348] THE THEORY OF SCREW-CHAINS. 379 



a case of &quot;Rational Transformation&quot; (see Salmon s Higher Plane Curves, 

 Chap. VIII.). The theory is however here much simplified. In this case 

 none of the special solutions are admissible which produce the critical cases. 

 Consider the equations U. 2 = Q, ..., U 5 = 0. They will give a number of 

 systems of values for lf ..., 9 5 equal to the product of the degrees of 

 C/2 &amp;gt; U 5 . Each of these 6 screws would be a correspondent to the same 

 &amp;lt;j) screw 1, 0, 0, 0, 0. But in the problems before us this &amp;lt;f&amp;gt; as every other &amp;lt; 

 can have only one correspondent. Hence all the functions U lt U 2 , etc. 

 Ui, 17%, etc. must be linear. We may express the first set of equations thus : 



fa = (11) 0, + (12) a + (13) e 3 + (14) 4 + (15) 6 t , 

 fa = (21) 1 + (22) 2 + (23) e. A + (24) 0, + (25) 9 5 , 

 fa = (31) 0, + (32) 9, + (33) 3 + (34) 6, + (35) t , 

 fa = (41) 0, + (42) 0, + (43) 3 + (44) 4 + (45) t , 

 fa = (51) 0, + (52) 6, + (53) B, + (54) 9, + (55) . 



For the screw (j&amp;gt; to be known whenever 9 is given, it will be necessary to 

 determine the various coefficients (11), (12), &c. These are to be determined 

 from a sufficient number of given pairs of corresponding screws. Of these co 

 efficients there are in all twenty-five. If we substitute the co-ordinates of one 

 given screw 9, we have five linear equations between the co-ordinates. Of 

 these equations, however, we can only take the ratios, for each of the co 

 ordinates may be affected by an arbitrary factor. Each of the given pairs 

 of screws will thus provide four equations to aid in determining the co 

 efficients. Six pairs of screws being given, we have twenty-four equations 

 between the twenty-five coefficients. These will be sufficient to determine 

 the ratios of the coefficients. We thus see that by six pairs of screws the 

 homography of two five-systems is to be completely defined. To any seventh 

 screw on one system corresponds a seventh screw on the other system, which 

 can be constructed accordingly. 



348. Application of Parallel Projections. 



It will, however, be desirable at this point to introduce a somewhat 

 different procedure. We can present the subject of homography from 

 another point of view, which is specially appropriate for the present theory. 

 The notions now to be discussed might have been introduced at the outset. 

 It was, however, thought advantageous to concentrate all the light that 

 could be obtained on the subject ; we therefore used the point-homography 

 of lines, of planes, and of spaces, so long as they were applicable. 



The method which we shallnow adopt is founded on an extension of 

 what are known as &quot; parallel projections &quot; in Statics. We may here recall 



