Mk. DE morgan, on triple algebra. 245 



in such manner as to satisfy 



Be^^.^^.= B,^A^,+ B^^A,^ - C,^C^, {M). 



Here Ag^ is a species of cosine of {Q, <p), and Bg^ and C,^ are two different species of sines. 

 The second sides of (i/) must admit the interchange of and fi, and also of d) and v- That tiiis 

 and all other conditions of self-consistence are satisfied, will appear as follows. We have 



■"8.1. — -^s.o-^o.v + ■"0.1.-^9,0 ~ C^9,0^0,>'' 



Again, Ag^.i^j,^y = "$+^^0^0.4,+^ + ■"o,i(.+v^9+n,o + -"9+/i,o-"o.if+i. 



= ("s.O -"m,0 + B^oAgj, — Cg„Ci^„) (Co.cf^O.i' + t,0,.'-"0.(, " "e.^^Hv) 



+ (■°«,o^n,o+ "iJ.,oCe,a + -^e,o-"M,o) (■"o,4>Co,.' + "o,vCo^+ A„^A„^^). 

 Develope these products, and the results will be seen to be identical with 



+ (Bii„A„, + B„^Ai,a - C^.oCo,..) (^9,0-^0,0 + C„j,Ag„ - Bg„B„^) 



+ (-"9,0 ^0.* + "0,4.^9,0 + -"9.0-"0,*) (^(i.C^O,!- + "o.v^ll.O + -"f.,0-^0,v)' 



which is •Sfl^.C'^.. + B^^Cg^ + Ag^A^^. 



The other equations may be treated in the same way. 



I am able to find the solutions of all three varieties of this system by means of that in 

 which the modulus is v'C"'^ + b' + c^ + ab + ac — be) ; in which case the equation answering to 

 sin^ 9 + cos' = 1 in common trigonometry is 



■^6,41 "f" "0,4> "*■ ^e,4t + -"8,4i-"8,0 + Ag^Cg^ — Jigj^Cg^ = 1. 



We have 



/ b + c\^ /b — n\' 

 a' + b^ + c^ + ab + ac - be = la+ J + 3 I ^1 . 



Assume Ag^ + ■L(Bg^+ Cg^) = cose, a(^8,<,-C9*) = -y-. 



Then we must have equations of the following form 



Ag^ = COS0+ Lg^, 



B =?^-/ 



'* a/3 •*' 



C ^ ""^ / 



