Now, 

 and 



DR WALLACE ON A FUNCTIONAL EQUATION. 



T = "-("-7)' 



I a J + sin a sin I a I : 



629 



cos = COS a COS 



c 



X y 



cos — =-i- 

 c a 



that is, 



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



y=fi?) = acos — : 



here a is the value off(x) when a;=0, and c is an arbitrary constant. 



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

 together, is this, 



,(dy\ _ ydx 

 '^ \dl) ~ ,P ■ 



d^ ' y (? 



dy 



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



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



dy 



dx 



; and we have 



_-i = 0, or b = ^ 



we may assume, as before, that a is the value of ^ when a;=0: we have now 



{dy\'^_y--a^ 



J- dx dy 



and — =__£__ . 



To integrate this equation, let us assume that 



» = !-('*¥)■ 



y-.= -(..-2.J,)=^(„-i)% 

 and by substitution in the differential equation. 



therefore 



VOL. XIV. PART II 



1 M^-1 2 

 = ^«- 5- • — 



T-^du = — ; 



f—i u 



dx du I X . , , 



— = — , and — = Iog«+log6. 



6a 



