92 



Prof. Cayley. On the 



[June 19, 



+ A 1 ) 2 +4A 1 (A 8 -A 2 )} 

 = B 3 *{(A S +A 1 )'-4A 1 A 2 }, 



this becomes 



whence 



cos B= 



sinB= 



and with these values we verify 

 tanh a = tanh c . cos B, 



sinh 6 = sinh c . sin B, 



which are the expressions for the sides BC, CA, in terms of the 

 length BA, = c and angle B, which are arbitrary. I have not thought 

 it necessary to give the direct verification of these equations for a 

 more general position of the right-angled triangle : we already know, 

 and it appears a posteriori by the following number, that the verifica- 

 tion really extends to any right-angled triangle whatever on the 

 surface. 



17. The pseudosphere is homogeneous, isotropic, and palintropic, 

 viz., this is the case when bending is allowed ; in other words, the 

 surface is applicable upon itself, with three degrees of freedom ; con- 

 sidering any infinitesimal linear element Ax, the point A may be 

 brought to coincide with an arbitrary point A' of the surface, with 

 the element Ax in an arbitrary direction A'*' through A', and the 

 area about A will then coincide with the area about A'. The analy- 

 tical theory is at once derived from that for the sphere, viz., we have 

 a rectangular transformation 



ft 



ft" 



where the coefficients are such that identically 



