SOME SPECIAL ALGEBRAIC TRANSFORMATIONS 

 REALIZED BY LINKAGES*'^ 



By Arnold Emch 



1. In a paper on "Algebraic Transformations of a Complex 

 Variable ReaKzed by Linkages," recently published in the Tkansac- 

 TiONS*") of the American Mathematical Society, I have shown that 

 any number of algebraic relations between n complex variables may 

 be realized by a plane linkage. 



I have considered in particular the algebraic relation 



f{u,z)=^ 



between the complex variables u and z, which may be realized by a 

 suitable combination of linkages for the relations 



Z — ^j-J-S'j, Z^^Z-^Z^. 



In conclusion linkages for these relations were described. Since 

 the publication of the paper I have found that Professor P. Somoff, 

 of Warsaw, in an article'^) on some applications of the kinematics of 

 variable systems to linkages, described the same linkage which I have 

 devised for z=^z^-\-z^. The realization of z = z^z.^ was based upon 

 Kleiber's linkage (K)^*^ as shown in Fig, 4 of my article in the 

 " Transactions." The compound linkage however is not perfectly 

 general. If the triangle AoA2A.^ is kept fixed, then the point A % 

 describes a circle and can, consequently, not reach every point of the 

 Gaussian plane. In a recent letter to the writer. Professor Kleiber 



(1) A part of this paper, with the title. "A Linkage for u^^e^ Z and Its Applications," has 



been presented to the Am. Math. Soc, Chicago, Sept., 1902. 



(2) Vol. 3, No. 4. pp. 493-498, October. 1902. 



(3) Zeitschrift fur Mathematik und Physik, Vol. 46, pp. 199-217, 1901. 

 (*) In what follows I shall designate Kleiber's linkage simply by K. 



