294 PROFESSOR C. J. .TOLY ON QUATERNIONS ANT) PRO.TECTIYE OEOMETRY, 
To exhillit tlie nature of this curve, observe that 
0 = S . F„ {q)[p,]X2lh] = fph, fpp-^ ■ 
for all quaternions [ [V\Pd^^. in the notation of Art. G5. 
(4:30) 
((7> fipiq), fiipq), fip^q): fippi))) = o.( 4 . 37 ), 
whenever (435) is satisfied. But we have seen that (437) represents a curve ot 
oi'der m = TO and rank 7 ' = 40 (278), which is common to all the quartic surfaces 
obtained by deleting one quaternion within tlie double brackets (436). 
The solution may be ex])ressed in a more explicit form by means of the identity 
'l(f(lh9),f{lh9),f(lvA/(rvj)) = i±f{lVl){q,/{/hq)^f{lVl\f{lh'l)) ■ (438). 
SO that we may write (434) in the form 
P (p) P- 2 psPi) = ^ ±Pi {q, fipp]), fipF/), f{PFl)) ; ^ (?) . (439). 
T09. When the qxjmt lies on the critical cw've it is generally a united point 
of every function of a determinate tivo-systeni. 
In this case the solution of (433) is (Art. 15) 
pJ' {q)=tGfq)-\- Ff 2 '>) .(440); 
P = + /(iV/) = 0, t^Jfq) .... (44 T). 
Tluis^ may be any point on the line joining the point G,j{<i) io p>Q — the Jacobian 
correspondent of q ; and consecpiently a determinate two-system exists, everv 
function of which has q for a united point (compare Art. 123). 
n o. Similarly for the conjugate four-system/''(^>/-), a point r is a united point 
of a definite function, unle.ss it happens to lie upon the conjugate critical curve 
F:{r)=0 .(442), 
where Ip is the auxiliary function of ( p) = f'f p?*), Imt we must ol)serve that f^ 
is not the conjugate of fg. 
Now the reciprocal of a united jioint of ( pr) (the conjugate to r of /’(ju’)) is a 
united plane of the original four-system. And thus an arbitrary plane is the united 
])lane of some definite function, hut if the plane belongs to the developahlc surface 
(442) it is a common united plane of a definite two-system of functions determined by 
P = ''rf (r) -f xqf, f” fpfr) = 0 .(443). 
I en of these singular planes pass through an arbitrary point; tlie order of 
the flevelopahle surface is r = 40; and the order of the cuspidal curve^'^ is 
n — 3 (r — 71%) + — 90. 
‘ Tliree Dimensions,’ Art. 327. 
