STRAINED ELASTIC SOLIDS 485 



G = 



a6 di as 



= aia2a3 + 2a4a5a6 — aiQi"^ — a^aC" — aza^. 



(See equations [431], [432] [433], [435].) 



This equation in p has three roots pi, p2, ps, functions of course 

 of E, F and G; if one of these roots is substituted for p in any 

 two of the equations [429a] above we can solve for the ratios 

 «Vt', fi'/y' and thus, using the condition a'^ + ^'^ + y"^ = 1, 

 determine a , /3', 7' for one of the axes; the remaining two 

 values p2, P3 determine similarly the other two axes. 



It remains to interpret the physical meanings of pi, p2, ps, 

 and that offers no difficulty. We saw above that if r is the ratio 

 of elongation parallel to any direction a, /S', 7' then 



^2 = a^a'^ _|_ a2/3'2 + 037'^ + 2a4i8'7' + 2a57'a' + 2a,a'^' 

 = {a,a' + ae/S' + a57')«' + («6a' + ag/S' + aa')^' 

 + (asa + a4i8' + a37')7'. 



If now a, jS', 7' is the direction of the first principal axis, then, 

 since aia + ae/S' + 057' = pia', etc., it follows that 



= Pi- 



Similarly p2 = r^"^, pz = ri^. The remaining steps now follow 

 easily. By the well-known relations between the roots and 

 coefficients of an equation of integral order in one unknown we 

 have 



Pi + P2 + P3 = -E", 



P2P3 + psPi + P1P2 = F, 

 P1P2P3 = Gf 



and these are just equations [439], which we obtained in this 



