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XX. 



ASTATICS AND QUATERIS'ION FUNCTIONS. By CHARLES 

 JASPEE JOLY, M. A., F. T. C. D. ; Andrews Professor of 

 Astronomy in the University of Dublin, and Royal Astronomer 

 of Ireland. 



[Eead May 8, 1899.] 



[Absxeact.] 



In his applications of quaternions to the statics of a rigid system, 

 Hamilton has used the quaternion equations 



^a/3 = {c + y)'$/3 = C + fx. 



In these, a is the vector from an arbitrary origin to the point of 

 application of the corresponding force /3 ; c is the pitch of the resultant 

 wrench, and y is the vector to a definite point on its axis which 

 Hamilton called the Getteral Centre ; - C is the virial, and /u. is 

 the resultant couple for the arbitrary origin as base-point. 



If we take the first of these equations and suppose the forces 

 rotated as a rigid system round their points of application, each vector 

 )8 may be replaced by q(^q'\ and the equation becomes 



^aq(3q-' = {c ^ y) q%Pq-\ 



c and y now referring to the rotated system. 



If the linear function %aq(B (5/3)"- is briefly denoted by fq, the 

 equation becomes, when multiplied into q (2/3)"', 



fq = {c + y) q. (A) 



This contains in a very simple manner the essentials of the various 

 systems of forces when the body is fixed while the forces move. 



If the body is rotated while the forces are fixed in magnitude and 

 direction, and if the rotation is specified by q~^ ( ) q, we find, when 

 each vector a is replaced by q~^ aq, the equally simple equation, 



f9 = 9{c + y), (B) 



applicable to the discussion of the force systems when the directions 

 of the forces are fixed in space. 



