136 
ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE 
[127 
MM'x 2 = (aa' — bb' — cd — dd') x x + ( ab' + ci'b + cd' — c'd) y x 
+ ( ac + a'c + db' — d'b) z x + ( ad' + a'd + be — b'c) w Xi 
MM'y 2 = (ab' + a'b— cd' 4- c'd) ^ 4- (— aa' + bb' — cc' — <PT) ^ 
+ ( 60'+ &'c — da' + d'a) z x + ( bd' + b'd — ac' + a'c) w, 
MM'z 2 = (ac' 4- a'c — db' + d'b) x x + ( be' + b'c — ad'+ a'd) y x 
4- (— aa' — bb' 4- cc' — dd') z x + ( cd' 4- c'd — ba' + b'a) w x , 
MM'w-i = (ad' + a'd— be' + b'c) x x + ( bd' + b'd — ca + c'a) y x 
+ ( cd' + c'd — ab'+ a'b) z x + (— aa' — bb' — cc' + dd') w 1} 
values which of course satisfy identically x 2 2 + y 2 2 + z 2 2 + w 2 2 = x 1 2 + y 1 2 + Zi+w 1 2 , and which 
belong to an improper transformation. We have thus obtained the improper trans 
formation of the surface of the second order a? + y 2 + z 1 4- w 2 = 0 into itself. 
Returning for a moment to the equations which belong to the surface xy — zw = 0, 
it is easy to see that we may without loss of generality write a = /3=a / = /3'=0; 
the equations take then the very simple form 
MM' x 2 = <y'8x 1 , MM'y 2 = yh'y l , MM'z 2 = — 77'w x , MM'w 2 = — 
where MM' = V — 7S V — y'S'; and it thus becomes very easy to verify the geometrical 
interpretation of the formulae. 
It is necessary to remark, that, whenever the coordinates of the points Q are 
connected with the coordinates of the points B by means of the equations which 
belong to an improper transformation, the points P, Q have to each other the 
geometrical relation above mentioned, viz. there exist two plane sections 6, cp such 
that P, Q are the opposite angles of a skew quadrangle upon the surface, and having 
the other two opposite angles in the sections 6, (p respectively. Hence combining 
the theory with that of the proper transformation, we see that if A and B, B and 
C,..., M and N are points corresponding to each other properly or improperly, then will 
N and A be points corresponding to each other, viz. properly or improperly, according 
as the number of the improper pairs in the series A and B, B and G, M and N 
is even or odd; i.e. if all the sides but one of a polygon satisfy the geometrical 
conditions in virtue of which their extremities are pairs of corresponding points, the 
remaining side will satisfy the geometrical condition in virtue of which its extremities 
will be a pair of corresponding points, the pair being proper or improper according 
to the rule just explained. 
I conclude with the remark, that we may by means of two plane sections of a 
surface of the second order obtain a proper transformation. For, if the generating 
lines through P meet the sections 6, cp in the points A, B respectively, and the 
remaining generating lines through A, B respectively meet the sections cp, 6 respec 
tively in B', A', and the remaining generating lines through B', A' respectively meet 
in a point P'; then will P, P' be a pair of corresponding points in a proper trans-