130 Mr. R. F. Gwyther on 



easily seen. The remarkable point is that it only contains 

 15 of the 21 constants. 



A quartic surface was also given by Haughton, and 

 called by Rankine tasinomic. Its equation is 



K u a* + K 222 / 4 + K S3 z 4 

 + 2(K 23 + 2K u )yV + 2(K 13 + 2K 65 >V + 2(K 12 + 2K 66 ) x*f 

 + 4(K U + 2K 56 » + 4(K 25 + 2K i6 )xifz + 4(K 36 + 2K 4B )^2 

 + 4K 24 j/ 3 2 + 4K 34 3 8 2/ + 4K 3B 3 3 a; + 4Kj B ,x 3 z + iKi 6 x 3 y + 4K 26 ^ 3 a: = 1, 



which contains fifteen terms, and introduces all the 

 constants. Both surfaces have been considered by 

 Haughton, Rankine, and Saint-Venant. 



Of a character remarkable in the same way as is the 

 orthotatic ellipsoid, is the equation of the ellipsoid 



(K 23 - K 4 > 2 + (K l3 - K BB )2/ 2 + (K 12 - K 66 > 2 



+ 2(K B6 - K u )yz + 2(K d6 - K, 5 )zx + 2(K 45 - K 3e )xy = 1. 



Rankine gives this ellipsoid the name heterotatic. 



If, also, we write 



P' for x 2 , Q' for y 2 , B/ for z 2 , S' for yz, T' for xz, U' for xy, 



we notice that the operator £2 X acting on a function of x,y, z 

 becomes 



2S'( ; 



d d \ ,^. _, x d „, d d 



K dU' ~ dR'J + < R ' - q) ds' - u '^f' + T dV' 



and hence we conclude, comparing with (8), that 



(K u + K 12 + K 13 )P + (K 12 + K 22 + K 23 )Q + (K l3 + K 23 + K 33 )R 



+ 2(K 14 + K 24 + K 34 )S + 2(K 15 + K 26 + K 3B )T + 2(K 16 + K 26 + K 36 )U, 



and 



(K 23 - K 44 )P + (K 13 - K B5 )Q + (K 12 - K 66 )R 



+ 2(K B6 - Km)S + 2(K 46 - K 2B )T + 2(K 4B - K 36 )U, 



are also expressions enjoying a permanence of form. 



The question of the number of the elastic constants 



