STOUFFER: INVARIANTS AND COVARIANTS. 69 



whence it follows that 



(40) 



Q'ik = />/xa (Q'ik — 2 5/ Qik), 



These results show by direct substitution and by differentiation 

 that J\, K\, L\, and their derivatives for the transformed equa- 

 tions have the values 



r - 1 — 1 



J\ = (^')4 Ji, J'l = ( fcM-^ ( J'l — 4r/ Ji), 



(41)<; (t ) 



I-l = T^(Ll-5r,K'i + 5r;V"l + 15r,2Ki-25r/Vl 



+ 25r/Jl). 



If the transformation (20) is made infinitesimal by putting 



where <f{x) is an arbitrary function of x and i^t is an infinitesi- 

 mal, the infinitesimal transformations of J\, Ki, L\, and their de- 

 rivatives are found by direct substitution in ( 41 ) to be 



f oJi = -A0'Ji>Jt, 

 <Wi = (-Sc'J'i - 4f"Ji)'5^, 



,7j"l = (_ 6cr'J"l- 9f"J'l)ot, 



oKi = {-6<f'Ki-2f"J'i)'U, 

 '5K'i = i-lf'K'i - 6cf"Ki - 2v'" J"i) H, 

 L'^Li = (- 8v''Ii - 5<f"K'i)'>t. 

 The resulting system of partial differential equations whose 

 solutions are invariants of {D) under the transformation (20) 

 with /' = contains two independent equations. There are 

 therefore four such absolute .nvariants involving the variables Ji , 

 J'\, J" I, Ki, K'l, L\. The five relative invariants may be taken 

 to be 



^ ^4 = Jl, «4.1 = 9(J'l)2 -8JlJ"l, 



^10- (J'l)'' - iJlKl, W,5= SWioJ'i - 2W'ioJl, 



"wiH= 1 (J'i)'^-4JiKi iL + Ki(J"i - 2Ki)2 + Ji(K'i)''' 



-J'xK\{J'\-2 Kx). 



(42)^ 



(43) 



