254 PKOFESSOli C. J. JOLY ON QUATERNIONS AND PROJECTIVE C4EOMETRY. 
where c = A, V. By (155) it is seen that the quaternion e of this formula satisfies 
the three equations 
Sr/hi = h, = 0, 
(1^3). 
and is therefore one of the intersections of three quadrics. 
42. Tn particnlar the equation of the curve of intersection of tlie original quadrics 
(1G8) is 
q = {/rYi + where Sq/ja = Safla = BqQ= o . . (174), 
as may he proved by direct transformation from the genei'al residt (172), or perhaps 
more shortly hy assuming the form q = (./’+ and determining f, or by verifica¬ 
tion. remembering (158). 
Hence the et[nation 
(.47)"^ + = fi.(175) 
determines the eight ])oints of intersection of the three quadrics 
Sq/hy = 0, SqYq = 0. 
SECTION VII. 
The Family of Curves q = {f-\- t)’‘'a and their Deyelofables. 
Art. " Pasre 
O 
43. Some memliers of the family.254 
44. The tangent line and the developable.254 
45. The oscnlating plane.255 
46. The intersections of the curve with the osculating plane.256 
43. Instead of v'riting dovni and discussing the equations of the circumscribing 
developable and of its cuspidal edge of the quadrics (139), which are in fact of the 
same form as (163) and (167), except that f = f~Y -2 is self-conjugate, we shall 
devote a fe^v i-emarks to the fimiily of cui'ves 
A = (/+4"«.(176) 
and their developables, m being a scalar, a a constant quaternion, t a scalar variable, 
and/an arbitrary linear quaternion function. This family includes the right line, the 
conic, the twisted culiic, the quartic intersection of two quadrics, the quartic which is 
not the intersection of two quadrics, and the cuspidal edge of the developable 
circumscribed to trvo quadrics ; the corresponding values of m being vi — 1, 2, —1 
or 3, 4 and f. 
44. The equation of a tangent to the curve (173) is 
(/+/(./■+.('^’ 7 ), 
