776 Proceedings of Royal Society of Edinburgh. [sess. 
7. We have, 
W 2 + 4 = V VM - - g-A) 2 ; or, 
W 2 = VtyM + iAA + /4*4 - m&lvl - ZgliAyA - • (10) 
The symmetry of this is obvious, for 
{^y - (/Vi + h v 2) 2 } {py - (wi - / x 2 v 2) 2 } = 
= { 1 - f4 - A - Vl - v\ + {fljV 2 ~ /* 2 iq) 2 } { 1 - (A - 14 - v\ - v\ + (lX Y V 2 + /^iq) 2 } ; 
1^3 ~ (v^ + V^) 2 } {AJ - (v lf X 2 - V^) 2 } = 
= { - 1 +/^i + /x 1 + v? + ^-(/a 1 v 2 + /a 2 v 1 ) 2 }{- 1 +^? + ^2 +v? + ^-(/x 1 i/ 2 -/x 2 v 1 ) 2 } 
When W vanishes, be it observed, equations (8) and (10) give us 
the well-known rectangular system. 
8. It will assist brevity in the calculations to define as fol- 
lows : — 
\ = So£S£cSy8; c 4 = SofSjBSSyc; e x = Sa£S/?yS8e ;\ 
\ = SacS^fSyS ; C 2 = SaeS/?yS8£ ; e 2 = SaeS/3SSy£ • I 
6 3 = SaSS/?ySe£; C 3 = Sa3S/3£Sy€ ; e 3 = SaSS£eSy£;V. (11) 
6 4 = SayS/?SSe£ ; C 4 = SayS/ScS S£ ; e 4 = SayS/?£SSe • i 
& 5 ==Sa/?SySSe£; C 5 = SajSSyeSSf ; e 5 = Sa/?Sy£SSe / 
Bi — Cj - ; 
Ci = Ci — ; 
Ex = 6 4 - 
C 1 > 
1 
ii 
<M 
M 
C 2 = e 2 — ^2 > 
B-2 “ ^2 
C 2 ’ 
= C 3 — e 3 > 
C 3 — e 3 - b s ‘ 
E 3^3 _ 
C 3 > 
B 4 = c 4 - e 4 j 
C 4 = c 4 - & 4 ; 
E 4 =6 4 - 
c 4 ; 
•^5 = e 5 — C 5 j 
C 5 = ^5 — ^5 ; 
E 5 = C 5- 
h\ 
so that 
0 = B 1 + C 1 + E 1 = B 2 + C 2 + E 2 = . . . — B 5 + C 5 + E 5 . (13) 
9. Then we have, 
Y = E 1 + C 2 + B 3 ; ] 
XfV= -B 2 -C 3 -E 4 ; X|V= -E 3 -B 1 -C 4 ; A|V= -C 4 - E 2 - B 4 ; ! 
^?Y=-E 5 -E 2 -E 3 ; ^-C^-Cs-C,; ^V=--B,-B,-B,; ^ (U) 
vfV= Ej + E 4 + E 6 ;4V- C 5 + C 2 + C 4 ; vfV = B 4 + B 5 + B 3 
