208 THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 



quintuplet, by completing the vertical circle : 



« =1325468709 

 ^. = 3014876592 

 </, = 093865742i 

 b =9206745813 

 e =2197584630. 



Next 



5, = ^^,= 1843705296, 



which completes our fifteen substitutions of the second 

 order, which are all of the same form. 

 The substitutions of the third order are 



a=«A = 89i7520364, 

 ^=a,c, = 276490i583, 

 y=a Ca = 7940268 153, 

 h=a 6^= 1752894306, 



€ = fl!,6 =7281954063, 

 f=«,e, =5936802174, 

 ^=^. = 4162357089, 



^=6,c, =4613250798, 

 \=c ^, = 5163248970, 



;t^ = 6e, =5612349807, 



a 



3680594127 = ft,e, 



^''= 7 104832956 = aifi?, 



7* = 850396i724=ae„ 



S* = 1487302569 = a d^, 



6^ = 420769i358=«^c, 



r=873oi49526=«A, 

 7?^ = 245i6379o8 = Z>,c„ 

 ^" = 3541628097 = 6,^,, 

 ^ = 2546139780 = ^ e, 

 //,* = 34561 20879 = Z> ^' 



Three more quintuplets are 



«i = 63245i0987 

 ^z= 1843705296 



</, = 093865742i 



c =5289176340 



^2 =7492061835, 



«, = 63245i0987 

 b =9206745813 



c, = 8059367 142 



^,= 1578293460 



^,=4731082695, 



«z = 62354i9078 

 ^z= 1843705296 

 d =0674923851 

 e =2197584630 

 ^z = 8059367142. 



The powers of ab^, ajb^, ac^, aj)^y afi, aj)^ are the twenty- 



