39 



a, a, . . are drawn, as from a common origin, to the points of 

 application of the various forces, /3, )3', . . This requires 

 that the two following conditions should be separately satisfied, 



2/3 = 0; SF.a/3 = 0; (2) 



which accordingly coincide with the two equations marked 

 (18) of the abstract just referred to. The former of these two 

 equations, S/3 = 0, expresses that the applied forces would 

 balance each other, if they were all transported, without any 

 changes in their intensities or directions, so as to act at any 

 one common point, such as the origin of the vectors a; and 

 the latter equation, 2F. a/3 = 0, expresses that all the couples 

 arising from such transport of the forces, or from the introduc- 

 tion of a system of new and opposite forces, - /3, all acting at 

 the same common origin, would also balance each other : the 

 axis of any one such couple being denoted, in magnitude and 

 in direction', by a symbol of the form V. aj3. When either of 

 these two vector-sums, Sj3, SF. aj3, is different from zero, 

 the system cannot be in equilibrium, at least if there be no 

 fixed point nor axis; and in this case, the quaternion quotient 

 which is obtained, by dividing the latter of these two vector- 

 sums by the former, has a remarkable and simple signification. 

 For, if this division be effected by the general rules of this cal- 

 culus, in such a manner as to give a quotient expressed under 

 the original and standard form of a quaternion, as assigned 

 by Sir William R. Hamilton in his communication of the 13th 

 of November, 1843 ; that is to say, if the quotient of the two 

 vectors lately mentioned be reduced by those general rules to 

 the fundamental quadrinomial form, 



Q^ = iv + ix+jy + kz, (3) 



where i,j,k are the Author's three co-ordinate imaginaries, or 

 rectangular vector-units, namely, symbols satisfying the equa- 

 tions, , ^ 

 i^=y^ = k'^ijk = -\, (4) 



