JoLY — Asiatics and Quaternion Functions. 367 



In order to illustrate the use of these equations, we find from (A), 



c = Sfqq-\ and y = Vf(iq-\ 



which give without ambiguity the pitch and the centre when the 

 lotation is assigned, and hence the wrench is completely determined, 

 and the equation of its axis is 



p = y + tq^PfK 



Or again, we may regard c and q as unknown ; and on solution 

 of (A) we find four values of q determining four rotations which place 

 the forces so that their centre is at the extremity of the given vector y. 

 Or we may seek the rotations which produce wrenches of given pitch, 

 and we see that if 



Vq „_ 



K = — - = tan £ y . UVq, 



K must terminate on the quadric 



^ ^ ^ /(I + 4 , or c(l -K^) = >Sf(l -k)/(1 + /<); 



1 + K 



and from this it appears that if we rotate the forces round a direction 

 parallel to a radius vector of this quadric, and through double the angle 

 Avhose tangent is equal to the length of that radius, the pitch of the 

 resultant wrench will be equal to c. 



Or we may seek the locus of centres of wrenches of a given pitch c. 

 This is determined by the equation of the latent quartic of fq - yq, 

 when the given value of c is substituted therein. The locus is a 

 cyclide. Or, if we turn to the equation 



y = W<ir\ 



it is seen without trouble that the locus of y is a region circumscribed 

 by the cyclide corresponding to the wrenches of zero pitch, and by a 

 cylinder of the second order. This cylinder touches where it meets 

 all the cyclides. 



Or again, if q is of the form qi + tq^, where q^ and qz are given 

 quaternions and t a variable scalar, we find, as t varies, that tbe 

 rotation takes place round a definite direction but through a varying 

 angle, and that 



y = ^(^1 + k^) ■ (?i + k^y 



describes an ellipse, while the corresponding axes of the resultant 



