ON stokes' theorem. 301 



That is to say, Stokes' theorem, which is true as long as u, r, tv, 

 are single-valued nnd continuou.t functions all over the surface. 

 This is the necessary and sufficient ci>ndition. 



Now, coming back to our function — 



Xrfr + Ydij + Zdz 

 sujjposed integrable, and assuming that X, Y, and Z are single- 

 vahied and uniform within a certain region of space, let us con- 

 sider within the same region a closed curve. 



Within that region a surface having the curve as bounding 

 edge can generally be constructed, and, for that surface, Stokes' 

 theorem being applicable gives (taking into account the conditions 

 of Integra bility) — 



J\Xdx + Ydy + Zdz) = o 

 the integral being taken along the closed curve. 



If the region of space considered is like the inside of a sphere, 

 or like the body bounded by the two sheets of a wave surface 

 (a cyclic region), a surface can always be constructed enclosed by 

 the region, and having as bounding edge any closed curve within 

 that region, so that always in that case 



fX dx + Y di/ + Zdzzzz o 



along any closed curve within the region considered; but if the 

 region is like a ring, that is to say, with hole or holes piercing 

 through (cyclic region), such a surface cannot always be constructed 

 for any curve whatever. These curves, which enclose one ,^^or 

 several holes, are excepted (irreconcilable curves). Stokes' 

 theorem is no more applicable for them, and therefore for them 

 /X dx + Y dy + Zdz:^: o. 

 Remark. — I have not thought necessary to examine in detail the 

 several demonstrations given of Stokes' theorem, amongst which 

 stand pre-eminent the following, viz. : — Clerk Maxwell (A Treatise 

 on Electricity and Magnetism, th. lY., of preliminary chapter) : A 

 demonstration by means of curvilinear co-ordinates. ^Thomson 

 and Tait ( Natural Philosophy, § 1 90 fjj). ^Tait— On Green's and 

 Other Allied Theorems (Trans. K.S., p]din., 1872, p. 69) : This 

 demonstration by means of a network is certainly the best of all. 

 Minchin (Statics, vol. 2) *■. A simple comparison with the demon- 

 stration I have given will easily show Avhat I intend to criticise. 



11.— FROM NUMBER TO QUATERNIONS. 



By G. FLEURI, Licoicie cs-sciences 3Iathi'»uifiqiies and Licoicie cs-scienees 

 Fhysiqiies. 



•Miiichin's ileiaonsUatiims luo ri'markablt- for their inaccuracy. In the theoiems 1,2, 

 and 3, passes 241-2-15, he does not state a single time what conditions must be fulfilled by his 

 functions ^ and -i^ and u, v, w. 



