341 

 In fact, we shall have 



J (5.r' — 12/r + 4)K(7.7r' — 6^ 2)(5.r*— 2.^4-3) ~ J 



dt 



(5.r' — 12/r + 4)K(7.7r'-6.f + 2)(5.r*— 2.^+3) J t\/\—t' 



As I remarked in the beginning, the principal condition for the 

 redncibility has been given by Bolza and by Igel. I will now show 

 that the invariant relations they deduced, may be derived without 

 diflficulty from the results obtained here. 



Bolza and Igel both introduce the anharmonic ratios >',,,, >^,i, Aj,, 

 formed by the roots of each pair of the quadrics i|'j, i|7j, tp,. 



The anharmonic ratio P.,^, formed by the roots of if;, and ip,, is 

 given by the equation 



and putting 



the constant j/j is related to the invariants A^^, A^^, ^,, by the 

 equation 



[/a a a.. 



Now we have 



J\.„~ -A,, -<4.. 



ca — g^ ah — A" gh — af 

 and hence by equations (13) 



A A A 



where s stands for « + /? + y. 

 In this way we get 



and we may take 

 Similarly Ave obtain 



'^'■81 ^ 



y« 



(/?4-r)(«+/J) 



i/a,,+i 



f*3=77== 



1 y (y-f«)( 



i/X-_i »^ (y-f«)((^+Y) 



