Sir W. Rowan Hamilton on Quaternions. 217 



if the two diagonals ae, bd' of the quadrilateral be equally long ; 

 so that a general and characteristic property of the ellipsoid, 

 sufficient tor the construction of that surface, and t'or the in- 

 vestigation of all its properties, is included in the remarkably 

 simple and eminently geometrical formula 



AE=BD'; ,(3.) 



the locus of the point E being an ellipsoid, which passes 

 through B, and has its centre at A, when this condition is 

 satisfied. 



This formula (3.)} which has already been printed in this 

 Magazine as the equation (10.) of article 30 of this paper, may 

 therefore be deduced, as above, from generally admitted prin- 

 ciples, by the Cartesian method of co-ordinates ; although it 

 had not been known to geometers, so far as the present writer 

 has hitherto been able to ascertain, until he was led to it, in 

 the summer of 1 846 *, by an entirely diiFerent method ; namely 

 by applying his calculus of quaternions to the discussion of 

 one of those new formsf for the equations of central surfaces 

 of the second order, which he had communicated to the Royal 

 Irish Academy in December 1845. 



36. As an example (already alluded to in the 32nd article 

 of this paper) of the geometrical employment of the formula 

 (3.), or of the equality which it expresses as existing between 

 the lengths of the two diagonals of a certain plane quadrilateral 

 connected with that new construction of the ellipsoid to which 

 the writer was thus led by quaternions, let us now propose to 

 investigate geometrically, by the help of that equality of dia- 

 gonals, the difference of the squares of the reciprocals of the 

 greatest and least semi-diameters of any plane and diametral 

 section of an ellipsoid (with three unequal axes). Conceive 

 then that the ellipsoid, and the auxiliary sphere employed in 

 the above-mentioned construction, are both cut by a plane abV, 

 on which b' and c' are the orthogonal projections of the fixed 

 points B and c ; the auxiliary point d may thus be conceived 

 to move on the circumference of a circle, which passes through 

 A, and has its centre at c' ; and since AE, being equal in length 



* See the Proceedings of the Royal Irish Academy, 

 t In reprinting one of those new forms, namely the following quater- 

 nion form of the equation of the ellipsoid: 



a slight mistake of the press occurred at p. 459, vol. xxx. of this Magazine, 

 which however, with the assistance there given by the context, can scarcely 

 have embarrassed the reader. In the preceding page, for a hyperboloid of 

 one sheet, touching the same cylinder in the same sheet, should have been 

 printed, .... in the same ellipse. 



