537 
And in like manner, if T1,.. 11; be any four collinear planes, of which 
consequently, by (25), the symbols z).., are connected by two other 
linear relations, such as 
Ry = to Ro -- te fig, Rz = iG Ro + ls R2, (82) 
we have then this other very useful formula of the same kind, for the 
anharmonic of this pencil of planes, 
tot 
(Ty Th, Th Hs) = >; (33) 
to be 
it being understood that the anharmonic function of such a pencil is the 
same as that of the group of points, in which its planes are cut by any 
rectilinear transversal : so that we may write generally, for any six points 
A..F, the formula, 
(EF . ABCD) = (a’B’c’D’), (34) 
if any transversal eu cut the four planes Era, . .’EFD in the four points 
a’,..D’; or in symbols, if 
A'=GH' EFA,..D/= GH’ EFD. (35) 
[11.] The expression of fractional form, 
Va+my+nvst+rwrs'o fl 36) 
le+ my+nze+rot+s f?’ ( 
in which the ten coefficients, 7..sand/’ .. s', are supposed to be given, 
and to be such (comp. (22)) that 
7+..+8=0, and /+..+s8'=0, (37) 
may represent the quotient of any two linear and homogeneous func- 
tions, f and f’, of the coordinates x..¥v of a variable point P, or rather of 
the differences of those coordinates (comp. [5.]); and if we assign any 
particular or constant value, such as k, to this quotient, or fractional func- 
tion, ¢, the equation so obtained will represent (comp. (21)) a plane 
locus for that point, which plane II will always pass through a given 
line A, determined by equating separately the denominator and nume- 
rator of g to zero. Hence the four equations, 
? (zyzwv) = 
f=0, J'=f, f =9, J’ =Ky, (38) 
which answer to the four values, 
=a, g=1, o=0, Q=kh, (39) 
represent a pencil of four planes Ty. . 113, of which the quinary symbols 
(23) may be thus written :— 
Ry = [lmnrs|; R2=(Vm'n'r's']; Ry=R,-Ro; Bs=Ro—hR; (40) 
and of which the anharmonic is consequently, by (33), the same quotient, 
, 
(Hi, 111,11) =(k=9 =), (41) 
as before. We have therefore this Theorem :— 
PROC. R. I. AA—VOL. VII. 4rF 
