72 DEFINITE INTEGRALS. [INT. III. 



If fj, const, is a new integral, let us by introducing the value 

 of z just found, express JJL in terms of x, y, X, 



fi = fi(a} t y t X). 



We shall distinguish the partial derivatives of //, thus expressed 

 from its partial derivatives when expressed in #, y, z, by brackets, 

 so that we have 



ty = [VI |~ VI ? dp = ra/ti rev] ax a^ = r^i ax 

 dx ~ [_a^J [_ ax J ^ ' 9 2/ ~ L ? 2/ J L^J ty* Tz~ [axj a^ * 



Accordingly we obtain for the values of A, B, C 



_ 

 ~te\?y\' a^L^J' 



Now /x being expressed in terms of x, y, X, we have 



and since X = const, is an integral, d\ = 0, 



But from the values of -4 and B 







ax 5 dy ax' 



a^ 



, Bdx Ady 



so that cut = - ... y . 



CA, 



a? 



But since ^1 = J/Z, B = MY, 



this becomes c?/i, = r- ( FcZa; JT 



a* 



Accordingly although the expression 



is not a perfect differential, the factor 



M 



ax 



dz 



