PROFESSOR C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 287 
95. Otherwise, if the quaternion variable q is a function of four j^arameters, x, y, 
z, iv, we may replace the arbitrary differentials in terms of the deriveds of q with 
respect to these parameters, and then (394) becomes 
where 
07 07 
7.- = 
07 
0z ’ 
qw = 
07 
dw 
In particular, if these four deriveds satisfy the six equations 
= 87,7^ = ^qAlw = ^qyqto = 87,7,„ = 0 
it easily appears that the symbolic equation (395) reduces to 
D = ^ 4- ^ d- - d- 
87,2 0X ^ 87/ 0^ ^ 87,2 02 ^ 87/ 0?C ■ 
(395) , 
(396) . 
(397) , 
(398) . 
More particularly if 7 is referred to the vertices of a tetrahedron self-conjugate to 
the unit sphere, so that 
q = ax hy cz clw, and if 8 a® = 85® = 8 c® = 8 (:/® = 1 . (399) 
for suitable selection of tlie weights of these four points, tlie operator takes its 
simjjlest form 
^ = a - -\-b -{-c -{-d .(400), 
ox oy oz dw 
while 
8L,= = ( g; 1 + 
^ 0'0 
+ hfo + 
0 " 
0 a’, 
.(401). 
If, on the other hand, 
the operator reduces to 
7 — i ~1“ “ix “h yy “b kz .(402), 
D = ^ - V 
dt 
(403). 
96 . It may be useful to collect a few formulse which may serve as examples of the 
application of the operator. We therefore give the following ; 
D7 = 4 ; DK7 = - 2 = KD7 ; 1)87 = I = 8D7 ; DV7 = 3 = VD7 ; 
D8a7 = a; D8.7® 2 q ; DT7® = 2K7 ; D7® = 4 (7 + 87); D (V7)® = 2 Yq ; 
DT(,/ + a) = KU(</ + «); = (,/■+/')</. 
To these we may add 
D®T (7 -f a)® - 4 = TD®8 (7 + a)®; TD®T (7 + a)® = 8 = D®8 (7 -f a)® ; 
TD®. T7" = hKI ). K7T7"-® n (4T7"-® -b (n - 2) 7X7X7"-“*) = n (w + 2) T7"-®. 
And again 
I)®(8.7®)" = 2nD.7(8.7®)"-i = 8n (8.7®)"-* + 4 n, (n - I) 7® (8.7®)"-®; 
