DR WALLACE ON A FUNCTIONAL EQUATION. (541 



From this last we form the following table of formulae : 



* tv 



/(3^)=^{4j/3_3a^y}, 



tv 

 (I 



/(6^) = ^{ 32^-48 a2y + 18a\^^-«6}, 



/(7a:) = ^{ 64^-112 aV + 56aV-7«s.?/}; 

 and in general,* 



By this formula, supposing any number of ordinates to stand at equal dis- 

 tances along the axis x ; and the parameter a, also p the first ordinate, to be 

 given ; then all the remaining ordinates, to the last, may be found. 



28. It has been found (Article 12), that ,r=CQ, and ?/=PQ, being co-ordinates 

 at any point P of the curve, and a=BC, the least ordinate, then 



dx c 



Now PK being a straight line that touches the curve at P, and meets the axis 

 CE in K ; and (p denoting the angle PKQ ; in all curves 



-— ^ = tan 9 ; 

 ax 



therefore, putting t to denote tan (p, 



c 

 Hence again f —c^fl= a^, and y dy=<? tdt. 



Now, tdx = dy, and y tdx=ydy = (? tdt; therefore, c^dt=yix. 



We have now cdt=y — , dy=tdx=ct — . 



c c 



And again, from these equations, 



d y ->r c d t~{ij -\- c t) — \, 



dx 



dy — cdt= {y — c() ; 



c 



* For the mode of deduction, see the paper just quoted. 

 VOL. XIV. PART II. 6 D 



