( 361 ) 



a' v' s'' a —<y 



+ --!-- = /«', with R — 



a^ ' h"" ' c" h^ — c" 



Likewise [g = ^ , «^ negative j the system of hyperboloids with 



two sheets 



— ^r=/i', within = 



c^ a" b' b'-^c' 



and also ( (j = — , a^ positive J the system of hyperboloids with one 



sheet 



.2 „,2 „2 



z' /ir ir , , _ c' — a' 

 - H = h\ with R = . 



§ 6. The "curves of a complex" are curves whose tangents are 

 rays of the complex. The coefficients of direction («, ^, y) in a 

 definite point {,v, ?/, z) must therefore be proportional to 



0^ dz 



of one of those surfaces through that point. From this ensues that 

 ij = and q = , whilst .v, ii, z, p and a must satisfy the 



^ y ^ y 



equation : 



X I y R 



= 0. 

 p R—l q \—R 



So the quantities x, y, z, a, /?, y must satisfy : 



y ^ 72—1 "^ a R—l 

 Let a curve of the complex be given by : 



.^■=/i(-^). y=f.(^)^ ^^A(^)^ 



where x need not of necessity represent the length of the arc, then: 



ƒ»(--) , fA^) R , fM 1 ^Q 



Amongst others all curves for all values of p to be represented by 



X = ).{l-{' s)/', y — li {m 4- «)/', ^ = V (n + s)P 

 satisfy this equation if only 



I — n 



^R, 



m — H 

 which condition can be satisfied by putting I = B, m = C, n-= A, 



