116 Proceedings of the Royal Irish Acadetny. 



dimensions, but it is distributed in a definite manner in a definite 

 number of spaces of three dimensions; the pitch must then be a 

 function of the units symbolizing these spaces, or of their volume 

 vectors. 



To take a concrete example, suppose the origin chosen so that 

 V^^'^V = 0, and suppose the quadratic r reduced to the canonical 

 form, so that 



r = ffnilh + yJsii + . • . + ff2m-l, 2>j2n^lh,n- 



Assuming $ = x^ii + ^24 + • . . + ^zm+iizm+i, 



it is evident that, in order Fi^"T should vanish, ^ must reduce ta 

 ^2»i+i^"2»»+i ; foi'j consider the part linear in «"i and 4 in Fj^-T ; it is 

 ffi2 (^i4 - ^2h) to a numerical factor, and its vanishing requires 

 a^i = ^2 = 0. "We may thus take as the canonical forms of a force 

 and couple 



^ = i2mH, and r = yi2?>2 + ffsii^ii + &c. 



In this case the pitch is 



and the essentials on which it depends are the numerical coefficients 

 g^o, ffsi, &c. {m in number when 2m + 1 or 2m + 2 units are involved),^ 

 and the symbols of the various spaces of three dimensions (%44)«+i)j 

 (444m+i)> &c. (which all contain the force). It should be noticed that 

 a wrench is always expressible in terms of an odd number of units, sa 

 that a wrench cannot Jill (or exhaust all the units of) an even space.^ 

 A wrench in four dimensions is a wrench in three. As an example,. 



let To = fi!io?i4 + asJsii, and let i = xj-^, + x-^i^. 



(This expression for ^ is perfectly general, since i^ may be taken to 

 be parallel to the component of f in the definite plane («'i?2), and % 

 may be taken to be parallel to the remaining component of ^.) 



With these values. 



and V = V,-^V^t'V,= 



x^ + .zv 



is a product of two vectors pei-pendicular to one another, and to ^; 

 but three independent vectors are involved in r and ^. 



1 Of course the/orce is supposed not to vanish. 



2 Confirmation maybe obtained from Art. 12, where it was shown to be generally 

 impossible to assign p so that V\qip should vanish in an even space. 



