Prof. Cayley. On the [June 19, 



and these lead to all the trigonometrical formulae, viz., any one of 

 these is deduced from the corresponding formula of spherical trigo- 

 nometry by writing therein at, bi, ct for a, b t c respectively ; or, what 

 is the same thing, by changing the circular functions of the sides into 

 the corresponding hyperbolic functions. 



Jn particular for the right-angled triangle ACB we have 



tan A tan B = _* , 

 cosh a cosh b 



which for a and b real is<l, that is, A + B<a right angle, or 

 A + B + C < two right angles, and thence also in any triangle what- 

 ever A + B + C<two right angles. But the points A, B, C of any 

 such triangle ABC on the imaginary sphere, and the lines BC, CA, AB 

 which connect them, are imaginary : the meaning of the proof will 

 better appear on passing to the pseudosphere. 



8. We have to consider the imaginary spherical surface as bent into 

 a real surface. This is, of course, an imaginary process, as any 

 process must be which gives a transformation of imaginary points 

 and lines into real points and lines ; but the notion is not more 

 difficult than that of the transformation of imaginary similarity, 

 consisting in the substitution of ix, iy, iz for z, y, z respectively. 

 We thus pass from imaginary points of the imaginary sphere 

 x 2 -)-7/ 2 -t-z 2 = 1. to real points of the real sphere x 2 + / 2 -i-z 2 =l ; or, 

 again, from imaginary points of either of the real hyperboloids 

 x^-ft/ 2 z 2 = 1, x*+y* * 2 =1 to real points of the other of the 

 same two real hyperboloids. 



9. I consider the formulae for the flexure of the imaginary sphere 

 X 2 + Y 2 + Z 2 = i f i n to the pseudosphere z=log cot0 cos 0, 

 v/t/-+z 2 =sin0: it would be allowable to dispense with Beltrami's 

 subsidiary variables u, v, but I prefer to collect here all the formulae. 

 We have 



v 



A 



values which give X 2 + Y 2 + Z 2 = 1. And observe that taking u, v 

 to be real magnitudes such that t* 2 +v 2 <l, we have X a pure 

 imaginary, but Y and Z each of them real. We consider on the 

 imaginary sphere points having such coordinates X, Y, Z ; any such 

 point corresponds as will immediately appear to a real point on tl 

 pseudosphere, and (the distances and angles being the same for the 

 pseudosphere as for the original imaginary spherical surface) it henc 

 appears that (notwithstanding that the points on the imagine 

 spherical surface, and the lines joining such points, are imaginary) 

 the distances and angles on the imaginary spherical surface are real. 



