62 



The President stated that the remarkable researches re- 

 specting algebraic triplets, made lately by Professor De 

 Morgan and John T. Graves, Esq. in England, and here by 

 the Rev. Charles Graves, had led him to perceive the follow- 

 ing theorem : 



If the three symbols |, rj, Z, or rather their squares and 

 products, be supposed to satisfy the three following " equa- 

 tions of signification :" 



and if, by the help of these three equations, we eliminate any 

 three of the six quadratic combinations ^^, if, Zf, ^rj, r\Z„ t,^, 

 from the development of the " formula of multiplication," 



{ul -\-vn + wl) {x"l + y"^ -I- z"l) 

 - {xl + ^„ -I- zX,) (x% + y'^ -h z%), 



and then treat the three remaining combinations of the same 

 set (^^, &c.) as three entirely arbitrary and independent mul- 

 tipliers : the three separate equations thus obtained between 

 the fifteen real quantities abcuvwxy z x' y' z' x" y'' z" will be 

 such, that whether we project the four lines (m, v, w), (x, y, z), 

 (x', y', z'), (x", y", z"), on the axis {a, b, c,) itself, or on the plane 

 perpendicular to that axis, the four projections thus obtained 

 will in each case form a proportion ; the proportionality of 

 the projections on the axis being of the kind considered in 

 ordinary algebra, and the proportionality of the projections 

 on the plane perpendicular to the axis being of the kind con- 

 sidered by Mr. Warren ; that is to say, the lengths of these 

 last projections are proportionals in the usual sense, and the 

 rotation from the first to the second is equal to the rotation 

 from the third to the fourth. 



Sir W. Hamilton has been able to prove this theorem by 

 treating the three real equations between the fifteen rectan- 

 gular co-ordinates a, b, c, &c. according to the known methods 



