(il 



hasi^, hat it is well lo select one n-hirh hua the mod powers thai do not vanish." Tlii:s in 

 bp, we. take l.^ S. ( ) A. Ill, — /j S. ( ) \. IJ.l^, whose siiuare is—/. y.( )A. /,/,/., 

 tlu' cube vanishing. This ;ilgt'br;i is then of second order, if .4, 7/ are anv two 

 expression.s of it, 



A' B + AB'+ A 11 A y B A B ^ o. 

 These examples are sui^iicient to show tlie use of these forms in interjireting tlie 

 subject. It remains only to show iiow thtv may be applied in a few eiises. Tiiere 

 are of cour.se -for every one of them two fields of api)licalion at once suggested by 

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

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

 <p = x.l38.a(S()-^ a 8. ( y V y « + .- Y « /?) ( ). 

 This transforms /> = x, a -f y^ /3 -f z^ ; into 

 (p f) = xz^ /^ + " ( yyi + 22,) 

 = y.'/i « + 2i (2 « + a- ft)- 

 This maj' represent any point of the plane (ff, ft). Since the value of a'l does 

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

 nnition of the (a, ji) plane. Those lines parallel to a which cut the ( ft, y) plane 

 in a line parallel to /3, 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 

 parallel to a along a line parallel to the line y, produces a series of expansions of 

 the (fl, ft) plane from a point y y i^ as center. If both y, and 2, vary, subject 

 to a law, we have the configuration of tiie («, ft) plane 



/' = yy\ « + / (:vi) (2 « + ^ ft)- 



Again, consider the algebra a 3. Tlie general expression here, is 



,^ = x(aS. i3y( ) + /?S.7a( ) + )'S. «/M) ) +2/(«S.y «() + /? S. a /?()) 



+ 3 n 8. a /J ( ), 

 = a S. (X V /3 y + 2/ V )• a + 3 V a /?) ( ) + /? 8. (x V / a + .V V G /3) ( ) 

 + ; 8. a: V a /3 ( ) 

 p becomes (p p — n (j-Xi + y?/,'+ zz^) -\- ft (j-.Vi-f yzi) + xzi y. 

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

 G = ^a -f ,/ /? -|- g y 

 we have at once from 



(l> p = (7, 



xxi + yy,-\-zzi =i, 

 XZi =C. 



