320 EDWARD SANG ON CASES OF 



and that their intensities must be proportional to the sines of the opposite 

 angles, — this result being independent of the positions of the auxiliary points 

 A and C. 



Does it thence follow that we may omit those points altogether ? Assuredly 

 not ; for our whole investigation proceeded on the ground that each member 

 transmits from its one end to its other end the strain with which it is 

 accredited. 



Each additional point needs for its establishment three new linear members, 

 so that in any self-rigid open structure, if n be the number of the points, there 

 must be 3n — 6 linear connections; this formula failing only in the extreme 

 case, M = 2. 



Hitherto we have been considering the self-rigidity of structures, and may 

 now proceed to treat of the laws of stability in relation to the ground, taking 

 first the case of a self-rigid structure to be kept firmly in position. 



In every case the support must be derived from points in the ground, which 

 points necessarily form by themselves a rigid structure, so that our problem 

 assumes the general form of " how to connect one rigid structure with 

 another." If/ be the number of the points in the foundation, and n that of 

 those in the supported structure, we have in all /+ n points in the compound, 

 which must clearly be self-rigid. Hence the total number of linear members 

 must essentially be 3f+3n — 6. But of these 3/— 6 are virtually included in 

 the foundation, wherefore the number of the members above ground must in 

 all possible cases be 3w. Of these, however, 3w — 6 are already included in the 

 supported structure, and thus we arrive at the important general law, " that 

 the number of linear supports must be neither more nor less than six when the 

 supported structure is self-rigid." This most elementary of the laws of sup- 

 port seems almost to be unknown, the enunciation of it takes even professional 

 engineers by surprise. 



All our portable direction-markers, our theodolites, alt-azimuths, levelling 

 telescopes, have to be supported above the ground at a height convenient for 

 the eye. It is essential that the stationary part of each be firmly held ; yet in 

 every case, with not one exception in the thousand, our geodetical instruments 

 are set upon three slender legs, diverging almost from a point. In such an 

 arrangement the steadiness in direction is derived exclusively from the stiffness 

 of the legs, which, however, are very flexible. The well-known result is, that 

 any strain in handling the instrument, even the pressure of a slight breeze, 

 deranges the reading. 



More than fifty years ago an instrument maker in London, Robinson by 

 inline, placed his beautifully made little alt-azimuths on a new kind of stand. 



