Sir W. Rowan Hamilton on Qiiaternioris. 221 



tudes fl], fig, . . shall be respectively etjual to the angles of the 

 spherical polygon which is formed by their representative 

 points R,, R^, . . taken in their order. To fix more precisely 

 what is to be understood in speaking here of these angles, 

 suppose that R^ is the representative point of the mih quater- 

 nion factor, or the mih. corner of the polygon, the next pre- 

 ceding corner being R;„_i, and the next following being 

 R»j_)_i; and let the angle, or (more fully) the internal angle, 

 of the polygon, at the point R^, be denoted by the same sym- 

 bol R^, and be defined to be the least angle of rotation through 

 which the arc R^ R»8-fi must revolve in the positive direction 

 round the point R^, in order to come into the direction of the 

 arcR^R^_j. Then, the rotation 27r — R,„ would bring 

 ^m ^fw-i to coincide in direction with R^ Rm+i > ^"^ there- 

 fore ihe rotation tt— R^, performed in the same sense or in 

 the opposite, according as it is positive or negative, would 

 bring ihe prolongation of the preceding arc R^j-i R^ to coin- 

 cide in direction with the following arc R^R^_^i; on which 

 account we shall call this angle tt — R^, taken with its proper 

 sign, the external angle of the polygon at the point R^. The 

 same rotation tt— R^ would bring the positive pole, which we 

 shall call P,„4.i, of the preceding side R^_i R^ of the poly- 

 gon, to coincide with the positive pole Fm+2 o^ ^he following 

 side R^ R^+i thereof, by turning round the corner R^ as a 

 pole, in an arc of a great circle, and in a positive or negative 

 direction of rotation according as the external angle tt — R^ 

 of the polygon is itself positive or negative ; consequently, by 

 the last article, we shall have the formula 



f*m Qm = COS R^ + /r,„ sin R^ = ip^^^ Zp^^2. 



Multiplying together in their order the n formulas of this sort 

 for the n corners of the polygon, and attending to the associa- 

 tive character of quaternion multiplication, which gives, as 

 an extension of the formula (P.), the following, 



^Pi h, ' ip, % • — iPn «'p, = (- l)"j • • • (P'O 

 we see that under the supposed conditions as to the amplitudes 

 we have this expression for the product of the n quaternion 

 factors, 



Qi Q2 Qs ... Q„ = ( - 1 r /*i /A2 /*3 ... /A„ ; 

 from which it follows, that for awj/ spherical polygon R| Rg •.•R„, 

 (even with salient and re-entrant angles), this general relation 

 holds good : 



