514 Proceedings of the Royal Society of Edinburgh. [Sess. 
may write temporarily in the form 
n + 1 sets of equations like 
-fr 
c)oc 
R^-4 , he obtains by solving 
J I 
ox of + dx d/j ^ 
+ fa djn 
df dx df\ ox 
dfn dx 
dx df , dx of, 
— -z — + — 1 + . . 
+ faf¥n 
df dx 1 of l dx ] 
dfn Ctej 
dx df dx df\ 
+ dx df n 
¥ dX n ¥l fan 
dfn dx n 
and by using the same sets of equations after row-by-row multiplication he 
obtains 
y / + dxdxj foA > y / + df_ dj\ df^\ = , 
df n J " \~ dx dx 1 ’ dxj 
or, say, SR = 1 . 
Further, he notes that as a consequence of these two theorems there results 
~df[ 
dx. 
fdx k _ 
L0/J 
= fa*.?/* 
Of: dx { 
which he might well have generalised by changing the i , k of the second 
factor of both sides into r , s. 
In dealing with the “ fundamental lemma ” his order of procedure is the 
reverse of Jacobi’s, that is to say, he deduces the form of 1844 from the 
original form of 1841. Thus, using the latter in regard to S, he has 
whence, because of R being the reciprocal of S, he obtains 
so that on substituting 
there results 
which on further substituting 
for 
becomes the form desired. 
ojk 
