The Principles of the Calculus of Quaternions. (:)53 



A represenrative of the last claj^s has not been found so far, 

 and very probably will never be found; if below a lower 

 critical point additional pressure made the homogeneous 

 liquid split into two layers, the two saturation-curves would 

 lie outside each other and a rise of temperature towards the 

 critical point would make them approach each other ; but this 

 is in contradiction with the general rule * that the saturation- 

 curves in the r-.r diaoram contract on heatino;. 



Univei-sity College, \ 



Dundee. 



LXXXI. -.1 Method of estahlisldng the Principles of the Cal- 

 culus of Quaternions. By Charles Jasper Joly, M.A., 

 JD.Sc, Fellow of Trinity College^ Dublin, and Royal Astro- 

 nomer of Ireland f. 



I 



DO not remember ha vino- seen the folio win or method of 

 establishing the principles of the calculus of quaternions. 

 If it is new it may be of interest, as it is simple and it clearly 

 exhibits the etfect of the associative property — the property 

 which sharplv separates quaternions from other systems of 

 geometrical analysis. 



Defining Sa;8 to be —ah cos 6 where a and h are the 

 lengths of the vectors a and /3 in terms of some assumed 

 unit, and where 6 is the angle between the vectors ; defining 

 also Ya/3 as a third vector at right angles to both and of 

 length ah sin 6 in terms of the assumed unit ; it is easy to 

 see that Sa^S and Ya/3 are both distributive with respect to 

 the vectors. I define the product of a and /3 to he a linear 

 function of Sa/3 and Va/3, or 



a/3=m^a^-{-nYcc/3, (A) 



^vhere ni and n are certain constant numbers, characteristic 

 of the calculus and independent of the vectors a and j3. This 

 insures the distributive property of a product of vectors. 



Again, I impose the condition that the associative law is 

 obeyed, or that a .a^ = a^ . p. If the product of a vector 

 into the sum of a scalar and a vector is distributive, we have 



a.ay8 = amSa/3 + a./iVa/3 (B) 



Now a, . nYa/3 = n'^YaYa^ by (A), and by the same equation 



* Van der Waals, Die Continuitdf , vol. ii. p. 101, sq. 

 + Communicated by the Author. 



Phil. Mag. S. 6. Vol.' 6. No. 36. Dec. 11)03. 2 X 



