193 BOOK OF MATHEMATICAL PROBLEMS. 



938. The limiting values of 

 d* x \" +1 '<*" 



are respectively 0, 0, 1, if n be odd ; and 



{1.3. 6. ..(-!)}, (-l) f Lz?,andl, 

 if n be even. 



939. If/(a) = and <(a) = 0, and if the limit, as a? ap- 



proaches a, of -~ be finite; the limit of {/(#)} <W will be 1. 



940. If /(a) = 1, < (a) = oo , the limiting value of {/(ar)}* W 

 as x approaches a, is e 7 ^^ ^ being the limit of (x- a) <fr(x). 



941. If 



z = 



then will 



d__d__ -A (d__d__ \ ( d _ _^__j\ z = $ 

 dx~dy )\dx dy )'"\dx dy J 



942. If 



then will 



(p + q-V)(p + q-2)...(p + q-n)z = 0, 

 where tt *a denotes 



943. If x, y be co-ordinates of a point referred to axes in- 

 clined at an angle o>, and u any function of the position of 

 the point ; . 



1 jcFu + ^ _ 2 CQ8 w rf> ) 1 (d?u tfu _ fd*^ Y| 

 Bin'w (dx* dy" dxdyy sin* to \dx* dy* \dxdyjy 



will be independent of the particular axes. 



