62 THE UNIVERSITY SCIENCE BULLETIN. 



Let us rewrite transformations ( 1 ) and ( 2 ) in the form 



(13) r fin ^22 - (3i2 fin ,^0. 

 \z^ (B21Y+ P22Z,. 



(14) c=|(x), 



and find the most general nature which these transformations 

 may have and still leave {B) in the semi-canonical form. By 

 these transformations ( S ) is converted into 



d- Y d^ Z d Y 



(/3i2r' + 2/3'i2l')-— +(y8"ii + gu/?ii + gi2i82i)y + 



d c 



(15)^ 



(/8"l2+ 911/812 + 912 /?22)Z= 0, 



/321 (^r^ + /?22 (r)^ ^ + (/32ir + 2 /?'2ir) ^ + 



(/?22c" + 2/?'22l')TT- +(/?"21+g21/8ll + 922^321)7 + 



(/8"22 + ^21 ^12 + 922 /?22 ) Z = . 



This system is in the form of system ( J5 ) if and only if 



^iir + 2Ai^^' = 0, {i,j = 1,2), 



that is, if 



h 



(16) A, = :^, (i,i = i,2), 



where 6ij are constants. If these values for /3ij are substituted into 

 (15) that system may be written in the form 



rd'Y 

 -7Zr + QnY + Qi2Z = 0, 



1 d^' 



(C) -<! . 



'^''^-+Q2iy + Q22Z = 0, 



V d ?' 



if we put 



( 17) D Qik + 7^ '- By. [ ( Hr - y2r/)k^+ i feik^ii ], (I, k = 1,2), 



where f/ = -j and where By. is the minor of by^ in the determinant 



D = 611 &22 — &12&21. 



