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XIU.—On Minding’s System of Forces. By Professor CHRYSTAL. 
(Received January 3, 1880. Read January 5, 1880.) 
Minpine has proved a remarkable theorem concerning a variable system of 
forces defined as follows :—the points of application of the different forces and 
their magnitudes are given, while the directions are such that a pencil of rays 
through any given point parallel to them moves as a rigid body. 
Besides Muinp1ne’s original investigation, several others have been given 
since. The last of these, due to Professor Tart, rests on purely quaternion 
methods, and is so elegant and concise that I was led to reinvestigate the 
whole subject by ordinary methods in the hope that the analysis might have 
some points of interest. Two methods of arriving at Minp1ne’s result are 
given, and a variety of other conclusions are arrived at by means of the second 
method, sufficient to indicate the course of a full investigation of the complex 
formed by the central axes, and of the congruency formed by the single 
resultants of MINDING’s system. 
First Method. 
The components of force and couple are found in terms of the RopRIGUES 
co-ordinates Awv, which determines the position of the rigid pencil representing 
the direction of the forces. Let the rigid pencil be referred to moving rect- 
angular axes O€, On, OG fixed relatively to itself, which again are referred to 
the fixed axes Ox, Oy, Oz, by the system of direction cosines 
ae Uz 
é dy [Sl = tA 
7 | Xy be 
C As [2B Pla 
We shall have occasion to use the well-known relations between these 
clirection cosines, and also their rational values in terms of RODRIGUES’ co- 
ordinates. These last are (see Satmon, “ Lessons on Higher Algebra,” art. 44) 
as follows :— 
14+—p?—v’, = 2(Aw +0), 2(Av — 2) 
2(Ap—v), 1+p?—)?—r?, 2(uv +d) 
2(\v + p), 2(pv—h), 1+r?—N—p? 
+ 1424+ p24+r 
' VOL. XXIX, PART II. 65 
