Ml 



RECIPROCAL FIGURES, FRAMES, 



of the force acting on its substance is in the ratio of o- to (1 <r)' in the 

 two canon, the internal forces will be the same in every part, and will be 

 independent of the actual values of the coefficients of elasticity, provided the 

 strains are small. The solutions of the cases treated by Mr Airy, as given in 

 his paper, do not exactly fulfil the conditions deduced from the theory of 

 elasticity. In fact, the consideration of elastic strain is not explicitly introduced 

 into the investigation. Nevertheless, his results are statically possible, and 

 exceedingly near to the truth in the cases of ordinary beams. 



As an illustration of the theory of Airy's Function, let us take the case of 



J ?T = 7* cos 2p6 (22). 



In this case we have for the co-ordinates of the point in the diagram corre- 

 ponding to (sy) 



= c =r 9 - 1 co8(2p-l)0, vj = -r*- 1 sin(2p-l)0 (23), 



and for the components of stress 



< fF__ . ^ ( 2 4 



If' we make G--r f coap0, and H=-r p smp0 (25), 



i 1 * fl L 7V~f ** 



then 



"XX 



(26). 



Hence the curves for which G and H respectively are constant will be lines of 

 principal stress, and the stress at any point will be inversely as the square of 

 the distance between the consecutive curves G or //. 



If we make 

 then we must have 



=/3 C os and i; = psin <f> } 



and ^-(2p-l)0J .................... 



= *~ l 



If we put q for -^ then i + 1 = 2 and (2p - 1) (2q - 1) = 1, 



