174 



THE ROYAL SOCIETY OF CANADA 



The fundamental magnitudes of the first and second orders have, 

 therefore, the following values: 



V dp/ \ di)/ \ da/ \ da/ 



L = 



dp/ \ dp/ \ dq 



V di? da/ 



dp dq 

 From what has been said above and the form of MV, it is clear 



^1 — ^ — I can not vanish, except for certain critical values of 

 dp dq/ 



p and q. 



By making use of a known theorem in the theory of determinants, 



viz.. 



A' +B^ +C2 AA'-{-BB'+CC'\^\AB\ 

 AA'-^BB'+CC A''- H-B'2 -\-C^ \a'B'\ 



we find 



F2=£G-F2 = 



V dp/ V dp/\ dq/ 



\ dp/\ dq/ \ dq/ 

 V ^ dp/ \ ^ dp/ 



To reduce this expression we make use of a theorem on minors, viz. 



a h c 



If D = 



, then 2 



B' c 

 B"C" 



= {a+h+c)D, 



a' b' c' 



a"b"c" 



where B' , B" , etc., are the minors of b' , b" , etc., in D. If we make use 

 of this result, the above expression for V^ becomes 



dp dq 



Gauss's measure of curvature K is given by the formula 

 ^LN-AP 1 



V dû da/ 



K 



=Q'= 



72 (^^2^.^22^^32)2 



The expressions for T, T', T", A, A', A" can now be found from (5); 

 they can also be calculated quite readily from their definitions by 

 making use of the above theorems in determinants. The values of 

 these svmbols are found to be: 



