1873. 



to express w\w\ in terms of iv^w^, so that 



e{w\,w'Jt 0K,M/3)r d{w\,w',)l 0K,m;,)|' 

 and also e(w\, w'^)f 3 _ 0(Wi, wj! , 3 



Section 2. — For the purpose of solving these equations, a Table similar to 

 that we have endeavoured to explain at the end of oui- remarks on section 3 

 of Konigsberger's first memoir is constructed ; using the same notation, we 

 have 



ofPiQi =-iiql +2nql -plqls+pUql, 



elF.Q^ =-p'S +piqU+Flql ~pUq\> 



These three equations, combined with the last three equations of section 1, 

 manifestly give the following : 



e(t«', + Wj, if'j + M'J, d{iv\—n\, iv\—w^\ =0, 



6(w\+iv^, iv\+wX^^e(iv\—tv^, tf'j— tfj)i,3 = 0, 



which reduces the problem to the solution of 



0(w', — ii'„ w\—w^\ =0, 



e{iv\—iv^, w\—w^^\ =0, 



«(w\— Wj, w',— w,\ 3 = 0. 



To resolve these equations Konigsberger enunciates the following proper- 

 ties : — 



If e^e^ are quantities which satisfy the three equations 



then also the three following equations are true : — 



e(u, + e^, t(:, + e^)l ^ epf,, u^y ^ 6(u^ + e^, u^ + e ^)l ^ e(u^, wjf 

 e(Mi + e„ ttj + tfjl d{u^, ujf e(u, + e^,u^ + e^)l d\u^, m jf 



These three formulae are fully proved by Konigsberger, and present no diffi- 

 culty. They are the result of the equations at the end of section 3 of the 

 first memoir and of those at the beginning of this section. We therefore 

 pass on to the theorems next enunciated, namely : — 





