64 THE UNIVERSITY SCIENCE BULLETIN. 



(25) qn Un + 921 C/21 + ^n Un + 922 TJ22 = 0. 

 Since the system contains four variables there are just two so- 

 lutions. These are easily seen to be 



/ = qn-\- q22, J = qnq22 — q\2q2\. 

 Since the coefficients in (19) are constants the transformations 

 of the various derivatives of qw, will be of exactly the same form 

 as the transformations of q,^. The differential equations for the 

 functions involving g'ik as well as q\^ are simply (23) with terms 

 of the same form in q'w, added. The relations (25) ceases to hold 

 so that there are just three more solutions. These are evidently 



I',J',K=~q'nq'22-~q'i2q'2i. 

 In the system of equations for the functions involving also 

 ^'ik there are just three independent equations and four more va- 

 riables so that there are four more solutions. These are evi- 

 dently _ _ _ _ 



, J , K , Li = q nq 22 — q 12 q 21. 



A continuation of this process shows that all the desired func- 

 tions involving higher derivatives of gik can be obtained by form- 

 ing the successive derivatives of 7 , J , K, L. 



Let us now substitute in I , J , K, L and their derivatives the 

 expressions for gik, q'\k, q"\k given in (7), (9) and (11). A com- 

 parison of these equations with (21) and its derivatives shows 

 that ^ik is expressed in terms of Wik, q'ik in terms of Vi^, and Q^'^ik in 

 terms of W\], in exactly the same way that Qik is expressed in terms 

 of ^ik, Q'ik in terms q'±, and Q"ik in terms of q''^,, respectively, ex- 

 cept of course that "ik replaces 6ik. If now in /, J, K, L or in 

 their derivatives we replace q.^, q'ik, q"ik by Qik, Q'ik, Q"ik respec- 

 tively, we obtain the original functions of q'ik , q'\k , q"ik ■ It follows 

 therefore that if in I , J, K, L and their derivatives we replace 



q\k, q'\k, Q"\k by Wik, ^ik, w^ik, respectively, we obtain the result of 

 substituting (7), (9), (11) into these functions. In other words 



^ I =Un-\-U22, J ^UnU22 — U\2U2\, 



I' = xin-\-V22, J' ^UnV22-\-U22V\\ —U\2V2\ — U21 V\2, 



(26)-< I" =Wn-\-W22, J" = 2K-\-UnW22-\-U22Wn— U12W21 — U21W12, 



K = Vll?;22 — ?^12t^21, L = WnW22 — Wl2W2l, 



K'= VnW22 -{- V22Wn — ?;i2i02i — i'2iw;i2. 



The expressions (26) and their derivatives are all semin vari- 

 ants of the system (A) and moreover they form a complete 



