But the first factor of the first member of this equation being the determinant of the system adx+bdy+cdz = 0, a'dx + b'dy + c'dz = 0, Adx + Bdy+Cdz = 0, expresses when equated to zero the condition that if in the system (1), (2), (3) dy' vanishes dx' shall also vanish; and de and dy being independent, this condition cannot be satisfied, so that (9) reduces to whence 1 1 (L2 + M2+ N2) § ̄ ̄ (L'2 + M'2+ N'2) § = 0, L'2 + M12 + N12 − L - M - N 0........(10), = and this, with (7), will fully express the conditions of similarity. 2. If we multiply (7) by 2 √1, and add and subtract the result from (10), we obtain the equivalent system (L' + L √ − 1)2 + (M' + M√— 1)2 + (N' + N√−1)2 = 0) (L' — L √ — 1)2 + (M' — M√— 1)2 + (N'— N√−1)2= ....(11). = dy B. D. E. II. dz dy dFd (x' ± y' √−1)__dF d (x' ± y' √−1) dF dy_dF dy) √1 15 to which we may give the somewhat more convenient form 2 dF 2 dF 2 (du)2 + (du)"} dy dz dF du dF du 2 dy dy + dz dz) = 0.....(I). 2 {(d)' + (dy')'" + (d')"} {(dc)' + (dc)" + (dv)"} dx dz dv 2 dx dy dz These are partial differential equations of the first order, serving to determine u and v as functions of x, y, z. But it is not necessary to solve the equations in their general form. For, x, y, and z being connected by the equation of the surface, the above equations may always be so reduced as to involve only two independent variables. As latitude and longitude determine the position of a point on the earth, so two co-ordinates of any given species will determine the position of a point on the given surface, and these co-ordinates, when fixed upon, become the independent variables of the problem. Let s and t represent such co-ordinates, and let their expressions in terms of x, y, z give 8 = P1 (x, y, z), t = 4, (x, y, z), 2 which equations combined with that of the given surface will reciprocally determine x, y, z as functions of s and t. Then 1st the differential coefficients of F which in the equations (I), (II), are functions of x, y, z may be transformed into functions of s and t; 2ndly, we have and as ds dt ... are known functions of x, y, z, they also are expressible in terms of s and t. The result of these substitutions will then be to convert (I) into a partial differential equation in which u is the dependent and s and t the independent variables, and this equation being, like (I), of the first order and second degree in the differential coefficients of u, will be of the form For v we shall have an exactly similar equation with the same coefficients. The above equation is, by the solution of a quadratic, resolvable into two equations of the form To these correspond the respective auxiliary equations dt+λds =0, dt +λds=0..... Now being determinable by an equation of the same form as u, it follows that of the above two values of u one must be assigned to v, so that the solution of the problem will be contained in the system u = $ (S), v = ↓ (T), or in the system u = $(T), v = ¥ (S). The particular forms of the arbitrary functions and will depend solely upon the nature of the problem under consideration. One other point remains to be noticed. The first members of (12) are essentially positive, being composed of squares; so are then the first members of (I), (II); and so, if the intermediate transformations are real, is the first member of the equation whose coefficients are P, Q, R. Hence the quadratic determining λ,, λ, will have imaginary roots of the form a +/-1. Ultimately therefore it will suffice to integrate one equation of the system (13) and then to deduce the solution of the other by changing √1 into -√-1. 3. Application of the above formula when the given surface is an oblate spheroid, such as the earth. Let the plane of the equator. be that of projection, the centre being the origin. Let the co-ordinates x, y pass through the meridians of 0 and of 90° respectively, and z through the poles. The equation of the surface will be where a is the earth's equatorial, b its polar radius. Let also the latitude of the point x, y, z be represented by s, the longitude by t. We have |