304 
ON THE THEORY OF INVOLUTION. 
[348 
which may also be written 
/3 
7 
L' 
M 
N 
DL + hJ)L' 
DM + kJJM' , 
DN + k L DN 
or, more symmetrically, 0 being any quantity whatever, it is 
« /3 , 7 
L + eu , M + dM' , N + 0N 
DL + ^DL' , DM+kDM' , DN+ kJM' 
or, substituting for I) its value, the equation of the tangent is 
a , /3 , y 
L + 0L' , M+0M' , N+0N' 
(a 4- lea', .fX, F, Z), (h + Jch', .$X, F, Z), (g + kg’, .][Z, F, Z) 
= 0. 
27. Now if the diacritics touch, this equation should be independent of a, ¡3, 7. 
Putting for shortness a + h x a! = a, &c., and also taking as we may do 0 — 0, the parts 
multiplied by a, /3, 7 respectively are 
M(gX + /F + cZ)-N (hX + bY +/Z), 
N(aX 4-hY+ gZ) — L (gX + fY+ cZ), 
L (hX + bY+fZ) -M(aX + hY+gZ), 
and we ought therefore to have 
Mg - Nh : Mf- Nb : Me - Nf 
= Na — Lg : Nh — Lf : Ng — Lc 
= Lh - Ma : Lb - Ml, : Lf - Mg, 
equations which are in fact satisfied; for take any one of them, for example, the 
equation 
Mg - Nh _ Mf-Nh 
Na - Lg ~ Nh - Lf ’ 
MN (gh - af) - N (h* - ab) - LM(fg-fg) + LN (fh - bg) = 0, 
or, omitting the term in LM, and throwing out the factor N, 
L (hf— bg) + M (gh — af) + N (ab — h 2 ) = 0. 
But the equations L + JcL' = 0, &c., give 
ax + liy + gz = 0, 
lix + by +fz = 0, 
gx +fy + cz = 0,