NEWSON: NORMAL FORMS OF PROJECTIVE TRANSFORMATIONS. 133 
eileen: interchanging y and z, 
a8) 
X,—=x and y,=y-t«. 
The final forms in all cases agree with those given by Franz 
Meyer in the volume of papers read at the Chicago Congress, 
page Igo. 
Part Il1—Space. 
We come now to the consideration of the types of projective 
transformations in space. The normal forms of the thirteen types 
are determined from geometrical considerations. The expression 
of the results in determinant forms is not essential and indeed is 
sometimes a trifle strained. Each determinant when equated to 
zero is the equation of some invariant or otherwise essentially im- 
portant plane connected with the invariant figure. Sometimes it 
is possible to express the equation of a plane in another form dif- 
ferent from that here given and equally simple. 
EVE ols 
! 
> ON et Aaa Ke Vom 2a el Kea enol ba ane ATIC 
a, by Cy I a, by Cy I ja, by cy 1 ja, b, ¢, 1 
ae Di cGa I ain tDeen Ga, oT aie DeiAGate T a) Aleas (a r 
ae DC) law Deere. er aes Da "ears Eze oy snmeex ites 
—k 5 —kit —} 
| | | 
eg 24.2 eee Yee) SOD ie ha Arkh ob Ye en reat 
Ae MGIC ox bl ay beige T cle gl Ora ner Ieee y iChn 
dit DiatCat Ml ales DighGae t lag by cy 1 lag by Cy I 
| | 
Se Oe pee aap Dye erm I la, bs c, 1 a, b,c; I 
IK. Ya 21 1 eae ae er A a 
as DCist Aleit "Cie 1 
ay Se eore a! a Dae 1 
PUY © cet a la Dar aan 
Se —_—]1+-r-rs aes 
[Merge Nase ZeeyeeB Kal Aig Zeal aT 
| | 
\diag lO Om “i aay. Da eae eT 
a, bi eat Bee Gre al 
ja, by Cy 1 a, b, c, 1 
The invariant tetrahedron is ABCD; the coordinates of A are 
(a5b,c); those of B are’(a,,b),¢c,);) those of Care (a,,b,,¢,);, those 
of D are (a,,b,,c,). kis the anharmonic ratio of the transforma- 
tion along AB; k!* is that along AC; k!'rrs is that along AD. 
