STOUFFER: INVARIANTS AND COVARIANTS. 



69 



whence it follows that 



(40) 



Q'ik = 



{i'y 



(Q'.k-2>;Q,k), 



Q"ik = jjyy (Q"ik - 5 ^ Q',k + 5 V^ Qik) . 



{i,k = 1,2), 



These results show by direct substitution and by differentiation 

 that Ji , Ki, Li, and their derivatives for the transformed equa- 

 tions have the values 



r — 1 — 1 J, . J 



J\ = (^'^4 Ji,J'i= i^^'\b {J 1 — 4';Ji), 



(4i: 



J"i = 

 Ki = 





(J"l- 9r, J'l + lSr; Ji), 

 iKi-2r,J'l + ivJi), 



K'x= -±^{K\- 6>;Ki-2'; J"i + 15r;Vi-20r/Vi), 

 Ii = ^-^(Li-5r,K'i + 5r;j"i + 15v^Xi-25r,Vi 



+ 25'iVi). 



If the transformation (20) is made infinitesimal by putting 

 ^ = X -\- <f {x) ot 

 where <f{x) is an arbitrary function of x and 'jf is an infinitesi- 

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

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



f ,)Ji = -4:C'Jl0t, 



8J\ = {-5c'J'i - 4.<f"Ji)'U, 



,W\ = (- 6cr'J"i- 9c"J'l)ot, 



oKi = {-6c'Ki -2c"J'i)'>t, 

 dK'i = (-Tcr'K'i- ec"Ki- 2 c" J'\)<n, 

 ^'>Li= i-S<s'Li- 5 c" K'i)ot. 

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

 to be 



' ^4 = Ji, ^4.1 = 9(J'i)-^ -8.JiJ"i, 



^10= (J'l)'- 4JiKi,^i5 = 5 ^10 J'l - 2^'ioJi, 

 I fc^i8= j (J'i)--4JiKi f L + Ki(J"i-2Xi)2 + Ji(K'i)2 

 1^ - J'iK'i(J"i-2 Ki). 



(42)^ 



(43)^ 



