0« the Equation Q = g- (w, a?, y, 2) = ttJ + ix -{-jy -\- kz, 1 35 



The iron and manganese were separated from the lime by 

 ammonia after the addition of bromine. 



The separation of phosphoric acid was complete; not a 

 trace could be discovered with the bases by means of molyb- 

 date of ammonia, nor could a trace of lime be discovered by 

 treating the 2MgO, PO^ with sulphuric acid, evaporating and 

 dissolving in alcohol. 



The results obtained in two analyses from crystals from 

 different parts of the specimen were in 1 00 parts, — 



I. II. 



Lime 53'38 52-81 



Oxides of iron and manganese 2*96 3*22 



Phosphoric acid 4)r34 4;r80 



Fluorine and loss 2*32 2*17 



100-00 100-00 



This composition corresponds very nearly to that of fluor 

 apatite, Ca Fl + 3(3CaO, PO^), in which the lime is partially re- 

 placed by the protoxides of iron and manganese. This com- 

 position would give per cent, (see Rammelsberg's Handwor- 

 terbuch der Mineralogiey p. 37), — 



Lime 55-88 



Phosphoric acid . . . 42*02 

 Hydrofluoric acid . . 2*10 



100-00 



And these analyses of Francolite confirm, by the direct esti- 

 mation of the PO^ by an accurate method, the results obtained 

 by Gustave Rose in an elaborate investigation of several va- 

 rieties of apatite from various localities, published many years 

 ago*, in which the phosphoric acid was estimated from the 

 loss. 



XVin. On the Equatioti Q = q['w, .v,i/, z)='w + ix+j7/ + kz. 

 By William Spottiswoode, M.A., of Balliol College^ 

 Oxford^, 



THE theorem expressed by the above equation is of consi- 

 derable importance in the calculus of quaternions, and in- 

 deed essential for the application of that method to geometrical 

 and physical problems. Sir W. R. Hamilton in his researches 

 (Transactions of the Royal Irish Academy, vol. xxxi.), has 

 effected the transformation by means of the symbolical division 

 of numeral sets ; but since nothing, which may throw light 



* PoggentlorfF's Annalen, vol. ix.j and Berzelius, Jahresbericht, 1828. 

 t Communicated by the Author. 



