1. SHEW how to obtain a first integral of the differential equation

Rr+Ss+ Tt+U (s2 — rt) = V,

when it has a first integral of the form F(u, v) = 0, where R, S, T, U, V, u, v, are functions of x, y, z, p, q.

Obtain the complete integral of the equation

z (1+q3) r − 2pqzs + z (1+p2) t − z2 (s2 − rt) + 1 + p2 + q2 = 0.

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(Boole, Differential Equations, Supplementary Volume, pp. 125 to 141).

Comparing the proposed equation with the standard form, equation (21), p. 133 becomes

2.2

m2 - 2mpqz + p2 q'z2 = 0,
2mpqz+p°q° 2

the roots of which are each equal to pqz.

In this case it is possible to find three integrals of the system of differential equations of p. 139, which become

and the last two, by reason of (1), reduce to

dx+zdp + pdz = 0 .....

dy + zdq + qdz=0....

The integrals of (2) and (3) are

x+pz=a, y+qz=b,

and substituting for dx and dy from (2) and (3) in (1)

(1+p2+q2) dz + (pdp+qdq) dz = 0,

the integral of which is

z" (1+p*+q")=c*.

Eliminating p and q from the three integrals, we have

(x − a)2 + (y − b)2+z2 = c2,

which represents a sphere.

Now

a=$(c), b=¥ (c), F(a, b)=0,

are all first integrals of the given equation, and the complete integral is found by eliminating c between the equations

and

{x − $ (c)}* + {y − ¥ (c)}2 + z3 — c2 = 0=f(x, y, z, c)

The complete integral is therefore the equation of a tubular surface, the central line of which is in the plane of xy.

A first integral of the equation is

F(x+pz, y+qz) = 0.

2. State the criterion for the selection of the combination weights of n independent measures of, a magnitude. Determine the probable error of the result in terms of the probable errors of the n measures.

In the observation of the zenith distances of stars for the amplitude of a meridian divided into four sections by three stations intermediate between the extreme stations, a stars are observed at the first, second, third stations only; b stars at the second, third, fourth only; c stars at the third, fourth,

U

fifth only; and the probable error of every observation of a star is e. Shew that there are only three independent modes of measuring the whole arc, and obtain equations for determining the combination weights of the three measures. In the case when a=b=c, prove that the square of the 10e2 probable error of the result is

За

(Airy, Errors of Observations, § 64-70, and for the rider 80-82).

The mean actual errors of possible measures of the whole

(4, – 4,)+ (B - B)+(C, - C)...........(3),

which represent three independent, though entangled measures of the arc.

Any other measure can be expressed in terms of (1), (2), and (3), for instance

(4, − A ̧) + (C, − C2) = (1) — (2) + (3).

(x + y + z) A ̧ − (x + y) A2-zA ̧+(x+y) B-(x-z) B.- (y + z) B ̧ + xСz + (y + z) C−(x + y + z) %

3

x + y + z

x + y + z

3

2

1

1

{(x + y + z)2 + (x + y)2 + z2 } } + {(x + y)* + (x − 2)2 + (y + z)* } } + {x2 + (y + z)2 + (x + y + z)°}

a