246 Mr. DE morgan, ON TRIPLE ALGEBRA. 



Substitute these values in the first of equations (M), and we have 



ifl+^.^+v = SL^^L^,, + cos e L^^ + cos ,i Lg^ + i sin sin ^. 



Assume Lg^ = ^(Pg^- cos9) which gives Pe+^,j,+v "^ Pe,iP^,^ the only solution of which 

 is /> =£''»+P*, giving 



-"e* - -jj-cos w + -gf , 



Bei, = ^cos0 + -^sin0-le'''+^*, 



C,^ = icos9---^sin0-le-''^^*. 



This gives Jg^ - B,^ - C«^ = e""-^^*, 



-"e.^) "e,* "^e,* ''■''9,,()-"8.*'-9,* ~ * 



We can now get solutions on the supposition that the other moduli are used. If we take 

 I = a — b — c, we have 



/cos 9 



«->-(T- 51) •-'••*"'-*• 



_ ^c^ sjn0\ „,,^^, 



But if we use /^(n' — 6' — c' — 3o6c), we have 



Jg^ = f cos . e-^<-'-^^« + leSM+P*), 



We must remember that, of any two solutions of {M), either must be the other multiplied by a 

 solution of Pg^^^^^ = Pg^ P^^; and any solution of {M) multiplied by one of the last is also a 

 solution of (M). And the form of the solutions might be generalized, but in appearance only, 

 by writing cos (a'6 + l3'(p) and sin (a 6 + /3'0) for cos 9 and sin 0. But by the same consideration 

 it appears that the system is not less complete if we write <p for a9 + fi(p- Adopting this simplifi- 

 cation, the equations of connexion between a &c. and I &c., are at full length as follows : 



I = y/' (a^+ b^+c^+ ah + ac - be), 



o = i {|cos0 + le*}, a + \{h + c) = I co%e, 



1 /s 



b = l |icos0 + — ^sin0-le*|, ^ (6 - c) = /sin 0, 



c = l jicose ^sin0- ^e*}, a - (b + c) = le*. 



From these premises it follows that the product of a + 6ij + cT and a'+ b'tj + c'^, or of \l, 9, ^] 

 and [I', ff, 0'] is \Jl', 9 + 9', <p + <p']. And it is certain that this is the only simple cubic system, 

 except that noted under case A, which as will afterwards be seen, is deceptive : also that this is the 



