320 PROCEEDINGS OP THE AMERICAN ACADEMY 



where w is the hist vector of the series. Hence, the product of such 

 a series of quateruious will be 



q^... qyq = T (^. . . . q"q'q)^~. 



The principal logarithm of such a product is 



l(y_..jY'?)=^1Ty + Z:UV~ 



or \Uq = ^Tq + Z Ilq-VVITq. (28) 



The order of the factors must be carefully observed in using this for- 

 mula. 



12°. Let us consider the logarithm of the product of a series of 

 complanar vectors. 



First, suppose the number of factors to be even. Pair the succes- 

 sive vectors, and regard these pairs as quaternions. Since they are 

 complanar quaternions, we have, by (20), (20)', and (27), 



1 (u,a.,.(t^a^.a.a^ . . . '^2,i-i«2n) = 



i ITa -|- [(n _ 2m) Q — 2: Z "2 1 1 UYa,a.„ (29) 

 1*1- 1 "2/;-! J 



(„_ 2/«4-l)0> (^Z2l_^ )> («— 2 m— 1)9, 



where 



and where the angles are to be taken positively or negatively according 



as they agree or disagree \vitli -. 



Secondly, if the number of factors be odd, then regard their product, 

 down to and excluding the last vector «2,„ as the product of a given 

 number of complanar quaternions. This i:)roduct, it will be observed, 

 is a quaternion q whose plane is the same as that of the vector factors. 

 The remaining odd vector lies in the plane o{ q ; so that qa2n is a 

 vector lying in the plane of q. Therefore the product of any odd num- 

 ber of complanar vectors is a vector complanar with those vectors. 

 Hence, if «o«i«2 • • • *^2r. be the given vectors, 



1 («o«i«„ . . . a^J = i 1T«, + I Vila,. (30) 



^0 



a formula true for any odd number of comjilanar vectors. 



