312 PROCEEDINGS OP THE AMERICAN ACADEMY 



Let ^„, ^„, ^'„, represent the sums of the first n terms of the 

 series which define the functions 6^, G^', 6^ + ^': — 



^.=1 + , + '-;+ '"- 



2!^* " (« — !)!' 



■'«=l + ?'+|^ + 



ym — 1 



Also, represent the functions 6''''^, 6"^^', 0'''« + '^5'', down to n terms, 

 by 



^„_l + T^ + -^ + ...^--^,, 



' ^ ^ 2! ' (« — 1)! 



^-„=i + (T,+T,')+... ''^^,:j(i';-' . 



Of this last series, take the sum of the first (2n — 1) terms : — 



By developing the numerators of S"^ and S" 2n_x, and effecting the 

 product »S'„^'„, we shall discov^er (since the multiplications are commu- 

 tative) that all the terms of S''^ are to be found in the product S^S'^y 

 and that all the terms of this product are completely represented by 

 corresponding terms in S"2n — i • Hence 



"When n increases indefinitely, S"„ and S"2n — \ tend to tlie same limit 

 QT9 + T5'. 



.•. hm (^„5'„— S\) = 0. 

 n = 00 



The property, that the tensor of the sum of any number of quater- 

 nions is as small as the sum of the tensors, together with the fact that 

 the terms of the expression (^„^'„ — -^"„) have for their tensors the 

 corresponding terms of {S^S'^ — 'S'"„), gives 



and . •• lim T{Z„I'„ — ^\) = 0. 



n = 00 



