516 Sir W. Rowan Hamilton on Quaternions. 



articles46and47),tothe equation of those lines for the ellipsoid, 



from which, when combined with the general equation S.vtisO, 

 the formula (63.) has been deduced, and geometrically inter- 

 preted as above. 



55. Another mode of investigating generally the directions of 

 those tangential vectors t which satisfy the system of the two 

 conditions in art. 51, may be derived from observing that 

 those conditions fail to distinguish one such tangential vector 

 from another in each of the two cases where the variable nor- 

 mal V coincides in direction with either of the two fixed cyclic 

 normals, » and x ; that is, at the four umbilical points of the 

 ellipsoid, as might have been expected from the known pro- 

 perties of that surface. In fact if we suppose 



v = mi, S.«T = 0, (64.) 



where w is a scalar coefficient, that is if we attend to either of 

 those two opposite tmibilics at which v has the direction of », 

 we find the value 



vt»tx = w(»t)^x, (65.) 



which is here a vector-form, because by (64.) the product »r 

 denotes in this case a jnire vector, so that its square {like that 

 of every other vector in this theory) 'will be a negative scalar, by 

 one of the fundamental and ■peculiar'^ principles of the present 

 calculus ; the scalar part of the product vtjtx therefore vanishes, 

 or the condition (49.) is satisfied by the suppositions (64.). 

 Again, if we suppose 



v = m^)i, ....... (66') 



m' being another scalar coefficient, that is if we consider either 

 of those two other opposite umbilics at which v has the direc- 

 tion of X, we are conducted to this other expression, 



vr»Tx = »i'xT<Tx; (67.) 



which also is a vector-form, by the principles of the 20th 

 article. In this manner we may be led to see that if in general 

 we decompose, by orthogonal projections, each of the two 

 cyclic normals, » and x, into two partial or component vectors, 

 i', j", and x', k", of which •' and x' shall be tangential to the 

 surface, or perpendicular to the variable normal v, but i" and 

 x" parallel to that normal, in such a manner as to satisfy the 

 two sets of equations, 



, = ,' + ."; S.i'v = 0; V.,"v=0; \ ^ ^ ^gg^^ 

 x = x' + h"; S.x'v=0; V.x"v = 0;J 



* See the author's letter of October 17, 1843, already cited in a note to 

 article 64. 



