Permanent Forms of Mathematical Expressions. 1 3 1 



will not be settled by these conditions of form, but there 

 are points in which they may serve for a guide. For 

 example, Saint-Venant, though holding the rari-constant 

 opinion that 



K B6 = Ki 4 , K 46 = K 2 5, K 45 = K 36 and K 23 = K 44 , K l3 = K B5 , K 12 = K 66 , 



endeavours to make an investigation more general by 

 taking only 



K56 = Kj 4 , K 4 6 = K 2 5, K 4 s = K S 6 

 and assuming 



-^-23 - &13 -K-12 _ ■ 

 -IV 4 4 -K-55 -K-66 



But, if the first of these equations holds, the heterotatic 

 ellipsoid becomes a sphere, and 



K 23 - K 44 = K 13 - K 55 = K 12 - K G 6 = an invariant. 



This is, therefore, the proper assumption intermediate 

 between the rari-constant and multi-constant hypothesis.* 



If we write 



K 23 = K44 + X 23 , Kg6 = K14 -1- ^U56 , 



K-ls = Kss + Xl 3 , K 46 = K 2 s + ^Lt 4 6 , 



K 12 = K.66 + Xl 2 j K 4 5 = K 3 6 + /JLtf , 



the part of the operator A x which affects the X's and 

 fis only is 



o [ d d \ , . d d d 



and we see that in order that the p's may remain o, we 



must have the X's equal. If the X's are taken equal, they 

 will only remain so if the fis vanish. 



I have taken the illustrations chiefly from the subject of 



* I have not made myself clear on this point. Saint-Venant is treating of 

 " ellipsoidal symmetry." This will require that the orthotatic and heterotatic 

 ellipsoids should he co-axial or that the heterotatic ellipsoid should be a 

 sphere. As his conditions are not those requisite in the first case, I have 

 expressed those necessary for the latter. — Feb. 27, 1895. 



