480 THOMAS BOND SPRAGUE ON CERTAIN CURVES. 
and the curves intersect in two points, as shown in fig. 24. Comparing this 
with fig. 23, we see that, whereas the infinite branches of P then touched the 
asymptotes (1, —4), (2, 3), (4, —1), (—2, —83), they now touch the asymptotes 
(1,, 2). (3. — ola e 1) (—3, —4). It follows that there must be a transi- 
tion position in which the branches touch the asymptotes (1, 3), (2, —1), 
(4, —1), (2, —3), so that P has two double points. 
If we now examine the relations of the P and Q curves in figs. 20-24, we 
see that they satisfy the conditions laid down at the outset, bearing in mind 
that the x-axis must in each case be considered, for this purpose, a part of the 
Q curve. 
Since the foregoing was written I have met with some remarks of the late 
Professor DE MorGan on the curves P and Q, contained in a paper of his read 
before the Cambridge Philosophical Society on 7th December 1857. He 
remarks that these curves “are such that two branches, one of each curve, 
“ cannot inclose a space.” ‘This is a particular case of the properties investi- 
gated in the early part of this paper. He also remarks that the curves “ always 
“ intersect orthogonally,” but he gives no proof of this. It may be proved 
as follows. 
Let h, k, be the co-ordinates of a point of intersection of P and Q, and 8, 6’, 
the angles which the tangents to P and Q at the point, respectively make with 
the z-axis. If ¢(xy)=0 is the equation to a curve, then for any point (a, y), 
oY ~$+o. Applying this formula to (P) and (Q) (p. 430) we have 
PW -SPW) + EL 
tan 0= : st gaps 
- ae 
hf"(h)—s Sh) + ates 
en LO aET OF ETO 
a  aaeae = 
aa h )— yt h) — 
Hence pithy “2 yny ae f(hy— 
tan @ tan = Bia : Ws = 
LIF Phy . 
and S@Z7"@)+ Sy) 
tan 6 tan  +1= eae in 
sf (h)— =i Ul 1) + om 
and the numerator of this fraction = 0, since h, /, satisfy the equation (Q). 
Hence tan # tan #’= —1, or the tangents are at right angles to each other. 
