660 MOORE. 



These give the values «i4 = =^ iciu- Hence the invariant planes are 



(16) (A-13 + h-n) ± i{ku - hs), (/u3 - hi) ± /(/m4 + hs). 

 Now if mi = ± 1112 



(17) ^1 = m{~[{ki2k-12 + hihd =^ (A-13A-42 + A-42A-13 + A-14A-23 + /^*23/i'l4)].' 



Hence the complex 



(18) P = Ch^kvi + «13(^'l3 ^ A"42) + rtl4(/'-"l4 =^ A-23) + 034^-34. 



(Both positive or both negative signs are to be taken together), is left 

 invariant for all values of an, Oi3, «i4, 034- The planes which belong 

 to this system of complexes are determined by the relation 



PxP = «iofl34 — 013- — a\i~ = 0. 

 Substituting in (18) we have for the invariant planes 



A'l = (ffi3- + au-)kn + rtl3a34(A;i3 =^ ^'42) + ai4a34(^'l4 =^ ^'23) + «34-^'34 



The plane 



Kl = a34'^'l2 — «13«34(A-13 =^ ^'42) " ai4ai3(^14 =^ ^'23) + (ai3'" + «14-)^'34 



also belongs to (18) and is completely perpendicular to K\. Hence 

 the planes left invariant by the transformation P' = P-^i form a two 

 parameter family and are completely perpendicidar in pairs. The 

 planes belong to a three parameter linear system of complexes and 

 so must cut two fixed planes. It is not difficult to see that these are 

 the planes (16). The first pair when /»i = mo and the second pair 

 when ///i = — »/2- We can now write 



3/ = mi{h2 + A-34) = . , '''\ , (A': + A'2). 



«34- + «i3 + au~ 



Hence the theorem may be stated: Any complex 2-vector can he 

 resolved into the sum of -iwo completely perpendicidar planes. If 

 M* 7^ ±iV the resolution is unique. If M* = ^M the resolution 

 can he made in 00 ^ ways. 



3. Complex 2-vectors in 5-space. In o-space a complex 2-vector 

 M can be resolved into the sum of two simple plane vectors. For, 



