242 Sir W. R. Hamilton on Quaternions. 



— cos R" = cos R cos R' — (a a' + ^ /3' + yy') sin R sin R', 

 a"sin R" =«sin Rcos R' + a'sin R'cos R + (/3y' — 7/3')sin Rsin R', 

 /3"sin R" =/3sin Rcos R' + /3'sin R'cos R + (y a'— a-/) sin R sinR', 

 y" sin R" =y sin R cos R' 4- y'sin R'cos R + (a /3' — /3 a') sin RsinR'; 



of which indeed the first answers to the well-known relation 

 (already employed in this paper), connecting a side with the 

 three angles of a spherical triangle. The three other equa- 

 tions (K.) correspond to and contain a theorem (perhaps new), 

 which may be enunciated thus : that if three forces be applied 

 at the centre of the sphere, one equal to sin R cos R' and di- 

 rected to the point R, another equal to sin R'cos R and di- 

 rected to R', and the third equal to sin R sin R' sin R R' and 

 directed to that pole of the arc R R' which lies towards the 

 same side of this arc as the point R", the resultant of these 

 three forces will be directed to R", and will be equal to sin R". 

 It is not difficult to prove this theorem otherwise, but it may 

 be regarded as interesting to see that the four real equations 

 (K.) are all included so simply in the single imaginary formula 

 (I.), and can so easily be deduced from that formula by the 

 rules oi ihe multiplication of quaternions ', in the same manner 

 as the fundamental theorems of plane trigonometry, for the 

 cosine and sine of the sum of any two arcs, are included in 

 the well-known formula for the multiplication of couples^ that 

 is, expressions of the form x + iy, or more particularly, cos 9 

 + /sin 5, in which 2^= ~ 1. A new sort of algorithm, or cal- 

 culus for spherical trigonometry^ would seem to be thus given 

 or indicated. 



And if we suppose the spherical triangle R R' R" to become 

 indefinitely small, by each of its corners tending to coincide 

 with the point of which the co-ordinates are 1, 0, 0, then each 

 co-ordinate a will tend to become = 1, while each /3 and y will 

 ultimately vanish, and the sum of the three angles will ap- 

 proach indefinitely to the value tt; the formula (I.) will there- 

 fore have for its limit the following, 



(cosR+2sinR)(cosR'+2sinR') = cos(R+ R') H-/sin(R + R'), 

 which has so many important applications in the usual theory 

 of imaginaries. 



7. In that theory there are only two different square roots 

 of negative unity, and they differ only in their signs. In the •• 

 theory of quaternions, in order that the square of w + ix +jy 

 •\-kz should be equal to —1, it is necessary and sufficient ' 

 that we should have 



TO = 0, a;2 + y2 4- 2^^ = 1 ; 

 for, in general, by the expressions (D.) of this paper for the 



