478 
A “ SMITHS PRIZE ” PAPER ; SOLUTIONS. 
[534 
and these are at once seen to give 
a : /3 : 7 : 8 = 234.341 : -341.412 : 412.123 : - 123.341, 
so that the required identical relation is 
012.234.341 - 023.341.412 + 034.412.123 - 041.123.341 = 0, 
in which 012, 023, 034, 041 stand for the triangles PAB, PBG, PCD, PDA, and 234, 
341, 412, 123 for the triangles BCD, CDA, DAB, ABC respectively. 
4. Find at any point of a plane curve the angle between the normal and the line 
drawn from the point to the centre of the chord parallel and indefinitely near to the 
tangent at the point. 
Examine whether a like question applies to a point on a surface and the indicatrix 
section at such point. 
Taking the origin at the point on the curve, the axis of x coinciding with the 
tangent and that of y with the normal; the equation of the curve taken to terms 
of the third order in x will by 
y = bx- + ca?, 
and if, considering x as a small quantity of the first order, and therefore y as a small 
quantity of the second order, we regard y as given, and find the two values x ly x 2 , 
each of the order f{y), which satisfy the equation, then, as will appear, x 1 + x. 2 is a 
small quantity of the order x 1 , and consequently Xl + X - w p[ have a finite value. And 
* 1 • 1 ( I /JQ \ 
if (f) be the required angle, then obviously tan <£ = AUt 1L 
V 
We have as a first approximation bx 2 — y, or say x = ^~r , 
whence to a second 
say 
and thence 
whence 
tan 
which gives the value of the angle ; it would be 
the differential coefficients 
d x y, d x -y, d x y. 
easy to express b, c in terms of