Sir W. Rowan Hamilton on Quaternions. 437 



X, give conversely these other expressions for the Jatter vectors 

 in terms of the former, 



. = T)jU(.j-fl); x = T9U(S-'->)-»): . (220.) 

 whence (it may here be noted) follow the two parallelisms, 



U,-Ux = U(>j-5) + U(.)-»-fl-') II U»j + U5; . (221.) 

 U* + Ux = U(,j-S)-U()j-»-fl-') II U>3-U3; . (222.) 



the members of (221.) having each the direction of the great- 

 est axis of the ellipsoid, and the members of (222.) having 

 each the direction of the least axis ; as may easily be proved, 

 for the first members of these formnlae, by the construction 

 with the diacentric sphere^ which was communicated by the 

 writer to the Royal Irish Academy in 1846, and was published 

 in the present Magazine in the course of the following year. 

 The equation (219.) may be verified by observing that it gives, 

 for the length of the perpendicular let fall from the centre of 

 the ellipsoid on an umbilicar tangent plane, the expression 



j9 = (d2_,,2)T(,j-S)->=:flC^»-'; . . . (223.) 



agreeing with known results. And the vector to of the um- 

 bilicar point itself must be the semisum of the vectors of 

 the centres of the two equal and sliding spheres, in that limit- 

 ing position of the pair in which (as above) they touch each 

 other ; this umbilicar vector oo is therefore expressed as follows: 



co = Yi + Q; (224,.) 



because this is the semisum of ft and x' in (145.), or of g>) and 

 gd when ^=2. (Compare the note to article 70.) As a veri- 

 fication, we may observe that this expression (224.) gives, by 

 (215.), the following known value for the length of an umbi- 

 licar semidiameter of the ellipsoid, 



u=Tco=T{ri + d)= Via^-b^+c^ . . (225.) 

 By similar reasonings it may be shown that the expression 



«;' = T>,Ufi + TflU)j, .... (226.) 

 which may also be thus written, (see same note to art. 70,) 



a,' = -T.»)fi.(»j-' + 9-0, .... (227.) 

 represents another umbilicar vector ; in fact, we have, by (224.) 

 and (226.), 



Tco'=T«>, (228.) 



and 



co-c«'=(Trj-Tfl)(U>]-U9)U * • • ^ •' 

 so that the vectors w co' are equally long, and the angle between 



