hasia, bat it is well to /select one ufiirh has the Jiiost powers thit do not i-anish." Tims in 

 bp, we talce /. ^- ( > A. / / /, — /, S. ( ) A. /./,/,,, vviui.sf square is— /. 8.( )A. l^l.J,,, 

 tlie cube vanishing. Tiiis algebra is then of secdiui oidei. If A, B are any two 

 expressions of it, 



A- B + A B' + .{ B A r I', A B ^ o. 

 These examples are sutiicient to sliow liie use of tliese forms in interpretin"- the 

 subject. It remains only to sliow liow iIkv may lie applied in a few cases. There 

 are of course for every one of tliem two ticlvls of applieatiun at once suggested by 

 this method of writing them, viz. : linear transformations and homogeneous strains. 

 E.g., the nilpotent algebra d^. The general expression of this algebra is 

 V.= X- (i^. aii{)+a^.(y\ ya +z\ a ,3) ( ). 

 This transforms P = Xj « + 2/i /^ ~F ^j ; into 

 (p p =z xzi ,i + a ( yy^ + 2z,) 

 = 2/yi « + 2i (sa + X t'i). 

 This may represent any point of the plane («, /3). Since the value of x^ does 

 not enter (p p, every straight line parallel to a is made to correspond to a config- 

 .uration of the (a, /i) plane. Those lines parallel to n which cut the ( /3, y) plane 

 in a line parallel to fi, correspond to a series of configurations of the («, ft) plane 

 produced by slipping it along the direction a. The movement of a line which is 

 I)arallel to a along a line parallel to the line >, produces a series of expansions of 

 the (rt, fi) plane from a point w?/i« as center. If both y^ and 2, vary, subject 

 :to a law, we have the configuration of the (", P) plane 



<P p = yvi « + / (:¥i ) (2 « + ^ /?)• 



Again, consider the algebra a^. The general expression here, is 



= x(«S. /37( ) + /?S. 7a( ) + yS. « /? ( ) ) + 2/ (« S. >- a ( ) + /^S. a/? ( ) ) 



-\r za^.a p {), 

 = a S. (x V /? y + 2/ V }• a + 2 V fl /?) ( ) + /? S. (x V / a + 2/ V a /3) ( ) 

 + y^.x\ ap{) 

 p becomes <p p = n (xx^-]- yyi + zz^) + ft (x2/i+ yzi) -f X2i y. 

 This strain operator will convert p into any other vector a, for if 

 <J = ia-\rVft + Zy 

 we have at once from 



(p P = (7, 



35X1+ ?///,+ 22 1 =^", 

 2-2/l + ?/2l ='/, 



xz,=C. 



