AND DIAGRAMS OF FORCES. 207 



In diagram IV a. : 



F=r 4 co*40, G= \r* cos 20, 11 = ^ sin 20. 



In diagram IV b. : 



/=fp* cos | <f>, i r=|p l cos|(#, ) & = ip ! sinf<. 



Diagrams V a. and V 6. illustrate Airy's theory of stress in beams. 



V a. is the beam supported at G and D by means of bent pieces clamped to the ends of the beam 

 at A and B, at such a distance from G and D, that the part of the beam between G and D is free from 

 the local effects of the pressures of the clamps at A. and B. The beam is divided into six strata by 

 horizontal dotted lines, marked 1, 2, 3, 4, 5, 6, and into sixteen vertical slices by vertical lines marked 

 a, b, c, &c. 



The corresponding lines in the diagram V b. are marked with corresponding figures and letters. The 

 stress across any line joining any two points in V a. is represented in magnitude by the line in V 6., joining 

 corresponding points, and is perpendicular to it in direction. 



These illustrations of the application of the graphic method to cases of continuous stress, are 

 intended rather to show the mathematical meaning of the method, than as practical aids to the engineer. 

 In calculating the stresses in frames, the graphic method is really useful, and is less liable to accidental 

 errors than the method of trigonometrical calculation. In cases of continuous stress, however, the 

 symbolical method of calculation is still the best, although, as I have endeavoured to show in this 

 paper, analytical methods may be explained, illustrated, and extended by considerations derived from 

 the graphic method. 



