645] 
a smith’s prize paper, 1877. 
43 
which when 
is small is = 
hence, however large a is, the chance of a deviation from equality not exceeding 
+ a, continually diminishes with n, and by making n sufficiently large becomes as 
small as we please. 
5. The point is represented in the given plane by two points which lie in 
lined with a fixed point (say 0) of that plane, viz. 0 is the intersection of the 
given plane by the line which joins the two projecting points. 
A line is represented on the given plane by two lines, viz. these are the 
projections of the line from the two given points; each point of the line is represented 
by the points of intersection of the two lines by any line through 0. 
A plane may be represented on the given plane by means of its trace thereon, 
and of the two points (in lined with 0) which represent any point of the plane. 
Thus any problem relating to points, lines, and planes, in space is converted into 
a problem of plane geometry. For instance, to find the trace on the given plane of 
a plane through three given points A, B, C, the three points are represented by 
means of the pairs of points A 1} A 2 ; B ly B 2 ; G ly C 2 , the points of each pair lying 
in lined with 0; the required trace passes through the intersections with the given 
plane of the lines d BG, GA, AB respectively, and we hence find it as the line 
through the three points which are the intersections of B 1 C ly B 2 C 2y of G 1 A 1 , C 2 A 2 , 
and of A 1 B 1 , A 2 B 2 respectively; that these points are in a line is a theorem of 
plane geometry, which, if not previously known, would have at once been given by 
the construction. 
6. The solution ought obviously to be obtained from the principle of virtual 
velocities; taking a, h, c for the lengths of the legs, /, g, li for the lengths of the 
strings, and z for the height of the summit, z is a known function of a, b, c, f g, h, 
where V, the volume of the tetrahedron, is a given function of 
a, b, c, f, g, h; and A, the area of the base, is a given function of f g, h 
Writing then F, G, H for the tensions, and W for the weight, and regarding 
z, f, g, h as variable, the principle gives 
Wdz + Fdf+ Gdg + Hdh = 0, 
that is, 
respectively. 
7. The ordinary case is when an isoparametric line has on one side of it 
larger values, on the other side of it smaller values of the parameter; the case 
where the isoparametric line is a line of maximum, or of minimum, parameter is 
excluded. 
6—2