Intersections of a Pencil of four Lines by a Pencil of two Lines. 501 



somewhat more than one-third of that of the earth. Between 

 this result, and what we already know of the motion of the solar 

 system through astronomy, there is no great divergence. 



I hope during the present year, however, to be able to con- 

 tinue my spectrum-experiments, and to have a better opportu- 

 nity of determining, numerically, the magnitude of the motion 

 of the solar system. In the present paper my object has merely 

 been to show the possibility of solving, optically, this interesting 

 problem in physical astronomy. 



LXVIII. On the Intersections of a Pencil of four Lines by a 

 Pencil of two Lines. By Professor Cayley, F.R.S.* 



pLUCKER has considered (" Analytisch-geometrische Apho- 

 r rismen," Crelle, vol. xi. (1834) pp. 26-32) the theory of 

 the eight points which are the intersections of a pencil of four 

 lines by any two lines, or say the intersections of a pencil of four 

 lines by a pencil of two lines : viz., the eight points may be con- 

 nected two together by twelve new lines ; the twelve lines meet 

 two together in forty-two new points; and of these, six lie on a 

 line through the centre of the two-line pencil, twelve lie four 

 together on three lines through the centre of the four-line pencil, 

 and twenty-four lie two together on twelve lines, also through 

 the centre of the four-line pencil. 



The first and third of these theorems, viz. (1) that the six points 

 lie on a line through the centre of the two-line pencil, and (3) 

 that the twenty-four points lie two together on twelve lines 

 through the centre of the four-line pencil, belong to the more 

 simple theory of the intersections of a pencil of three lines by a 

 pencil of two lines ; the second theorem, viz. (2) the twelve points 

 lie four together on three lines through the centre of the four- 

 line pencil, is the only one which properly belongs to the theory 

 of the intersections of a pencil of four lines by a pencil of two 

 lines. The theorem in question (proved analytically by Pliicker) 

 may be proved geometrically by means of two fundamental theo- 

 rems of the geometry of position : these are the theorem of two 

 triangles in perspective, and Pascal's theorem for a line-pair. I 

 proceed to show how this is. 



Consider a pencil of two lines meeting a pencil of four lines in 

 the eight points [a, b, c,d), {a!, b 1 , c', d 1 ) ; so that the two lines 

 are abed, a'b'c/d' meeting suppose in Q ; and the four lines are 

 aa ! , bb'j c6 \ dd' meeting suppose in P ; then the twelve points are 



* Communicated by the Author. 



