Theorem of Ph. Gilbert. 
141 
Looking first to the right-hand member, we see that the differentiation 
there indicated with respect to t, gives 
and that the Jacobian following this having c>R/()^ for the element in the 
place 4,4 may be partitioned into two parts, one consisting of those 
terms of the Jacobian which involve the element c)R/c)^, and the other 
being the aggregate of the terms which are independent of ()R/()i. The 
former, however, is equal to 
and so of the four expressions thus obtained, the first and third cancel 
each other, leaving the right-hand member transformed into 
^y_j^ 
^y. 
^y. 
^y. 
hi 
1 
^y^ 
^2/3 
^R 
;)R 
(a) 
Turning next to the left-hand member, and usiog Jacobi's theorem 
regarding the differentiation of a determinant, we can express it as the 
sum of three determinants, the first differing from ~d{y^, y^, y^)~dl(tu^, w^, w^) 
in having the operational symbol '^j^x prefacing every element of the first 
column, and the second and third having the same change made on the 
second and third columns respectively. This sum we may indicate by 
A ^y._ 
^X 'dlUj 
^y._ 
^ys 
But now every one of the elements here which involve differentiation with 
respect to x is transformable as follows : — 
