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# Extend the result to any number of variables 24 let f

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Extend the result to any number of variables. 24. Let f ( x ) be a function of x whose derivative is 1 /x and which vanishes when x = 1. Show that if u = f ( x ) + f ( y ), v = xy , then u x v y - u y v x = 0, and hence that u and v are connected by a functional relation. By putting y = 1, show that this relation must be f ( x ) + f ( y ) = f ( xy ). Prove in a similar manner that if the derivative of f ( x ) is 1 / (1 + x 2 ), and f (0) = 0, then f ( x ) must satisfy the equation f ( x ) + f ( y ) = f x + y 1 - xy .

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[VII : 164] ADDITIONAL THEOREMS IN THE CALCULUS 376 25. Prove that if f ( x ) = Z x 0 dt 1 - t 4 then f ( x ) + f ( y ) = f ( x p 1 - y 4 + y 1 - x 4 1 + x 2 y 2 ) . 26. Show that if a functional relation exists between u = f ( x )+ f ( y )+ f ( z ) , v = f ( y ) f ( z )+ f ( z ) f ( x )+ f ( x ) f ( y ) , w = f ( x ) f ( y ) f ( z ) , then f must be a constant. [The condition for a functional relation will be found to be f 0 ( x ) f 0 ( y ) f 0 ( z ) { f ( y ) - f ( z ) }{ f ( z ) - f ( x ) }{ f ( x ) - f ( y ) } = 0 . ] 27. If f ( y, z ), f ( z, x ), and f ( x, y ) are connected by a functional relation then f ( x, x ) is independent of x . ( Math. Trip. 1909.) 28. If u = 0, v = 0, w = 0 are the equations of three circles, rendered homogeneous as in Ex. 19, then the equation ( u, v, w ) ( x, y, z ) = 0 represents the circle which cuts them all orthogonally. ( Math. Trip. 1900.) 29. If A , B , C are three functions of x such that A A 0 A 00 B B 0 B 00 C C 0 C 00 vanishes identically, then we can find constants λ , μ , ν such that λA + μB + νC vanishes identically; and conversely. [The converse is almost obvious. To prove the direct theorem let α = BC 0 - B 0 C , . . . . Then α 0 = BC 00 - B 00 C , . . . , and it follows from the vanishing of the determinant that βγ 0 - β 0 γ = 0, . . . ; and so that the ratios α : β : γ are constant. But αA + βB + γC = 0.] 30. Suppose that three variables x , y , z are connected by a relation in virtue of which (i) z is a function of x and y , with derivatives z x , z y , and (ii) x is a function of y and z , with derivatives x y , x z . Prove that x y = - z y /z x , x z = 1 /z x .
[VII : 164] ADDITIONAL THEOREMS IN THE CALCULUS 377 [We have dz = z x dx + z y dy, dx = x y dy + x z dz. The result of substituting for dx in the first equation is dz = ( z x x y + z y ) dy + z x x z dz, which can be true only if z x x y + z y = 0, z x x z = 1.] 31. Four variables x , y , z , u are connected by two relations in virtue of which any two can be expressed as functions of the others. Show that y u z z u x x u y = - y x z z y x x z y = 1 , x u z z y x + y u z z x y = 1 , where y u z denotes the derivative of y , when expressed as a function of z and u , with respect to z . ( Math. Trip. 1897.) 32. Find A , B , C , λ so that the first four derivatives of Z a + x a f ( t ) dt - x [ Af ( a ) + Bf ( a + λx ) + Cf ( a + x )] vanish when x = 0; and A , B , C , D , λ , μ so that the first six derivatives of Z a + x a f ( t ) dt - x [ Af ( a ) + Bf ( a + λx ) + Cf ( a + μx ) + Df ( a + x )] vanish when x = 0. 33. If a > 0, ac - b 2 > 0, and x 1 > x 0 , then Z x 1 x 0 dx ax 2 + 2 bx + c = 1 ac - b 2 arc tan ( ( x 1 - x 0 ) ac - b 2 ax 1 x 0 + b ( x 1 + x 0 ) + c ) , the inverse tangent lying between 0 and π . * 34. Evaluate the integral Z 1 - 1 sin α dx 1 - 2 x cos α + x 2 . For what values of α is the integral a discontinuous function of α ? ( Math. Trip. 1904.) * In connection with Exs. 33–35, 38, and 40 see a paper by Dr Bromwich in vol. xxxv of the Messenger of Mathematics .

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[VII : 164] ADDITIONAL THEOREMS IN THE CALCULUS 378 [The value of the integral is 1 2 π if 2 nπ < α < (2 n + 1) π , and - 1 2
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