212 On the Transformation of a Surface of the Second Order. 



the points Q are connected with the coordinates of the points P 

 by means of the equations which belong to an improper trans- 

 formation, the points P, Q have to each other the geometrical 

 relation above mentioned, viz. there exist two plane sections d, ^ 

 Buch 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, cf) 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 correspond- 

 ing to each othei", viz. properly or improperly, according as the 

 number of the improper pairs in the series A and B, B and 

 C . . . 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 remain- 

 ing 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. 



1 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 

 eections 6, (f> in the points A, B respectively, and the remain- 

 ing generating lines through A, B respectively meet the sec- 

 tions <f>, respectively 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 transformation. In 

 fact the generating lines through P meeting the sections 6, (f) in 

 the points A, B respectively, and the remaining generating lines 

 through A, B respectively meeting as before in the point Q, 

 P and Q will correspond to each other improperly, and in like 

 manner P' and Q will correspond to each other improperly, i. e. 

 P and P' will correspond to each other properly. The relation 

 between P, P' may be expressed by saying that these points are 

 opposite angles of the skew hexagon PAB' P'A'B lying upon the 

 surface, and having the opposite angles A, A' in the section 6, 

 and the opposite angles B, B' in the section <p. It is, how- 

 ever, clear from what precedes, that the points P, P' lie in a 

 section passing through the points of intersection of 6, (p, i. e. 

 that the proper transformation so obtained is not the general 

 proper transformation. 



2 Stone Buildings, Jan. 11, 1854. 



