SLOCUM. — FINITE CONTINUOUS GROUPS. 93 



and 7; by 



X 



'\ = x\ + b.„ 



(15) 



J J. I *i fJ>, 



x'\:=.x'.e'^- + j{e''^- 1). 



Consequently the transformation Tf^ T„ is defined by 



X"i r= Xj + flo + b.., 



(16) , 



x"o = X. e"^ + *' + e''^- (e"' - 1) + r i^'' - ^)i 



«2 «2 



whence, if 7; T„ = T„ 



[e'^- (e"= - 1) + P (e^= - 1) ]= c^i («i, a^, i^, b,), 



ao + 6. . «! IN , ^1 /J, 



e"2 + ^2_ 1 *- a2^ ^-z 



(17) _ 



C2 = Oo + &2' = <^2 (^''ij 'J'2) ^1) ^2)' 



In this case there is but one system of functions ^. If, now, a.^ + 62 is 

 an even multiple of tt -\/— 1? <^2 is finite, but Cj is infinite ; that is, there 

 is no finite parameter c^ corresponding to tliis choice of the parameters 

 a and b. Consequently, if a^, -\- b^ ■= 2 k it \/— 1 =1= 0, T^ T^ cannot be 

 generated by an infinitesimal transformation of the group, and therefore 

 G2 is discontinuous.* 



Lie states that two groups having the same structure are (holohedri- 

 cally) isomorphic ; but the groups G.y and Gi are not properly isomor- 

 phic, except in the neighborhood of the identical transformation, since one 

 is continuous and the other discontinuous. Whence it appears that the 

 conception of isomorphism, as developed by Lie, requires modification. 



The parameter group of G^ is defined by the equations 



^ ^ a2 + a2-f2^-^V-l a, _ 1) ^ (^„, _ i)-j 

 (13 a) 



a 



2 



«2 + 02 + 2 Z: TT V— 1» 



For the group -1^, \(x\V\ — x<iP%)', which also has the structure 



(Xi, Xo) rr X^ , we have 



rt2 + h + ^ ^ -"V- 'l r t',«i / «, 1 \ , ^'1 / ^ 1 M — ^ / J i \ 



ci- ' TTb -r- [e -„-(.e 2- \) + jf{c »-!)] =<pAai,ao,b-^, h.^, 



g"2-|-"2 \ (1-2 O2 



C2 = Oo + 62 + ^'^w \/— 1; ^ <^2("i. %> ^i> h)' 



and. if r;^ + ft.j = 2 (2 k + 1) t /\/— 1, where /c is an integer, r^ is always infinite. 

 Consequently this group is also discontinuous. 



