Mr. A. Cayley on the Theory of Logarithms. 275 



ling, contained neither magnesia nor iron, which are doubtless 

 derived from the matrix. 



My specimens from the next locality, Dumbarton Moor, so 

 much resemble the Glenfarg specimens, that it is impossible to 

 distinguish between them. The specimens from this locality are 

 perhaps the tinest that Scotland affords. My analysis was made 

 on a mixed red and white speciuien : — 



Silica 46-96 



Alumina .... 26-908 



Lime 3-76 



Soda 12-83 



Water .... 9-50 



99-958 



This species is also said to occur at the Bin above Burnt- 

 island in Fife, and near North Berwick : I have not seen speci- 

 mens from these places, the above being all the Scotch locali- 

 ties I know. Glenfarg is, as far as I am aware, the only one of 

 these where Natrolite occurs distinctly crystallized ; the form is 

 m of Brooke and IMiller. These ci-ystals contain merely a trace 

 of lime, but it is singular that elsewhere in Scotland it should 

 always contain that base ; I should, however, mention that Dr. 

 Scott, who analysed the Bishoptown variety (see Edinb. New 

 Phil. Journ. for Oct. 1852), found no lime. I have, however, 

 examined certainly a dozen specimens from this locality, and in- 

 variably detected its presence. 



In this species, as in many others of the Zeolites, the quantity 

 of water varies considerably, even in specimens from the same 

 locality — sometimes as much as 1 per cent. 



XXXV. Note on the Theory of Logarithms. 

 By A. Cayley, Esq.'* 



AN imaginary quantity x-\-yi may always be expressed in 

 the form 



x-{-yi = r{cos. ^ + ?'sin d)—re^\ 



where r is positive, and 6 is included between the limits —it 

 and +7r. We have, in fact, 



and when x is positive, 



= tan-'^j 



X 



* Communicated by the Author. 



