g28 DK WALLACE ON A FUNCTIONAL EQUATION. 



10. We may consider this differential equation, and the functional equation 



as the representatives of each other, so that if x and y be co-ordinates of a curve, 

 the functional equation will express a property of that curve. Now, considering 

 2^ as a function of x, the function and its variable may either increase together, 

 or else p may decrease while x increases : therefore, it may be, that the function 

 which satisfies the equation (A) wiU have different forms ; for if y decrease while x 



d y 



increases, the differential coeflficient -~ wiU be negative ; if, however, y and x 



increase together, then it wiU be positive. 



11. Let us first consider the case in which y decreases while x increases. The 

 differential equation to be resolved may then be expressed thus, (putting & for a 

 constant), 



dx- y c^ 



or 



dy 



and, multiplying both sides by -j^ 



d (^\ =-il± ■ 



\dzj c" ' 



dy 



d!/ ^ (dy \ _ ydy 

 \dx ) (? " 



dx 



and taking the integrals, 



^dx) 



d V 

 Now, -~- expresses the tangent of the angle which a line touching the 



cui-ve makes with the axes to which ?/ is a perpendicular ordinate : and since 

 ■^ manifestly cannot exceed b, therefore y must have a maximum value, which 

 must satisfy the equation -^ = 0, and putting a for that maximum value of //, 

 «" rv ■, , a^ ,-, r. / d 1/ y d' — y- 



we have ^-^=0- and *=-^, therefore, ( 77 ) = c- '" 



and 



dz _ dy _ 



from this, by integration, we get 



cos (a-^) =^, or siu(a-l) = V(l-^) : 



We may assume that, when y is a maximum, and =a, then .«=0 ; therefore 



cosa=l, andsina=:0. 



