196 Mr II. Cox, On the application of Quaternions. [Feb. 20, 



From these formulae assuming the associative principle, the 

 relations between the sides and angles of a triangle can be obtained. 

 They are in I, 



cosh a = cosh b cosh c - sinh b sinh c cos A 

 sinh a _ sinh b _ sinh c 

 sin A sin B sin G 



with others similar. In this way all the properties of the different 

 kinds of space can be obtained by a purely algebraical method. 

 The chief distinctions between the three cases are pointed out. 



Passing to the geometry of three dimensions it is found that 

 seven distinct imaginary quantities are required which in case I. 

 will be identical with the symbols that occur in Sir W. Hamilton's 

 bi-quaternions. If /, J, K be the ordinary quaternion symbols 



XI+ YJ + ZK+J^l (LI+ MJ+NK) 



will represent a line or force in space provided that 



XL+YM+ZN = 0. 



If this equation be not satisfied it will represent a system of forces. 



Some applications are made to the theory of screws and the 

 equation to the cylindroid is found to be 



(pa -Pp) O 2 - Z 2 ) X\f = (1 +p a p(d (*? + */) OiZ. 



It is therefore a surface of the fourth degree as has already been 

 shewn by Lindemann. For points near the origin making co nearly 

 and x, y, z, pz, p/3 small, it becomes 



(Pa-pp)xy = z(x-+y 2 ) 



and coincides with the equation to the cylindroid in ordinary space 

 in the form given to it by Prof. Ball. It is shewn that the 

 properties of forces in space can be derived from those of forces 

 meeting at a point by taking for components of force instead of 

 x, y, z the imaginary quantities X + Leo, S + J\Ico, Z+ Nta 



where co 2 = — 1 in I, 



co 2 = in II, 

 a> 2 = 1 in III. 



Lastly, the systems of imaginary quantities for spaces of higher 

 dimensions are shewn to be obtained from the commutative pro- 

 ducts of lower systems. 





