Sir W. Rowan Hamilton on Quaternions. 



will doubtless be procured by the further action of reducing 

 agents upon nitraniline. We shall shortly ascertain this. 



C ' 2 {n6 } N + 6HS = 2C 6 H 4 N + 4HO + 6S. 



V r— 1 , , 1 



Nitraniline. Semianiline. 



The small quantity of resinous matter which remains be- 

 hind upon dissolving crude nitraniline in boiling water is 

 probably Zinin's semianiline. 



XLIX. On Quaternions ; or on a New System of Imaginaries in. 

 Algebra. By Professor Sir William Rowan Hamilton, 

 LL.D., Correspondhig Member of the Institute of France, 

 and Royal Astronomer of Ireland. 



[Continued from p. 122.] 

 28. 'THIE known and purely graphic property of the cone 

 ■■ of the second degree which constitutes the theorem 

 of Pascal, and which expresses the coplanarity of the three 

 lines of meeting of opposite plane faces of an inscribed hexa- 

 hedral angle, may be transformed into another known but 

 purely metric property of the same cone of the second degree, 

 which is a form of the theorem of M. Chasles, respecting the 

 constancy of an anharmonic ratio. This transformation may 

 be effected without difficulty, on the plan of the present paper; 

 for if we multiply into v. yy' both members of the equation 

 (3.) of the 24th article, and then operate by the characteristic 

 s., attending to the general properties of scalars of products, 

 we find, for any six vectors a a' /3/3' yy\ the formula 

 S (v. aa'. V. /3/3'. V. y/) = a. ayy'. S. a' j8/3' — S. alyy'. S. a/3/3'; (1.) 

 which gives, for any^w vectors a a! a" yy 1 , this other : 



S (V. a a'. V. a' a". V. y y') = S. a a! a", s.ya'y' . . (2.) 

 If, then, we take six arbitrary vectors a ex! a" a 1 " a lv a v , and de- 

 duce nine other vectors from them by the expressions 



u = y.au l , a l = v. a! a", a 2 = V. «"«'", 



a 3 =V. «"'a lv ,«4 = V.« lv a v ,« 5 = V.a v a, V . . . (3.) 



/3 = v. «o « 3 , /3' = v. «! a 4 , /3" = v. « 3 « 5 ; J 



we shall have, generally, 



S./3/3'/3" = S.a « 2 a 6 . S.a 3 « x « 4 — S.a 3 a 2 a 5 . S.aQaja^ 

 r=s.« a 1 a 4 .S.« 2 a 3 a 5 — S.a 3 a 4 aj .s.a 5 « a 2 | 

 = S.aa'a". S.« lv a'a v . S.a"a'"a lv . S.aV'a I u \ 



— S.a'"a lv « v . S.a'a w a". S.a v aa'. S.a'W 

 ssS.aa'a". S. a"a'"a lv . S.aa'"a v . s.a v a'a lv 

 -S.a«'"a". S.«"«'« 1V . S.a«'a v . S.« v «'"a lv . 



