KJJir W. Rowan Hamilton on Qtiaternions, 135 



series of quaternions, of which the last will (by the theorem) 

 have its scalar part equal to zero; while the vector part, or the 

 product itself, will be constructed by a right line with a cer- 

 tain definite direction, which will (by the same theorem) be 

 that of a certain rectilinear tangent to the sphere, at the point 

 or corner where the first side of the inscribed polygon begins. 

 [The tensor of the resulting vector, or the length of the pro- 

 duct line, will of course represent, at the same time, by the 

 general law of tensors, the product of the lengths of the lactor 

 lines, with the usual reference to some assumed unit of length.] 

 And conversely, whenever it happens that an odd number of 

 successive right lines in space, being multiplied together suc- 

 cessively by the rules of the present Calculus, give a liJie as 

 their continued product, that is to say, when the scalar of the 

 quaternion obtained by this multiplication vanishes, then those 

 right lines may be inferred to have the directio?is of the suc- 

 cessive sides of a polygon inscribed in a sphere. 



84-. Already, even as applied to the case of an inscribed 

 gauche pentagon, the theorem of the last article expresses a 

 characteristic property of the sphere, which may be regarded 

 as being of a graphic rather than of a metric character ; inas- 

 much as it concerns immediately directions rather than mag- 

 nitudes, although there is no difficulty in deducing from it 

 metric relations also : as will at once appear by considering 

 the formula which expresses it, namely the following, 



= S.(^-pJ(p4-p3)(p3-p,)(p2-pi)(pi-p). • (24-8.) 



(See the Proceedings of the Royal Irish Academy for July 

 18'l-6, where this quaternion theorem for the case of the in- 

 scribed pentagon was given.) For the theorem assigns, and 

 in a simple manner expresses, to those who accept the lan- 

 guage of this Calculus, a relation between ihejive successive 

 directions of the sides of a gauche pentagon inscribed in a 

 sphere, which appears to the present writer to be analogous 

 to (although necessarily more complex than) the angular rela- 

 tion established in the third book of Euclid's Elements, be- 

 tween \\\efour directions of the sides of a plane quadrilateral 

 inscribed in a circle. Indeed, it will be found to be easy to 

 deduce the property of the plane inscribed quadrilateral, from 

 the theorem respecting the inscribed gauche pentagon. For, 

 by conceiving the fifth side P4P of the pentagon P...P4 to tend 

 to vanish, and therefore to become tangential at the first corner 

 p, it is seen that the vector part of the quaternion which is the 

 continued product of the four first sides must tend, at the 

 same time, to become normal to the sphere at p ; in order 

 that, when multiplied into a tangential vector there, it may 



