I40 KANSAS UNIVERSITY QUARTERLY. 
TYPE XIIl. 
| ; | ‘ | | 
Pre Vigne X,Y, 2, 1) |x Zens 
FE Rao PRR or eevee viet tet eel ey aR om Nein lacey 1G,./e0) Me Epa aaa nee 
DDS = A Sy yor Oe 3 tee Poses Sora eer ue be EE fc ba. 
OF a eae herd es ole Meenas [OG re Da. P e O  ae ae 
—_ == -—-a. 
| Ra SN a OL | eZ Ber ye 2g oD ai] ys eee 
AHO hCG ait. 412m ae Cage a eis al ha Se anne 
heey (Gi) IO) Sa XO | 20} 16 AD Nl, MC) EGON PG 
Tepe Gy Ole py eae oO) Pi Pa GO| ll Rep aaa 
Kens Vee lac Ts 
Dae St 
| 
eee aera 
I I ORO eR Ll. aa 
—= +-ma -+—_a? tha, 
3 . | ane os | | ate 4 2 
MVE ea Ll Raye pee Rear een 
Ae isDeMG ST An oni ee meE Else de utton (ua 
1, (Gp igh Pape yay ene Sie | Shee onpe yes 
| | 
Por ay 2) Pa eRaTeaeS) Teer Sa ae 
The coordinates of the point A of the invariant figure are (a,b,c); 
p and q determine the direction of the line Al; p, and q, determine 
the direction of the line AB, every point of which is an invariant 
point. m and h have the same meanings as k and h in type III in 
the plane. m and h are the same for all planes passing through 
Al. 
If k be made equal to one in the equations of type IV, the result 
is equations of type XIII as above. 
CANONICAL FORMS. 
The canonical forms of the thirteen types of projective transfor- 
mations in space may be obtained in the same way as that employed 
for the forms in the plane. 
Ty pends aq == kx yg yg ee 
Oy peal ck = kee yet Ran ye ee a 
Type Wl (xsS kx hy — sya ke eo 
w 
Dy pe ,.5 slows yee ye pa na. 
Type V. -x,=*--e; y,=y- kexS kK tha; Z,—=z-+max-+ 
3 2 
_ +(hm-4 n) | Ga, 
D} 
ae < 
(km——--+na)y-+km 
2 
