136 Sir W. Rowan Hamilton on Qiiaternioiis. 



give a vector as the product. Hence the vector part of the 

 product of the four successive sides of an inscribed gauche 

 quach'ilateral pp,P2P3, is constructed by a right line which is 

 normal to the sphere at the first corner; and more generally, 

 either by the same geometrical reasoning applied to the theo- 

 rem of art. 83, or by considering the signification of the for- 

 mula (246.), we may deduce this other theorem, that /^<? wt^or 

 of the continued product of the sttccessive sides of mi inscribed 

 gauche polygon p . .v-im-u of any even number ofsides^ is normal 

 to the sphere at the first corner p. Suppose now the inscribed 

 quadrilateral, or more frenerally the polygon of 2»i sides, to 

 flatten into a plane figure ; it will thus come to be inscribed in 

 a circle^ and consequently in infinitely many spheres at once', 

 and the only way to escape a resulting indeterminateness in 

 the value for the vector of the product, is by that vector va- 

 nishing: which accordingly it may be otherwise proved to do, 

 although the present mode of proof will appear sufficient to 

 those who examine its principles with care. And thus we 

 shall find ourselves conducted to the well-known graphic pro- 

 perty of the quadrilateral inscribed in the circle, and more 

 generally to a corresponding theorem respecting inscribed 

 hexagons, octagons, &c., under the form of the following pro- 

 position in quaternions, which expresses a characteristic pro- 

 perty of the circle ; — The vector part of the product of the suc- 

 cessive sides of antf polygon^ issith any even number of sides, in- 

 scribed in a circle, vanishes; or, in other words, the product 

 thus obtained, instead of being a complete quaternion, reduces 

 itself simply to a positive or negative number. On the other 

 hand, it is easy to see, from what precedes, that the product of 

 the successive sides of' a triangle, pentagon, or other polygon of 

 any odd number of sides, inscribed in a circle, is a vector, which 

 touches the circle at the first corner of the polygon, or is parallel 

 to such a tangent. 



85. Although the precise law of the relation between the 

 directions of the sides of an inscribed gauche pentagon, hep- 

 tagon, &c., expressed by the first formula (247.), is peculiar 

 to the sphere ; yet it is easy to abstract from that relation a 

 jpart, which shall hold good, as a law of a more general cha- 

 racter, for other surfaces of the second order. For we may 

 easily infer, from that formula, especially when combined with 

 the other equations of art. 82, that if the first '2m sides of an 

 inscribed polygon of 2ot sides, p'p'jP'a • • p'2»i» ^^ respectively 

 parallel to the successive sides of another polygo7i of 2w2 sides, 

 pPj-.p^m-i, inscribed in the same surface, then the last side, 

 p'-imP'j of the former polygon, xdll be parallel to the plane which 

 touches the surface at the ^first corner p of the latter j^olygoji : 



