newson: continuous groups. 89 



eral projective group G^ contains cc'^ sub-groups G^.p, and also 

 oc- sub-groups Gg.;. Three conditions determine a linear element; 

 if the co-ordinaies of a linear element are fixed, the variation of 

 the other five parameters generates a five-termed group G j. If two 

 points of the plane are invariant, four parameters are fixed and the 

 variation of the remaining four produces the four-termed group 

 G^ij. If two lines of the plane are invariant, four parameters are 

 fixed and the remaining four produce the four-termed group G^ y. 

 If a point and a line are invariant, four parameters are fixed and 

 the remaining four produce the four-termed group G^ ,.. If two 

 points, their join and a line through one of them; or two lines, their 

 intersection and a point on one of them are invariant, five condi- 

 tions are satisfied; the variation of the remaining three parameters 

 generates a three-termed group G^. If three non-collinear points 

 or three non-concurrent lines are invariant, all six co-ordinate par- 

 ameters are constant and the two anharmonic ratio parameters 

 generate a two-termed group G^,. (If three coUincar points are 

 invariant, all the points of the line are invariant; but the transfor- 

 mations leaving all the points of a line invariant are of the kind S 

 and S'; the same is true of three concurrent lines. Groups of this 

 kind will be discussed later, j 



ij3 One-Termed Croups of Transformations of the Kind T. 



We shall next show that a two-termed group G (ABC) can be 

 decomposed into one-termed sub-groups. To do this we proceed 

 as follows: 



Let T^ be any transformation of the group G^, and let its char- 

 acteristic anharmonic ratios along the invariant lines x, y, z be 

 A., /X.J V, respectively. Let T.3 be another transformation of the 

 group Gj, and let its characteristic anharmonic ratios be A, /x., Vg 

 respectively. These two transformations are together equivalent 

 to another transformation of the same group G^, whose character- 

 istic anharmoic ratios are respectively A3 fji.^ v.j. But these two- 

 dimensional transformations each determine along the invariant 

 lines one-dimensional transformations. The three one-dimensional 

 transformations, one along each of the invariant lines, since they 

 leave two points of the line invariant, belong respectively to one- 

 termed groups of transformations of the points on a line. Hence 

 we have by theorem 5, part I, AjA.,=A.j; /x^fi^^fji^; v^v.,~v.^. (i) 

 But by means of the relations Aj/AjV, =^i, X.^fji.^v2 = i, and X^fx^v.^ = i, 

 we have Vj — (A,)u.j)-i; v.,={>^2l^-2'>'^' "3 = (-^3/^3 )"^- ^o^^ ^^^ us 



put a rr-\-'^ where a is some unknown constant; let us also put 



1 



ix,=.\-^, wdiere /> is another unknown constant. The three charac- 



