4 One great series of crust waves from east to west is crossed by a ' " Areal and mittlere Erhebung der Landflachen sowie der Erdkruste " in Gerland's Beitriige zur Geophysik, ii.
The crystallines are confined to the portion of the belt east of the Great Valley where Paleozoic rocks are always highly metamorphosed and occur for the most part in limited patches, excepting in New England and Canada, where they assume greater areal importance, and are besides very generally intruded by granites.
The corresponding equations in areal co-ordinates are readily derived by substituting x/a, ylb, z/c for a, 1 3, y respectively in the trilinear equations.
The circumcircle is thus seen Areal to be a 2 yz+b 2 zx+c 2 xy=o, with centre sin 2A, sin 2B, co sin 2C; the inscribed circle is A t (x cot ZA)+ (y cot 2B) nates.
- N / (z cot IC) =o, with centre sin A, sin B, sin C; the escribed circle opposite the angle A is - N I (- x cot ZA)+ -1 (y tan 2B) + -V (z tan 2C) =o, with centre - sin A, sin B, sin C; and the selfconjugate circle is x 2 cot A+y 2 cot B+z 2 cot C =o, with centre tan A, tan B, tan C. Since in areal co-ordinates the line infinity is represented by the equation x+y+z=o it is seen that every circle is of the form a 2 yz+b 2 zx+c 2 xy+(lx+my+nz)(x+y+z) = o.
The various forms in areal co-ordinates may be derived from the above by substituting Xa for 1, µb for m and vc for n, or directly by expressing the condition for tangency of the line x+y+z = o to the conic expressed in areal coordinates.