ee ed all 
275 
diverging from one common origin, may all be sides of one 
common cone of the second degree. ‘The summit of this cone, 
or the common initial point of each of these six vectors, being 
ealled o, let the six final points be ascpec’: the transformed 
equation of homoconicism (4) expresses that the ratio com- 
pounded of the two ratios of the two pyramids oaBc, OCDE, to 
the two other pyramids oavc, ocBE, does not change when we 
pass from the point c to any other point c’ on the same cone 
of the second degree: which is a form of the theorem of M. 
Chasles, respecting the constancy of the anharmonic ratio. 
An intimate connexion between this theorem and that of 
Paseal is thus exhibited, by this symbolical process of trans- 
formation. 
As the equation (1) expresses that the ¢hree vectors 
B PB’ B” are coplanar, or that they are contained on one com- 
mon plane, if they diverge from one common origin, and as 
the equation (4) expresses that the six vectors a,...a” are 
homoconic, so does this other equation, 
s.p(p— y)(y —B)(B —a)a=0, (5) 
express that the four vectors a, (3, y, p are homospheric, or 
that they may be regarded as representing, in length and in 
direction, four diverging chords of one common sphere. Thus, 
the arithmic part of the continued product of the five succes- 
sive sides of any rectilinear (but not necessarily plane) pen- 
tagon, inscribed in a sphere, is zero; and conversely, if in 
any investigation respecting any rectilinear, but, generally, 
uneven, pentagon ABCDE in space, the product aB x Bc X cD 
x pEXEA of five successive sides, when determined by the 
rules of the present calculus, is found to be a pure vector, or 
can be entirely constructed by a line, so that in a notation 
already submitted to the Academy (see account of the com- 
munication made in last December) the equation 
S.ABCDEA = 0, (6) 
