474 THE THEORY OF SCREWS. [427, 



and, from the second vector, 



*i = + ay, + ffy* + yy a + 8 y 4 , 



Substituting for y n y 2 , .Vs. y*, we obtain the following values for z lt z%, z s , z 4 . 

 The right vector, a, ft, 7, 8, followed by the right vector, a , /? , 7 , 8 



z a = + aa - pp - 77 - 65 ^ + + a/3 + /3a + 75 - 7 x 2 + ay - fid + ya +dp X 3 + + ad + Py - yp + da x 4 , 



z%= -ap -pa -y8 + dy x l + + aa - pp - yy - 85 a; 2 + a5 + 7 - 7(3 + Sa x 3 + - ay + pS -ya - 



= -a7 + /35 --ya -8/3 a; 1 + - aS - Py + 7/3 - da x. 2 + aa - pp - yy - 88 x 3 + + a/3 + /3a + 78 - dy x 4 , 



5a x 1 + +ay - pd + ya +dp x 2 + - a/3 - /3a - 76 + 67 x 3 + + aa - /3/3 - yy - 55 x 4 . 



The right vector, a, ft , 7 , 8 , followed by the right vector, a, ft, 7, 8 



!= + aa - /3/3 - yy - 55 ^ + + a/3 + /3a - 78 + 87 rc 2 + + a&amp;gt; + /35 + 701 - S/S x 3 + + ad - fiy + 7/3 + 8a x 4 , 



z. 2 = - a/3 - /3a + 78 - 87 x l + + aa - /3/3 - 77 - 55 x 2 + + aS - fiy + y{? + 8a x 3 + - ay ~ /35 - ya + 5/3 x t , 



~ 7* + 8/3 ^i + - a5 + fty - 7/8 - 5a J 2 + + aa - /3/3 - 77 - 55 x 3 + + a/3 + /3a - 78 + 87 x 4 , 



yp -8a x 1 + + ay + /36 + ya - 5/3 x. 2 + - a/i - /3a + 75 - 57 x 3 + + aa - /3/3 - 77 - 55 x . 



We thus learn the important truth, that when two or more homonymous 

 vectors are compounded, the order of their application must be carefully 

 specified. For example, if the object x be first transposed by the vector a 

 and then by a , it attains a position different from that it would have gained 

 if first transposed by a! and then by a. 



We see, however, that in either case two homonymous vectors compound 

 into a vector homonymous with the two components. 



We now study two heteronymous vectors, i.e. one right and one left. 

 The right vector, a, ft, 7, 8, followed by the left vector, a , ft , 7 , 8 



z l = +aa -pp -yy -88 x 1 + + a/3 + pa + yd - dy x 2 + + ay - p8 + ya + 5/3 x 3 + +aS + py - 7/8 + 5a z 4 , 



+ aa - /3/3 + 77 + 55 x 2 + -a8 -py - yp + da x 3 + +ay - pd -ya - dp 



= -a7 -/35 -7a + S/3 a; 1 + + aS - pS - yp - da x 2 + + aa + pp - yy + 55 x 3 + - a/3 + /3a - 78 - 8y x 4 , 



Z 4 - - a5 + 7 - 7/3 - Sa x l + - ay - pd + ya - dp x 2 + + a/3 - pa - 78 - 87 x 3 + + aa + pp 1 + yy - 55 x 4 . 



The left vector, a , ft , 7 , 8 , followed by the right vector, a, ft, 7, 8 



! = + aa - pp - 77 - 85 x 1 + -f a/3 7 + /3a + 78 - 87 x 2 + + 07 - /3S + ya + 8p x s + + aS + Py - yp + da x t , 



2 = - ap - pa + yd - dy X 1 + + aa - pp + yy 4 58 X 2 + - ad - Py - yp + da X 3 + + ay - pd - ya - dp X 4 , 



Z 3= -ay -pd -ya + dp x l + + ad - py - yp - da X 2 + + aa + /3/3 - 77 + 55 X 3 + - a/3 + )3a - 76 - 



z 4 = - ad + Py - yp - da X-L+ - ay - pd + ya - dp x 2 + +ap - pa -yd - dy x 3 + + aa + pp + yy - 88 z 4 . 



