339 



Now equations (9) and (3) give 



L=:2, ifr-_3, A'=l, 7. =4, ^, = -3, 9,-^-1 

 and then by equations (1), (4) and (5) we find 



T=co7ist.{hx'~-Qx^4), è4=co«s«.(7;l•-2)^ t|?,=cons<.(35.t-' — 64.r -16)) 

 The elements §ji|'i, ^,V',i è,V'r *4^4. ^'' of the invohition J being 

 known, we may put for instance 



_g,ip,_ x'jbx'- 12x^4) 



and obtain consequently 



t 1 1- < 



x' {hx'-l2x + A)~ (x-iy {hx'-'^^^ + 2) {1x^-6x^2) 

 375<— 32 8-3< 



= (7^_2)»(35.r*— 64;r— T6)~(5.r''-6^4-4)'' 

 The above transformation now will reduce four integrals of 

 deficiency p = 2, connected with the involution ./, and we may 

 write down at once 



J l/f5a;^-12.r + 4H5^^=2^ + 2)(7^'''-6^ + 2) ~"~ V^J 1/^(1-0(8=37) ' 



^dx - r '^ 



y^^^^^^^x^^^^^^^^^^^^^^)^)" ^ ^ P^f (32-3750(8 37)' 



r {x — \)dx 



J ''\/{ha^^\2x\^^x' -Qx + 2) {—?>^x* + QAx^\Q) 



_ r dt 



= — 1/2 - 



J \/ta—t 



JViha 



(1—0(32-3750(8-30 



X dx 



l/[5'^'-2^f2)(7^'--6.r-j-2)(-35^'' + 64..+ l6) 

 r dt 



~ J 1/(1—0 (32=3750 (8-30' 

 the constants — -^, etc. at the right hand sides being easily found 



by observing, that for small values of x we must have t = x\ 



As I have remarked before, the same transformation will also 

 reduce an integral of deficiency p = 3, connected with the involution J. 



In fact, we have 



(5^' — 6ar + 4)<ia; 



ƒ; 



l/(5;t'— 12^ + 4) (5a!'-2x + 2)(7;r'-6;r-|-2)(—35;i:' + 64;r+ 16) 



= '^^JFm 



dt 



\/t{\ -0(32—375 



