SPECIAL ALGEBRAIC TRANSFORMATIONS 213 



lar to the linkage K which is based upon AqCAjD. If one of the 

 above quadrilaterals is fixed then each of the remaining points of 

 Kj or Kj has one degree of freedom left in its motion. Designate 

 the points of Kj and K^ by the same letters as those of K and affix in 

 K, everywhere the index 1, in K2 the index 2. Then repeat the 

 links A2Ci2B,2C3, parallel and equal, on CjBj and B,C, and designate 

 the pivots of the last links by B^Cl^ByiC. In this manner two 

 similar polygonal lines 



-^oC'iiBiiC,BjCi*2Bi2C CO AjAjjA^jAjAjAjjAgaAj 



are obtained. If the polygon A2C3A3D3 is fixed then A22 may de- 

 scribe a circle with A3 as a center. We have now 



AAoB„Bj2 <r, AA,A„A22, 



no matter how the whole linkage may be distorted. These triangles 

 are otherwise independent of each other. Thus taking A^Ajj as the 

 first real unit of the complex plane and designating the points B*2, 

 A22 and Bjj respectively by 2, s'j and z.^, we evidently have 



i. e., the realization of complex multiplication. The range of effect- 

 iveness of this, as of any other linkage is, of course, limited to a cer- 

 tain finite portion of the plane. This range, although in some cases 

 small, always exists. (No interference between the parts of the 

 linkage is allowed.) 



By the linkage just described all the special cases may be 

 obtained. Thus for z^^=z^=u we get z^ii^ and by inversion %i= 



In the following section I shall show, how the problem of 

 many-valued functions may be solved in the case of the ;^th root. 

 Putting 22=constant and j^al^l we have the rotation of the point z^ 

 through an angle = argument of z^_. For multiplication by a constant, 

 Sylvester's pantograph may be used. Two years ago the writer has 



