OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
359 
21 . Accordingly, we can take as an algebraically complete aggregate, containing 
the 31 necessary members, the set which follows :— 
(i.) V, 
(ii.) 
(iii.) J (?ri, nr,) = — Fi/;io) f/)oi — — Cfi/zio) 
(iv.) w., 
(v.) H (h'o) = eg - 
(vi.) wA, 
(vii.) J {w.,, id. 2 ) = (EM — EL) — (EN — GL) + (FN — GM) 
(viii.) H (wA) = LN - or I (an, = EN - 2 FM + GL, 
(ix.) a A, 
(x.) J (?(n, aA) = {Eh — Frt) + ..., 
(xi.) H (icA) = ac — /r, or f (?fn, a/'.,) = Ec — 2 Fh + Ga, 
(xii.) id",, 
(xiii.) J (un, a/",) == (E// - FA) ^ 01 ^ + ■ • 
(xiv.) H {id",) = a!d - h'\ or I {io.„ w"d) = Ed - 2 F// + GA, 
(XV.) W 3 , 
(xvi.) F[ (^Cg) = (PR - Q 3 ) 
(xvii.) dJ {iv^) = (P-S — 3PQR + 2 Q®) + • • , or A {in^), 
(xviii.) J {iv.„ an) = (EQ — FP) An'" + • • •, 
(xix.) Ag, 
(xx.) H (a/g) = {km - P) + . . ., 
(xxi.) (R {idn) = {k-n — Slclm + -Jd) A)i® + • • •, or A (''ab), 
(xxii.) J {w,, Ag) = (E/ — F^■) An^ + • • •, 
(xxiii.) id'g, 
(xxiv. ) H(rAg)z=(P7A-A)(/>or + ..., 
(xxv.) <t> (io"g) = {k'~n' — SJc'l'm' + 2A) (^ 01 '" + • • •> or A (a/'g), 
(xxvi.) I {iv.„ w"s) = {El' - FP) An® + • • •, 
(xxvii.) 7 ^ 4 , 
(xxviii.) H ( 7 ^ 2 ) = {ay - /3') Ajf + • • 
(xxix.) I {w.,} = ae — 4j3S + 3y~, 
(xxx.) J (a;^) = aye + 2/8yS — aS^ — j3'^e — y'b 
(xxxi.) J {iv.„ W.J — {E/3 — Fa) (fyQ^^ + ... 
22. It is still necessary to satisfy equation (Ii)^b This will be effected as follows ; 
