MISCELLANEOUS EXAMPLES ON CHAPTER VII 
293 
and in virtue of the vanishing of the determinant /3y' - fly, yd — ay, aft — df3 
all vanish; from which it follows that the ratios a : ¡3 : y are constant. But 
aA + ¡3B + yC=0.] 
27. Let z he a function of x and y, and z x , z v its two first derivatives. 
Now let x be expressed as a function of y and z, and let x y , x z denote its first 
derivatives. Prove that 
Xy= — Zy/z x , x t — 1 ¡z x . 
[We have the approximate equations 
8z=z x 8x + Zy8y, 8x=x v 8y + x z 8z. 
The result of substituting for 8x in the first is 
8z = {z x x v +z y ) 8y+z x x z 8z. 
Since 8y and 8z are independent this can be true only if z x x y + z y =0, z x x z =l.\ 
28. Four variables x, y, z, u are connected by two relations. Show that 
y z u z x u x v u = -y z x z x y x v z = 1, x z u z x v+y z u z v x = 1, 
where y z w denotes the derivative of y, when expressed as a function of ,s and 
u, with respect to z. {Math. Trip. 1897.) 
29. Prove the formulae 
d_ 
dx 
jJ(t)dt=f{x), ^ fit)dt=f{x) ft{x). 
30. Find A, B, C, X so that the first four derivatives of 
f f{t)dt - x [Af{a) + Bf(a + Xx) + Cf{a +.r)], 
J a 
vanish for x=0\ and A, B, C, D, X, y so that the first six derivatives of 
j f{t) dt - x[Af{a)+B/{a + Xx)+Of {a+fix) + Df{a+x)~\ 
J a 
vanish for x = 0. 
31. If a > 0, ac - b 2 > 0, and x x > x 0 , then 
f •» dx _ = 1 arc tan i (xi-x<>) \i{ac-b 2 ) ] 
J x 0 ax 2 +2bx+o f{ac- h 2 ) \ax x x 0 +b (x x +x 0 ) + of ’ 
the inverse tangent lying between 0 and tt*. 
32. Evaluate the integral / -— a d f_— For what values of a is 
J _! 1 — 2XGOSa + X 2 
the integral a discontinuous function of a? {Math. Trip, 1904.) 
[The value of the integral is \ir if 2mr <a<{2n + \)ir, and — if 
(2n - 1) rr < a < 2mr, n being any integer; and 0 if a is a multiple of ?r.] 
33. If ax 2 +2bx+c>0 for .r 0 < x ^ x\, f{x) = J {ax 2 + 2bx+c), and 
y=f{x), yo =/(•%), yi=/(^i), A = {x 1 -x 0 )l{i/ 1 +y 0 ), 
* In connection with Exs. 31-83 see Mr Bromwich’s paper quoted on p. 236.