Permanent Forms of Mathematical Expressions. 129 



Qi, etc., as the operator upon the elements of the strain. 

 We then find, from the permanency of form, 



+ 4K "rfE; + 2 < K " - K ^ + (2K " - K " + Ki,) SK a 



" 4K »5I7 S - (2K " - Km + Kaa) <7K^ 



+ 2(K S4 - K 24 )^g- + (K s5 - K 25 - K 46 )^- + (K 36 - K 26 + K^)^-- 



d_ m „ v d 



^65 



~ 2K66 dK7 5 + ( K55_K66 ^K, 



+ ^ 6 VK 66 



From the equation Ai = o and the two similar equations 

 we obtain by solution the invariantal connections between 

 the elastic constants, and by adding to Ai the terms 



d d 



z~i — v~r , etc., 

 dy b cfe' ' 



we obtain the covariant surfaces which indicate elastic 

 qualities at the point. 



Among the most remarkable of these which have already 

 been found is the ellipsoid found by Haughton in 1846, 

 and called by Rankine orthotatic. Its equation is 



(K a + K 12 + K l3 )r> + (K 12 + K 22 + K 23 )/ + (K M + K 8 + K 33 > 2 



+ 2(K U + K 24 + K 3i )yz 4- 2(K 16 + K 25 + K 35 )sx + 2(K 16 + K 26 + K 36 )xy 



and it implies three invariantal relations. Its properties are 



