ffx 



Prof. Cayley. On the 



[June 19, 



Y=$( P + Q), the equation *X+/3Y + <yZ=0 becomes (-i i/3)P 

 + (_,- a +0)Q + ,yZ=:0, which will then be of the form AP + BQ 

 + CZ=0, with real coefficients A, B, C : viz., we have P, Q, Z=sin 0, 

 sin 0(02 -j-cosec 2 0), sin 0.0; and the equation thus is 



A + B(02 + cosec 2 0) + C0=0, 



which is the equation for a geodesic (or straight line) on the pseudo- 

 sphere. The equation A +C0=0, that is, 0=const., is obviously that 

 of a meridian. 



12. If the geodesic pass through a given point 0j, 0j we have, of 

 course, A+B(0 1 2 + cosec 2 1 ) + C0 1 =0, and hence also the equation of 

 a geodesic through the two points (0^ 0j), (0 2 , 0j) is 



1 , 02 + cosec 2 , 



2 2 + cosec 2 2 , 3 



=0. 



We may for 1 , 2 write 1 + 2n 1 7r, 02 + 2n27r respectively, nj, n 2 being 

 arbitrary integers ; and it would thus at first sight appear that there 

 could be drawn through the two points a doubly infinite series of 

 geodesies. There is, in fact, a singly infinite system of geodesies : to 

 show how this is, write for shortness A, A 1? A 2 , , j, * 2 for cosec 2 0, 

 cosec^, cosec 2 2 , 2n?r, 2^77-, 2 a w respectively ; then the equation of 

 the geodesic through the two points may be written 



1 , (0 +) 2 + A , +<* 



= 0, 



where the constant =2nir may be disposed of so as to simplify the 

 formula as much as may be, it is what I have called an apoclastic 

 constant. Taking /3 an arbitrary value, this may be transformed 

 into 



, (0 



, 



= 0, 



and then assuming =^ y3= a.^, this becomes 

 1 , 2 -I- A ,0 



=0, 



