Wires of varying Elasticity. 85 



Thus 



^= F & + - + fe-¥]- • • • (24) 



The total extension of the cylinder is thus 



- F2 i::(i) w 



du 



In any given layer, ~ is = — ^ = cr, and depends only on 



dz 

 the elastic properties of the layer considered. 



No assumption has been made as to the thickness of the 

 layers or the law of variation in the elastic properties ; thus 

 the previous results will be true when we suppose the layers 

 to become indefinitely thin, and the difference between the 

 elastic constants of consecutive layers to be indefinitely 

 diminished. Proceeding to the limit, we can deduce the 

 solution for a cylinder whose constitution varies gradually 

 from one end to the other, so that the elastic constants are 

 continuous functions of z the distance from the fixed end. 

 Thus, in such a cylinder at a distance z from the fixed end, 



n=-~Fr, 



>* * (26) 



where <r and M are supposed to be known functions of z. In 



the most general case ^ might be expanded in a series of 



sines and cosines by Fourier's method, and the above integra- 

 tion effected. If the variation be such that, expanded in a 

 convergent series of ascending powers of z, 



i = "i (1+w+ * i * ,+ --'" ) (27) 



we get 



w 



-Srfl+J^*jV+,.-.} • • • <»» 



It is to be remarked that, if a now denote the radius of the 



cylinder, the increase in radius — -^ Fa is different for the 



different materials, or for different sections of the continuously 

 varying material. Since, however, a is a proper fraction, and 



