Sir W. R. Hamilton on Continued Fractions in Quaternions, 371 



nine and sugar being produced in the largest quantity. In the 

 course of this paper I shall have occasion to mention circum- 

 stances in which a stdl greater preponderance takes place in the 

 amount of several of these substances formed over that of the 

 others. Whether it would be possible to confine the decompo- 

 sition of rubian entirely to one of these processes, or whether all 

 three are essential, is a question of the highest importance, not 

 so much in a theoretical, as a practical point of view. That 

 beautiful substance, alizarine, is the only one of these products 

 which is capable of yielding dyes. It is this body which in my 

 opinion gives rise to all the beautiful colours for the production of 

 which madder is employed. The others are not only useless, they 

 are positively injurious, as I have shown on a former occasion. 

 Though experimentally alizarine is formed in the smallest pro- 

 portion, it is nevertheless theoretically possible to convert rubian 

 entirely into alizarine, without the least quantity of the other 

 substances being produced. From this point of view the other 

 substances may be considered as formed at the expense of aliza- 

 rine. In fact, by adding together 1 equiv. of verantine and 

 1 equiv. of rubiretine, and subtracting 1 equiv. of water, we ob- 

 tain the elements of 2 equivs. of alizarine, for 



C i4 H 5 05 + C 14 H 6 4 =2(C 14 H 5 O 4 ) + HO. 

 Also by adding together 1 equiv. of rubianine and 1 equiv. of 

 sugar, and subtracting 16 equivs. of water, we obtain the ele- 

 ments of 4 equivs of alizarine, for 



C 44 H 24 O 20 + C 12 H 12 12 =4(C 14 H 5 O 4 ) + 16H0. 



If any chemist should succeed in changing rubian entirely 

 into alizarine, an undertaking in which there is no occasion to 

 despair of success, he would be the means of giving a great sti- 

 mulus to many branches of manufacture and adding a large sum 

 to the national wealth. 



LII. On Continued Fractions in Quaternions. By Sir William 

 Rowan Hamilton, LL.D., M.R.I. A., F.R.A.S. ^c, Andrews' 

 Professor of Astronomy in the University of Dublin, and Royal 

 Astronomer of Ireland*. 



1. TT is required to integrate the equation in differences, 



Ux+i{u a + a) = b, 



where x is a variable whole number, but a, b, u arc quaternions f. 



* Communicated by the Author. 



f The advertisement, that the present writer's Lectures on Quaternions 

 were to be ready in January last, was inserted, contrary to his wishes, 

 through the over-zeal of an agent : but the work in question is now nearly 

 all in type. 



2B2 



