342 



and we infer 



lhjh__ ^ 



Hence we have between fi, jj, fZj the set of relations 



«Ml + ^MaM» -f YMiMa = , 

 «M3M1 + /^M» + yf^aMi = ^ ' 



and by eliminating «, ^, r, we obtain as the invariant relation between 

 the quadrics \p^,xp^,xps Bolza's equation 



— 1 ~ (X,"- — fi,' — Ms' + 2 MiM^Ms = ^ 



Mj ^ M 



ft, Ml 1 



When this relation is satisfied for any one of the possible deter- 

 minations of the constants ii^.n^^Hs, the quadris tf',,t|%iV'8 are apt 

 to build up a degenerate integral. 



As we have 



./y-^Mi + l^ (Mi + ir 



■"~Mi-^ (Mi'-l)' 

 we have also 



" " "i/Cmi'-iKm,'-!)' 



Now it follows from Bolza's equation that 



(Ml* — i)(m,' — 1) = (MiM,— mJ' 

 and since 



a-\-3 s 



Ml = MiMj t MiM, — Mi =^-MiM, » 



r 



we get 



y(l4-Mi+M3 + MiMj 7 



^ ^•js'^si— = - t (yM,M, + yM8Mr-(« + |5)MiM» • 



«MiM, « «MiMjMj 



Writing out similar expressions for ^"Aj^A,, and t^O-jiA,,, we find 



by adding them Igel's equation 



^KX'. + ^^^+ ^07= > . 

 Again, if for any one of the possible determinations of the surds 

 this relation is satisfied, a degenerate integral can be constructed 

 by means of the quadrics tp^, if?,. i|^,. Both the equations of condition 

 given by Bolza and by Igel involve rather intricate surds, and I 

 should say that they are less adapted for examining the reducibility 

 of a given integral than the equation (12) deduced in this paper. 



