191 



f4=fq + cg_=r, (6) 



where c is an arhitrary scalar, I find that the new formula of solution, 

 or of inversion, may be thus written : 



m = n,; (7) 



where F, = F cG + cm c\ (8) 



and nc = n n'c -\- n"c^ + n'" c^+ c*; (9) 



G and ^ being the symbols (or characteristics) of two new linear opera- 

 tions, and r^, rl' , n" denoting three new scalar constants. 



8. Expanding then the symbolical product fcFc, and comparing 

 powers of c, we arrive at three new symholical equations, namely, the fol- 

 lowing : 



fG + F^ n' ; G = ri' ; f^II= n'" ; (10) 



by elimination of the symbols, F, G, H, between which and the equa- 

 tion (5), the symholical hiquadratic, 



0 = n-n'fi- n"p - n!" p + f\ (a) 



is obtained. 



B. E. Stonet, B.A., read the following paper : — 



Ol^ THE StEEITGTH OE LojfG PiLLAES. 



Amon^g the numerous difficulties encountered in designing large iron 

 structures, such as railway girders or roofs of large span, none perhaps 

 is of more importance, or requires greater skill to overcome, than the 

 tendency of parts under compression to deflect beneath the pressure, 

 and yield sideways, like a thin walking-cane, when the load is greater 

 than it can support without bending. 



To understand the matter clearly, we must recollect that the mode 

 in which a pillar fails varies greatly, according as it is long or short 

 in proportion to the diameter. A very short pillar — a cube, for in- 

 stance — will bear a weight sufficient to splinter or crush it into powder ; 

 while a still shorter pillar— such as a penny, or other thin plate of 

 metal — will bear an enormous weight, far exceeding that which the cube 

 vsdll sustain, the interior of the thin plate being prevented from escaping 

 from beneath the pressure by the surrounding particles. "We can thus 

 conceive how stone or other materials in the centre of the globe withstand 

 pressures that would crush them into powder at the surface, merely be- 

 cause there is no room for the particles to escape from the surrounding 

 pressure. 



It has been found by experiment that the strength of short pillars 

 of any given material, all having the same diameter, does not vary much, 

 provided the length of the pillar is not less than one, and does not ex- 

 ceed four or five diameters ; and the weight which will just crush a 

 short pillar, one square inch in section, and whose length is not less 

 than one or greater than five inches, is called the crushing strength of 



