94 Prof. Cayley. On the [June 19 



and conversely 



2X(v-l)Z 1 , 



where Q= l + i* 2 X^ : it can be at once verified that each of the 

 two sets of formulae does in fact give PiQi Zj 2 =PQ Z 2 . 



19. The pseudosphere is a surface of revolution having for its 

 meridian section the curve a;=logcot-0 cosfl, t/=sin0. This is a 

 curve symmetrical in regard to the axis of y ; and we obtain the 

 portion of it lying on the positive side of this axis, by giving to the 

 series of values 6=0 to 0=90; for 0=0, we have i/=0, 3=00, or 

 the axis of x is an asymptote ; for 0=90, a? = 0, y=l, the point being 

 a cusp of the curve. The geometrical definition is that the portion of 

 the tangent included between the curve and the axis of x has the 

 constant length = 1 ; the inclination of the tangent is in fact=0. 



We have dx= C08 ^ , dy=cosOde-, and thence ds= cot Ode, and the 



Fro. 2. 



length in question is -?=!. The curve may be constructed 

 dy 



graphically : take (fig. 2) the distance B0= I, on OB, B, very near to 

 B, and then BiOi=l ; on O^, B 2 very near to B lt and then B 2 2 =l, 

 and so on ; the curve is shown on a larger scale in fig. 3. 



But the curve may also be laid down numerically; writing 

 a.-=\TtQ (so that 3. is the inclination of the tangent to the axis of y) 



