62 THE UNIVERSITY SCIENCE BULLETIN. 



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



(13) ]_ )8ll/?22-)8l2;821?^0. 



(14) .'=Hx), 



and find the most general nature which these transformations 

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

 these transformations ( -B ) is converted into 



'^ dr Y df^ Z dY 



^"('')'-^ +A2(r)^— +(/8ll>^" + 2;8'nl')-^ + 

 (A2r + 2y8'i2l') -— - + ()8"ii+^iiySu + 9i2/?2i) y + 



(15)^ 



{^"i2+qnP\2 + q\2(i22)Z= 0, 

 ^2 Y ^^ Z dY 



/321 (.^')' T^ + /322 (s^')-^ — ^ + (/?21 I" + 2 /3'21 r) -^ + 



d" a c^ d c 



dZ - - 



(/^22r + 2/?'22.^') — + (/5"21+ 921/311 + 922)821 )y + 

 (/?"22 + 921 /8l2 + 922 iS22) Z = . 



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



/3ijr + 2/3ijr = 0, (i,i = l,2), 

 that is, if 



(16) Ai = r^, (^,y = i,2), 



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

 (15) that system may be written in the form 

 rd'^Y 



^^^ \ d-Z 



^ + Q2iy + Q22Z = 0, _ 



if we put 



(17) DQik + 7^ ^^Bii[(^''/'-3^'?')&ik+^'&ik9ji], (^,^ = 1,2),- 

 U )" i=i '=1 



where ry = — and where Bji is the minor of \ in the determinant ' 



D = 611&22 — &12621. 



