340 



Again, if we had used the second set of admissible values for «, ^, y 

 a = — 2, ^ = 3, 7 = 5, 

 we should have found successively 



1 —1 1 ' 4 10 —4' 



gi g» ^ ; 



4tp, + 8tf.,-8tp3 "91^,-271^, -8if?,-7if., + 25tp; 



(.i--2)' (6a; -3)" (5^ — 1)'' 



^^IT^ t^l7 - ^^, ^ 



.f— 2 —6.^ + 3 5^ — 1' 

 7. = 3, iï/=2, A' = 6, (^j = - 12, ^, = 6, ^3 = 6. 

 T =z const. (25^;^ — 16a' + 4), 5^ ■=z const .^j, ip^ = const. \\)^. 



Now we may apply the transformation 



whence we have 



9« 9 



{w—2Y (5.r» — I2.'c f 4) {6.7; - 3)'' (5.?;»— 2.7;+ 2) 

 9(l-«) _ 32— 27« 



~ (5A--l)^(7.r^ — 6a--f2) ~ (2 5.?;* — 16^+4)'' 

 and we shall obtain 



dt 



r {a.'—2)dx ^ r 



J 1/(5^^— 12.r + 4)(5^^— 2.r4-2U7.t-^-6^4-2) ~~ J I 



1/(5^^— 12.r+4)(5^^— 2.r4-2)(7.t-^-6^-f 2) J V^«(l -^(32— 27«) 



where the constant | is found by observing that x:=2-]-d implies 



* 7 29 " • 



The involution J, in this case, is somewhat special, because we 

 have now 



^4 — ^1 . 5, = ^1 • 

 In the corresponding system S the points A^ and A^ coincide, 

 and Ihe right line if'i passes through A^. Hence of the four reducible 

 integrals of deficiency p = 2 in the general case connected with 

 the involution ,/, three degenerate here at once into ordinary 

 logarithmic integrals. The integral 



Tdw 



ƒ, 



in the general case of deficiency p ^ 3, reduces here to an elliptic 

 integral of the third kind, but the transformation indicated above 

 effects still a further reduction, and we obtain another logarithmic 

 integral. 



