Sir W. Rowan Hamilton on Qitaierniorts. 433 



and signifies, when interpreted on the same principles, that 



sin ««'. sin «"«'", cos (««' «"«'")= cos«a". cos «'«'" 



— cos aa'". cos «'«" ; (189.) 



where the spherical angle between the two arcs from a to a' 

 and from a" to a!" may be replaced by the interval between the 

 poles of the two positive rotations corresponding. The same 

 result may be otherwise stated as follows: If Z/, L', L", V", 

 denote any four points upon the surface of an unit-sphere, 

 and A the angle which the arcs LL', L"L"' form where they 

 meet each other, (the arcs which include this angle being mea- 

 sured in the directions of the progressions from L to Z/', and 

 from Z/" to Z<"' respectively,) then the following equation will 

 hold good : 



cos LZ". cos L'L"'- cos LUK cos LIL" 



=z sin LL'. sin L"L"'. cos A. . . . (190.) 



Accordingly this last equation has been incidentally given, as 

 an auxiliary theorem or lemma, at the commencement of those 

 profound and beautiful researches, entitled Disquisitio7ies 

 Generates circa Superficies Curvas^ which were published by 

 Gauss at Gottingen in 1828. That great mathematician and 

 philosopher was content to prove the last written equation by 

 the usual formulae of spherical and plane trigonometry ; but, 

 however simple and elegant may be the demonstration thereby 

 afforded, it appears to the present writer that something is 

 gained by our being able to present the result (190.) or (189.), 

 under the form (188.) or (179.), as an identity in the quater- 

 nion calculus. In general, all the results of plane and sphe- 

 rical trigonometry take the form of identities in this calculus ; 

 and their expressions, when so obtained, are associated with 

 a reference to vectors, which is usually suggestive of graphic 

 as well as metric relations. 

 78. Since 



p>j-ep = S.f(>j-d) + V.p(>!+fl), . ... (191.) 



the quaternion prj—Qp gives a pure vector as a product, or as 

 a quotient, if it be multiplied or divided by the vector ri + Q 

 (compare article 68) ; we may therefore write 



pn-Qp=:\,{ri + $), ..... (192.) 



Ai being a new vector-symbol, of which the value may be thus 

 expressed : 



A,=p-2(>j + 0-iS.fip (193.) 



Phil. Mag. S. 3. Vol. 34. No. 231. June 1849. 2 F 



