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 

 called o, let the six final points be abcdec': the transformed 

 equation of homoconicisin (4) expresses that the ratio com^ 

 pounded oj" the two ratios of the two ■pyramids oabc, ocde, to 

 the two other pyramids oa.dc, 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 

 Pascal is thus exhibited, by this symbolical process of trans- 

 formation. 



As the equation (1) expresses that the three vectors 

 j3 j3'j3" arie coplanar, or th^t 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(^-7)(7-|3)(i3-a)a=0, (5) 



express that the four vectors a, /3, y, p are homosphceric, or 

 that they may be regarded as representing, in length and in 

 direction, /bur 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 abxbcXcd 

 X DE X EA 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 zz 0, (6) 



