64 THE UNIVERSITY SCIENCE BULLETIN. 



(25) qi2 Ui2 + 921 f/21 + qn Un + 922 U22 = 0. 



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

 lutions. These are easily seen to be 



/ ^ qn-\- q22, J = qnq22 — 912921. 

 Since the coefficients in (19) are constants the transformations 

 of the various derivatives of q,\, will be of exactly the same form 

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

 functions involving 9 'it as well as ^ik are simply (23) with terms 

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

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



r,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 _ _ 



I , J , K , Li = q nq 22 — q V2q 21. 



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

 tions involving higher derivatives of q^^, 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 q,],, g'ik, q"± given in (7), (9) and (11). A com- 

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

 that 9ik is expressed in terms of Wik, q'± in terms of v^^, and ^''ik in 

 terms of w^], in exactly the same way that Qik is expressed in terms 



of 9ik, Q'ik in terms g'ik, and Q"ik in terms of ^''ik, respectively, ex- 

 cept of course that '-(ik replaces 6ik. If now in 7, J, K, L or in 

 their derivatives we replace ^ik, g-'ik, 9".k by Qik, Q'^k, Q"± respec- 

 tively, we obtain the original functions of g-ik, (/'ik, q'\k- It follows 

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



q±, 9'ik, 9"ik by Wik, %, w^ik, respectively, we obtain the result of 

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



^ I =un+ U22, J = Un U22 — Uu U21 , 



r = Vn-{-V22, J' = Wll 2'22 + W22 «^ll — Wl2 «^21 — U2I V\2, 



(26)-<J 7" =w;ii + w;22, J" = 2K-\-u\\W22 + U22Wn— u\2W2\ — U2\Wn, 

 K — i'iif22 — vi2?^2i, L = i<;iim;22 — W12W21, 



K' = VnW22-\-V22Wu — ^^12^t;21 — ?^2lWl2. 



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

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



*^ 



