DR WALLACE ON A FUNCTIONAL EQUATION. f|41 



From this last we form the foUowing table of formulae : 



f(ox) = a, 



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



/(6a7)=-i-{32/-48rt2y + 18a\/-fl6}, 



/(7a;) = -i-{64y-112aV + 56aV-7o^y}; 

 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 y 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 ^=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 



dy 



— — = tan ffl ; 

 dx ^ 



therefore, putting t to denote tan (j), 



c 

 Hence again /-c^ f=a^, and y dy—c'tdt. 

 Now, tdx=di/, and y tdz=ydy=(^ Id r, therefore, c^dt=yix. 



We have now cdt=y — , dy=tdx=ct — . 



c c 



And again, from these equations, 



d f 

 dy->rcdt~{y -^-ct)-^; 

 c 



dy-cdt= -{y-ct); 



c 



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

 VOL. XIV. PART II. 6 D 



