752] 
237 
752. 
ON THE FINITE GROUPS OF LINEAR TRANSFORMATIONS OF 
A VARIABLE; WITH A CORRECTION. 
[From the Mathematische Annalen, t. xvi. (1880), pp. 260—263; 439, 440.] 
In the paper “Ueber endliche Gruppen linearer Transformationen einer Veränder 
lichen,” Math. Ann. t. xii. (1877), pp. 23—46, Prof. Gorclan gave in a very elegant form 
the groups of 12, 24 and 60 homographic transformations aX —.~ > . The groups of 12 
and 24 are in the like form, the group of 24 thus containing as part of itself the 
group of 12; but the group of 60 is in a different form, not containing as part of 
itself the group of 12. It is, I think, desirable to present the group of 60 in the 
form in which it contains as part of itself Gorclan’s group of 12: and moreover to 
identify the group of 60 with the group of the 60 positive permutations of 5 letters: 
or (writing abc for the cyclical permutation a into b, b into c, c into a, and so in 
other cases) say with the group of the 60 positive permutations 1, abc, ab .cd and 
abcde. 
Any two forms of a group are, it is well known, connected as follows, viz. if 
1, a, ß, ... are the functional symbols of the one form, then those of the other form 
are 1, AaA -1 , A/3A -1 , ... (where in the case in question A is a functional symbol of 
the like homographic form, fra? = ^■—-gj. But instead of obtaining the new form in 
this manner, I found it easier to use the values of the rotation-symbol 
cos - + sin - (i cos X +j cos Y + k cos Z) 
for the axes of the icosahedron or dodecahedron, given in my paper “ Notes on 
polyhedra,” Quart. Math. Jour. t. vn. (1866), pp. 304—316, [375]; viz. if for any axes,