PROFESSOE C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOxMETRY. 255 
when the scalar parameter s alone varies. When s and t both vary the equation is 
that of the developable of the tangent lines. 
If for suitable weights of the united points pj, pg, we write 
^ — 5^1 + 5'2 + 7.8 + ^ 1 -.(^^ 8 ), 
the equation of the developable becomes 
p = E (h + s) (h + .(W9). 
When — 1 is positive, the result of putting t — — is 
q = {u + s) (^2 - q)'" ^q, + (q + s) (q - q)"' % + (q, + 5) (q -q)»^ \ . . (180) ; 
and this represents a certain number of right lines in the united plane [cq^, pg, pj, the 
number being determined by the nature of m, being as we know 4 when the develop¬ 
able is circumscribed to a pair of quadrics, or when m = f. 
The remaining part of the intersection in the united plane is obtained by putting 5 
equal to — q, and its equation is 
<l = (h- h) (h + + ih - h) (h + + (h - <i) (*. + ■ • (181); 
or more simply 
7 = (/+ 0“”^’ where = (q — q) ry, + (q — q) q^ + (q — q) q^ . . (182). 
The plane curve is likewise included in the family (176), and for m = | it is a 
quartic (174), or rather a conic counted twice. 
x. 
U — t± 
u 
{h - hY 
3 "8 ~ 'U I -r 2 'U ^3 _L 
^ “ ('3 - 0= + * 
2 h, ~J3_ _ 0 
(183), 
as we see from (181) on putting q = x^q^ + a^373 + ^474- 
In case m — 1 is negative it is necessary first to multiply (179) by the product 
n (q -f i)^““ before putting — t equal to a latent root. Then, on making t = — 
we find only the point q^, which shows there are no right lines in the plane [fhqzq^, 
and which indicates multiplicity of the curve at the united points. 
45. Just as the equation of the tangent line was obtained in the last article from 
that of the curve, the equation of the osculating plane may be written in the form 
7 = (/+ w.) (/+ «) (/+ .(184); 
where t is sujDposed to remain constant, while s and u vary together. It is easy to 
verify that this plane contains two consecutive tangents to the curve. 
The reciprocal of the plane is the point (compare Art. 5) 
p = (/' + > 
a = [«,/(,/%] . . 
(185); 
