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LXIX. On Symbolical Statics. By the Rev. M. O'Brien, M.A., 



Professor of Natural Philosophy and Astronomy, King's Col- 

 lege, London, late Fellow of Caius College, Cambridge*-. 



Various applications of the notation ('(1 + m)U^ ivhich represents 



a force U acting at the point uof a rigid body. 

 (I.) nntlE principles upon which this notation is founded have 

 A been briefly stated in the former paper. U is supposed 

 to represent, in magnitude and direction, any force acting upon 

 a rigid body at a certain point (0), which is chosen as origin of 

 distances; u denotes, in magnitude and direction, the distance 

 of any other point P of the rigid body from ; and it has been 

 shown, that the symbol of the force U acting at P is 



(1+m)U: 

 which expression consists of two parts, namely, U and wU ; the 

 former denoting the mechanical effect which the force U acting 

 at produces on the rigid body, and the latter the efi"ect pro- 

 duced by translating U, without alteration of magnitude or direc- 

 tion, from to P. In fact mU denotes the couple consisting of 

 the forces, — U acting at 0, and U acting at P. 



(II.) The application which I made of the notation (1 +m)U in 

 the former paper manifestly assumes that parallel and equal 

 translations oi parallel and equal forces are mechanically equiva- 

 lent ; that is, that if the line AB and the 

 force U be respectively parallel and equal 

 to the line A'B' and the force U', then the 

 translation of U' from A' to B' produces 

 the same mechanical effect as the transla- 

 tion of U from A to B. It is easy to see 

 that this amounts to assuming, that two '^ 



equal parallel forces produce the same 



effect on a rigid body as their sum acting midway between them. 

 For, let C be the intersection of AB' and A'B ; then, if U be 

 translated from A to B, we shall have U at B and U' at A', which 

 produce the same effect as U + U' at C : again, if instead of U 

 we translate U' in a similar way, that is, from A' to B', we shall 

 have U at A, and U' at B', which produce the same effect as 

 U-hU' at C. It appears, therefore, that the translation of U 

 from A to B and that of U' from A' to B' produce the same 

 effect, because in both cases the change produced is equivalent 

 to making the two forces act at C. 



(111.) The following is a mode of proving the truth of the sym- 

 bolization (1 +u)U, without introducing the idea of translation. 



Assuming that U represents a force acting at the origin, and 

 * Communicated by the Author. 

 ^ L2 



