Page:VaricakRel1912.djvu/11

perpendiculars MP and MQ upon the coordinate axes, then the Lobachevskian coordinates of M are

$X=OP,\ Y=MP$

Those of Weierstrass are

$x=\operatorname {sh} \,MQ,\ y=\operatorname {sh} \,MP,\ l=\operatorname {ch} \,OM$

or

(21)	$x=\operatorname {sh} \,\xi ,\ y=\operatorname {sh} \,\eta ,\ l=\operatorname {ch} \,r.$

From the quadrilateral OPMQ of three right angles we obtain

$x=\operatorname {sh} \,X\ \operatorname {ch} \,Y$

in addition we have

(22)	${\begin{array}{l}y=\operatorname {sh} \,Y,\\l=\operatorname {ch} \,X\ \operatorname {ch} \,Y,\end{array}}$

and thus the Weierstrass coordinates are expressed by the Lobachevskian ones.

In the general case we have Fig. 5. N, R, S are the foot points of the three perpendiculars ξ η, ζ of M upon the coordinate plane, then

$X=OP,\ Y=PN,\ Z=NM$

are the Lobachevskian, and

(23)

${\begin{array}{l}x=\operatorname {sh} \,\xi =\operatorname {sh} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,Z\\y=\operatorname {sh} \,\eta =\operatorname {sh} \,Y\ \operatorname {ch} \,Z\\z=\operatorname {sh} \,\zeta =\operatorname {sh} \,Z,\\l=\operatorname {ch} \,r=\operatorname {ch} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,Z\end{array}}$

are the Weierstrass coordinates of point M.

From the quadrilateral MNRT we have^[1]

$\operatorname {sh} \,\xi =\operatorname {sh} \,NT\ \operatorname {ch} \,Z$

while we obtain from OPNT the equation

$\operatorname {sh} \,NT=\operatorname {sh} \,X\ \operatorname {ch} \,Y$

From these two relations we obtain the expression for x. From the quadrilateral MNPS we easily find the value for y. The limiting arcs MA, MB and MC are our x, y, and z. We find in addition

$l^{2}-x^{2}=\operatorname {ch} ^{2}Y\ \operatorname {ch} ^{2}Z$

1. F. Engel, Nikolaj Iwanowitsch Lobatschefskij. Zwei geometrische Abhandlungen, 1898, p. 347