DR WALLACE ON A FUNCTIONAL EQUATION. 629 



Now, T"""(""""^)' 



X ( ^\ • ( ^\ 



and cos — = cos a cos I a J + sin a sm I a 1: 



X y 



that IS, cos— =— . 



' c a 



Hence it appears, that one form of the function y is 



X 



!/=f{^) = «cos — : 



here a is the value of. f{x) when x=0, and c is an arbitrary constant. 



12. The second case of the differential equation, in which x and ^/ increase 

 together, is this, 



— ^ -=-5- or -'' -^^ -^ 



<^) 



d^ y (? \ dx ) <? 



d u 



hence, as before, multiplying both sides by -/- , and integrating, we get 



m-^-- 



Let a be the value of ^ when it is the least possible ; this must satisfy the equation 



-^ = ; and we have 

 dx ' 



_-ft = 0, or b=-^ 



we may assume, as before, that a is the value of ?/ when ai=0: we have now 



(dyy_ y^-a' 

 \dx) c2 ' 



and 



dx __ dy 



To integrate this equation, let us assume that 

 then dy = — ll -\ du = -^ . —^—du\ 



2 





a u' — l 

 u 



and by substitution in the diflPerential equation, 



dy 1 u^ — \ 2 u , du 



^(y—a^) 2 u^ a w — X u 



therefore — = — , and — = log m + log 6. 



VOL. XIV. PART XL 6 A 



