1898.] of a certain determinant, etc. 9 



Now R n contains no quantity r with a suffix n, and iR n con- 

 tains no r with either of the suffixes i or n ; .'. differentiating we 

 obtain 



•Vgf " * W*0 £ W«0 = J A (ug - *g)/*M. 



= i-ttji (-tt • i-ttjn i -"i ■ ■*%»//-"* -"»* 



where ^R^, denotes the second minor in i£, obtained by erasing the 

 t'th andjth rows and the ith and ?ith columns, and affixing a sign 

 according to the ordinary rule ; but 



■"in > -"i 

 -"j w > -"ij 



— it . i-tijn, 



• • it • t-"jjj "r -itj • -itjj], — "m . -Llij- 



dr- 



i w in t-> -p D /D2D 



i>i o „ — - f * / in • i-"n • -"#/ -"i -"»i' 



And, in particular, if i =j, 



, 3 (r'in) 



-"mi . i-ti n / -tli-tl n . 



m d (r in ) 

 Hence the ith row in the Jacobian J 2 ' has the constituents 



— RijRin . iR n /r' in Ri 2 R n . (j = 1, 2, . . . n — 1 ) 



Taking out the common factors and reducing we get 



91-1 



j 9 ' = (- if- 1 [ n (A/Mr)] R n -i^ rt+ (i3), 



1 



and, interchanging dashed and undashed letters, 



J 2 = (- l)--i [ ILQR'JR'r)] R' n - iin - 1} R n -> • • • (14). 



«-l 



n 



i 



But by some easy determinantal reductions we get 



iR' n = (1 - r\ n ) RiRJtr+lTlBj, 



i 



and R'i = RiR n - 2 /URj. 



i 



Substituting in (14) and reducing we obtain 



n-l n 



J 2 = (- l)n-i [ n (1 - r 2 iri )] . Br*-* RJ» [UR$ ... (1 5). 

 i i 



