J. W. Gibbs — EqulUbr'mn). of ireteroi/eueonn Substauves. ;i67 



ILiving thus proved the existence of lines, with reference lo :iny 

 particular strain, which have the properties mentioned, let us pro- 

 ceed to find the relations between the ratios of elongation for these 



. . dx dz 



lines (the principal axes of strain) and the quantities ^, , . • • -rj 



under the most general supposition with respect to the position of 

 the co-ordinate axes. 



For any principal axis of strain we have 



when a da' + ft' dfi' + ;/' dy' z= 0, 



the dilFerential coefficients in the first of these equations being deter- 

 mined from (420) as before. Therefore, 



a' da' ~ ff d/3' ~ y' dy' ' ^ ' ^ 



From (420) we obtain directly 



2 da' ^ 2 d/3' ^ 2 dy' ' ^ ^ ' 



From the two last equations, in virtue of the necessary relation 

 «'2 -j- ft'^ -|- )/'2 :zz 1, we obtain 



4?="'-' 4?=^^'-' ^V-''^^ ''■''' 



or, if we substitute the values of the difterential coefficients taken 

 from (420), 



,-, /ffe\~ ,j,^./dxdx\ . , /dx dx\ , „ 



«■:£ 



If we eliminate a', ji' ^ y from these equations, we may write the 

 result in the form, 



[dx\^ g >Y— — \ ^[dx_d^\ 



W/ "'' \d^dy) [dx'dz'J 



(dx dx\ /dxy _ 2 V /^ ^ \ 



\%'^7 W/ *" \dy'dz') 



, /dx dx \ y /dx ^\ ^ /^ V _ 



[d^'d^'J \ch'dy') \dz' ) 



Trans. Conn. Acad., Vol. III. 



= 0. (430) 



May, 1877. 



