170 
ON GEODESIC LINES, 
[508 
22. Changing the notation, and writing 
P-(a + p) (b+p) (c +p), 
Q =(a + q)(b + q)(c + q), 
<ï> = (<£ 2 — 7 2 ) (« + b — 2c — </)), 
the equation is 
yjr y^ yyj 
or if, instead of <£, we introduce the original parameter a, then, observing that 
2 da d(f) 
we at once find 
dp dq 4¡da 
7p + 7Q + VZ 
where 
% = 7 (1 + cr 4 ) — 2 (a + b — 2c) cr 2 , 
or, what is the same thing, 
£ = a (<r 2 — l) 2 — 6 (cr 2 + l) 2 + c . 4cr 2 ; 
viz. passing from a point (p, q) on the line cr to a consecutive point (p + dp, q + dq) 
on the line cr + da, the above is the relation between the variations dp, dq, dcr. If r 
be the parameter of the other line through the same point, then we have in like 
manner, say 
(viz. one of the radicals \JP, VQ mu st present itself with a reversed sign): and we 
thus have dp, dq each expressed in terms of da, dr; viz. we have the increments 
dp, dq when a point passes from (cr, r) to (a + da, t + dr). These results will be 
presently obtained in a more simple manner. 
Formulae where the position of a Point on the Surface is determined by means of the . 
two Lines through the Point. 
23. We may determine the position of a point by means of the parameters a, r 
of the two lines through the point. The equations of these are 
w _! 
cr