421 



order of the two quaternion-factors would throw the pro- 

 duct-point k" from the right to the left, or from the left to 

 the right of iir'. 



It results from these principles, that if rr'r" be any 

 spherical triangle ; if, also, a /3 7 be the rectangular coordi- 

 nates of R, a'fi'Y those of r', and a" )3" y" of r", the centre 

 of the sphere being origin, and the radius being unity ; and 

 if the rotation round -f x from -f- y to + 2; be of the same 

 (right-handed or left-handed) character as that round Rfrom 

 r' to r" ; then the following formula of multiplication, ac- 

 cording to the rules of quaternions, will hold good : 



^ cos R -f (ia+J^ + ky) sin R I { cos r' + {la' +j(i' + ky') sin r' \ 

 = - cosK" + (ia"+j(5" + ky") sin r". (1) 



Developing and decomposing this imaginary or symbolic 

 formula (i), we find that it is equivalent to the system of the 

 four following real equations, or equations between real 

 quantities: 



— cosr":=cosr cosr' — (aa'-j- /3/3'-(-y y')'"'" ■'s'""'! "1 

 a" sin r" := n sin R cos r' -f- a' sin r' cos R + (J3y' — y/?) sin R sin r' ; I 



/3"sinR"=:)3sinRcosR' -f- /3'sinR'cosR -f (ya' — ay') sin R sin r'; | \'^) 



y"sinR" ^ y sin R cos r' -|- y'sinR' cos R -\- {a^ — jSa') sinR sinR'. J 



Of these equations (k), the first is only an expression of the 

 well-known theorem, already employed in these remarks, 

 which serves to connect a side of any spherical triangle with 

 the three angles thereof. The three other equations (k) are 

 an expression of another theorem (which possibly is new), 

 namely, thai a force z= sinR", directed from the centre of 

 the sphere to the point r", is statically equivalent to the sys- 

 tem of three other forces, one directed to r, and equal to 

 sinR cosr', another directed to r', and equal to sin r' cos r, 

 and the third equal to sin r sin r' sin r r', and directed to- 

 wards that pole of the arc r r', which lies at the same side of 

 this arc as k''^. It is not difficult to prove this theorem other- 

 wise ; but it may be regarded as interesting to see that the 

 four equations (k) are included so simply in the one formula 



