In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions 'X=' by+ **cz**, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2.

In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions 'X=' by+ **cz**, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2.

If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and **cz**), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.

The complete covariant and contravariant system includes no fewer than 34 forms; from its complexity it is desirable to consider the cubic in a simple canonical form; that chosen by Cayley was ax 3 +by 3 + **cz** 3 + 6dxyz (Amer.

C or **cz** is pronounced as English ts; cs as English ch; ds as English j; zs as French j; gy as dy.

BOULEUTERIOry **CZ**: By permission from plans by F.de Billi.

These often agree with **CZ** against M, and the readings of **CZZ** are generally superior.

On OCi, 0C2 take ~ -**Cz** lengths 0A1, OAf, respectively pro- 0

Of connection cuts off from any two lines drawn from a given **CZ** A1