Doctrine of Extraneous Force. 123 



in the plane (xy), which may be regarded in two ways as a 

 simple shear of which the magnitude is Sot* (this being twice 

 the elongation in one of the normal axes). 



13. Going back to (10), (11), and (12), let a be so small 

 that o- 3 and higher powers can be neglected. To this degree 

 of approximation we neglect abc in (10), and see that its 

 three roots are respectively 



b' 2 c 2 (? a 2 a 2 b 2 



s4 ~^a~^^s4 ; ®~~M=M~~G=M % G ~~®^G~£4=G (20) ' 



provided none of the differences constituting the denominators 

 is infinitely small. The case of any of these differences 

 infinitely small, or zero, does not, as we shall see in the con- 

 clusion, require special treatment, though special treatment 

 would be needed to interpret for any such case each step of 

 the process. 



14. Substituting now for S4, t$, G, a, b, c in (20), their 

 values by (11) and (12), neglecting a 3 and higher powers, 

 and denoting by 8a., 83, 8y the excesses of the thz'ee roots 

 above «, /3, <y respectively, we find 



Sa = a | 2<rl\ + a 2 [\ 2 - -1- (n\ + lv) 2 - ^— (I ft + rnXf 1 | 

 8/3 = /3 [ 2amfi + a 2 [>- -^ (Ifi + mXf L ( m „ + nfl ) 2J j I 



Sy = 7 | 2<jnv + o- 2 \f~fi~ -(mv + nfif- -~( n \ + lv) 2 J J. J 



and using these in (19) , we find 



8E = a(Al\ + Bm/JL + Gnv) ^ 



+ \<? {A\ 2 + Bfi 2 + (V + L(w + n/j,) 2 + M(n\ + Ivf + N(fc + m\f\ t (22). 

 + 2a\ai 2 \ 2 + llm 2 fi 2 + In 2 v 2 ) ) 



where 



L= B r c^ ; K= ^M K _Ag-B. 



p— 7 7 — a a— (5 v J 



15. Now from (5) and (6) we find 



(mv + nfi,y = l-l 2 ~\ 2 + 2Q 2 X 2 -77i 2 /x 2 -n 2 v 2 ) . (24), 

 which, with the symmetrical expressions, reduces (22) to 



* Thomson and Tait's ' Natural Philosophy,' § 175, or ' Elements/ 

 § 154. 



(21 



