OF AETS AND SCIENCES. 313 



Now wheu T^- is null, q itself is null, and 



.•.lim(2'„2^„ — ^"„) = 0. 



Thus at the limit -S;2''„»= 1"^, 



or QiQ^' =zQ<l + <l'. (1) 



This proof includes the proof of 6''= 6^'''6^^''', since in the products 

 of scalars with vectors the commutative property holds. 



With diplanar quaternions, formula (1) is not true, since such qua- 

 ternions are not commutative with each other. 



3°. Defining the five quaternion functions 6S sin q, cos q, Sh q^ 

 Ch q, by the five series 



(«) s' = f ^! = i + ^ + It + It + --- 



^ ^ ^ (^« + l)! 3! ' 5! 



(c) cos^ = 2^(-r^ = l-f^ + ^- 



W «"? = -- ^ = i + It + .^ + --- 



we have, by adding (d) and (e) 



6^ = Ch ^ + Sh q, (2) 



and by taking for q the particular value Vq, 



e""' = ChYq + ShVj ; 



whence observing that 6'' = 6^*6^', we obtain the important general 



formula 



6' = 6'* (ChV^ + ShV^). (3) 



If i denotes any value of sj — 1 which is commutative with q, for- 

 mulae (i), (c), (d), and (e) give 



