296 KANSAS UNIVERSITY SCIENCE BULLETIN. 



termine the existence of 3 G 2 (ll'm),by taking twotranforrnations having 

 on 1 and 1' one common invariant point with reciprocal values of k, the 

 resultant is at once a parabolic transformation, and the space collinea- 

 tion is of type III. If we combine go 1 of these, in order to produce 

 x G.i(lmF), we have also combined oc 1 3 G 2 (ll'mF), giving us the group 

 3 G 3 (lmF). But this group we have already proved to contain 

 ; 'G 2 (lmpF), n Gi(/|uF), and n Gi(mAF). Hence, we may state with- 

 out further comment that these same three groups exist in 1 G4(lmF). 

 The existence in the fundamental group 1 G2(ll'mm'F) of both of 

 the dualistic groups 6 Gi(H'mm'wF) and 6 Gi(li'mmWF) of type VI, 

 and of the two one-parameter groups of type X, one on each system 

 of generators of the invariant surface, is apparent, the conditions 

 necessary to produce them being merely the proper choice of the par- 

 ameters r and r' of the group. The following table gives the group 

 structure of type I, first class, in symbolic language : 



^(ll'mm'F) = ^(ll'mm'F) + 10 Gi( IV /x F) + 10 Gi(mm' /i F) 

 + 6 Gi(irmm'wF) + 6 Gi(U'mmWF). 



K3*(ll'mF)-= oo 1 ^(ll'mm'F) + 3 G 2 (ll'mF) + 6 G 2 (11' mIF) 

 ■+ fi G 2 (ll'mVF) + 10 Gi(^>F) + 10 G 2 (mAF) 

 + n Gi(mXF). 



1 G 4 (lmF)= oo 2 H3i(U'mm'F) + 3 G 3 (lmF) + 6 G 3 (lm^F) 

 + e G3(ApF) + 10 G 2 (^F) + 10 G 2 (mAF) 

 + n Gi{m A F) + u Gi(l(xF) + u G 2 (lmpF). 



1 G 4 (irF)=-oo 2 'G^n'mm'Fl + oo 1 3 G 2 (ll'mF)-f oo 1 6 G 2 (ll'm~^~F) 



+ 00 1 6 G 2 (ll'mVF) + 10 G 1 (ZZ>F) + 10 G3(AF) 

 + QC 1 n Gi(mAF). 



1 G,(lF)==oc 3 1 G 2 (lTmm'F) + 3 G4(lF) + oo 1 6 G 8 (lm77F) 

 + 00 1 6 G 3 (ApF) + 10 G 2 (^F) + 10 G 3 (AF) 

 + u Gi{lnF) + az 1 11 Gi(mAF) + oo 1 y G 2 (lm77F). 



1 Ge(F) = cx^G 2 (TT'mm / F) + 2oc 13 G4(TF)-|-oo 26 G3(7m^F) 



+ oo 2 6 G 3 (a"FF) + oc 3 6 G 3 (pTF) + 2 10 G 3 (aF) 

 + 2 ex 1 11 Gi(mAF) + oo 2 9 G 2 (Tm"£"F). 



34. Second class. — We found (art. 21 ) that for any definite value 

 of r there existed a one-parameter subgroup of ^^(ll'mm'F). But 

 with r a rational number, we may have three cases, viz., r may be odd 

 over odd, even over odd, odd over even ; the fourth possible case, even 

 over even, degenerating always to some one of the other three. The 

 group structure in each of these three cases must be separately ex- 

 amined. For all continuous groups the three cases are identical; but 

 we shall see that, with regard to singular transformations, the three 

 are distinct. Hence, in our present statement, we shall merely indicate 



