BREWSTER: COLLINE ATIONS OF SPACE. 291 



§2. Type X. 



23. For combinations with collineations of the same type in the 

 ease of type X, the work has been given, save that in which the in- 

 variant systems are different. This evidently gives a collineation of 

 type I. 



In combination with transformations of type XI, only one is worthy 

 of special mention. 1(I T(^>F) and n T(wtXF) give along I and V 

 the combination of identical and parabolic one-dimensional trans- 

 formations, evidently a parabolic resultant, while along m the re- 

 sultant of loxodromic and identical gives a loxodromic transforma- 

 tion. The space collineation is of type III. 



The following statement is given for reference : 



10 T(W>F)+ 10 Ti(mm'AF)== 1 T 2 (ZZ'mm'F) 

 10 T(//>F) + 11 T(Z / aF) = 10 Ti(ZA/>tF) 

 10 T(^>F)H- u T(^i/xF)== 10 Ti(^'2/*F) 

 10 T(M>F) -f n T(w AF) = 3 T(ll'mF). 



§3. Type IX. 



24. It is worthy of notice that a transformation of type IX, in com- 

 bination with those of other types, may result in one of type III. 

 The resultants of type I are not very frequent in this case. 



9 T(lmp£F) -f- <J Ti(lmp*F)== 9 T,(lmp<!F) 

 9 T(lmp*F)4- 9 Ti(lmp*iF)== 9 T 2 (lmpfeF) 

 9 T(lmp*F) + 9 Ti(lmip^iF) = 1 T(ll'2m 2 m' 2 F) 

 9 T(lmp2F) + 9 Ti(limi P i*iF) == 1 T(] 8 l',mam'*F) 

 9 T(lmp^F) + n T(l/xF) = = 9 Ti(lmp6F) 

 ,J T(lmp;!F) + 11 T(l 1 ^F) = 3 T(] 2 l' 2 mF) 

 9 T(lmp*F) + 10 T(ll>F) = 3 T(ll' 2 mF) 

 9 T(lmp*F) + 10 T(hlVF) = 3 T(l 2 r 2 mF). 



§ 4. Type VI. 



25. The resultant of a transformation of type VI with another of the 

 same type has been discussed. Combined with another of any other 

 type, the resultant is easily seen to be always of type I, since no 

 transformation of another type leaves invariant all points on a polar 

 line, save a special transformation of type X, which will be treated 

 under the head of singular transformations,* or when combined with 

 one of type IX. This last case we will explain more fully. Take 

 two transformations, 6 T(ll'ram'wF) and lJ T(l'mp^F). On the quadric 



* Art. 35. 



