438 
ON THE TETRAHEDROID AS A PARTICULAR 
[700 
where, a, /3, y, a, /3', 7', a", /3", 7" being connected as above, the number of constants 
is = 6. 
The equations of the 16 singular planes are 
Z = 0, Y= 0, 
a (77"Y — /3'/3"Z) — W=0, ¡3 (a'a! , Z-y'y"X)-W = 0, 
a' (y" y Y-/3"/3Z)- W= 0, ¡3' (a"aZ-y"yX)-W=0, 
a" (77' Y-/3/3' Z)-W = 0, (3" {aot! Z-yy' X)-W = 0, 
z = o, w = 0, 
7 (/S'/TX - a a" Y)-W = 0, /3 7 Z + 7 a F+ a/3£ = 0, 
7' (/3"/3 Z - a"« F) - TF = 0, /3yZ + 7 VF+ a'/3Z = 0, 
y" (/3/3' X - act' Y) - W = 0, /3VZ + 7 "a" F + a"/3'Z = 0. 
Writing x, y, z, w as current coordinates, the equation of the Tetrahedroid is 
rri i nf‘ 2 x 4 ‘ 4- nH-g'Y + Pm 2 h 2 z^ 4- f 2 g 2 h?w i 
4- (If 2 — W'fg 2 — n 2 h 2 ) (lyz* 4- f 2 x 2 iv 2 ) 4- (— If 2 + mrg 2 — n 2 h 2 ) (m 2 z 2 x 2 + g 2 y 2 w 2 ) 
4- (— If 2 — m 2 g 2 + n 2 h 2 ) (n 2 x 2 y 2 + b?z 2 w 2 ) = 0, 
where, inasmuch as f, g, h, l, m, n enter homogeneously, the number of constants 
is = 5. 
The equations of the 16 singular planes, written in an order corresponding to that 
used for the 16-nodal surface, are 
* ny—mz+ fw = 0 
fx —gy—hz * = 0 
—mx—ly * +hw= 0 
nx * +lz + gw= 0 
— nx * +lz +gw=0 
mx+ly * +hw—0 
—fx +gy—hz * =0 
* -ny—mz-\-fw=0 
mx—ly * +hw=0 
—nx * —Iz +gw=0 
* ny+mz+fiv =0 
—fx —gy+hz * =0 
—fx —gy—liz * =0 
* ny — mz—fw=0 
—nx * +lz — gw—0 
mx—ly * —liw = 0. 
These equations can be made to agree each to each with those of the 16 singular 
planes of the 16-nodal surface, provided that we have 
m_n n _ l l 
y~P’ r 
m 
f= — laa'a.", g — — m/3/3'/3", h = — nyyy 
where observe that the first three equations give ol/3"y = a"(3y', which is the relation 
between the constants when the 16-nodal surface reduces itself to a tetrahedroid in 
the above manner. And if we then assume 
X =ny — mz + fw, Y— — nx + lz+ gw, Z = mx — ly + hw, W = — fx — gy — hz, 
the 16 linear functions of Z, F, Z, W will become mere constant multiples of the 
corresponding 16 linear functions of x, y, z, w\ the constants, by which the several