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II. 



INTEGRALS DEPENDING ON A SINGLE QTJATEENION 

 YARIABLE. By CHAELES JASPER JOLT, M.A., D. Sc, 

 F.T.C.D., Royal Astronomer of Ireland, and Andrews' Professor 

 of Astronomy in the University of Dublin. 



[Read April 28, 1902.] 



TABLE OF CONTENTS. 



AbTj Page. 



1. — Explanation of Hamilton's 

 method for single Quaternion 

 Integrals. Mode of passage, 7 



2. — Difference of integrals be- 

 tween fixed limits according 

 to different modes of pas- 

 sage expressed as a double 

 integral, .... 8 



3. — Conditions for independence 



of mode of passage, . . 9 



4. — Case in which the variable is 



a vector. Stokes's Theorem, 10 



5. — Double integrals with a single 



quaternion variable, . .10 



6. — Variation of double integral 

 corresponding to variation 

 in mode of passage, . .11 



7. — Difference of double integrals 

 for different modes expressed 

 as a triple integral. Con- 

 ditions for independence of 

 mode of passage, . .12 



8. — Triple integrals in a single 



quaternion variable, . .13 



Art. Page. 



9. — Integrals of higher order, . 14 



10. — Specification of the modes of 

 passage. Time and space 

 method, . . . .14 



11. — Hydrodynamical illustration 

 for single quaternion in- 

 tegral, . . . .15 



12. — Physical illustration of mean- 

 ing of quaternion double 

 integral, . . . .16 



13. — Electro-magnetic equations as 

 conditions of independence 

 of mode of passage for a 

 double integral, . . .17 



14. — Double integral expressed as 

 difference of two single in- 

 tegrals. Vector potential of 

 magnetic current. Quater- 

 nion potential, . . .17 



15. — Case in which the integral 

 depends on the mode of pas- 

 sage. Conducting dielectric, 18 



16. — Physical illustration for triple 



integral, . . . .19- 



iNTEODTTCTIOlf. 



In the '< Lectures on Quaternions," Hamilton devotes a brief series 

 of Articles (625-630) to the investigation of quaternion integrals. 

 It does not seem to have been observed that bis results lead directly to 

 the fundamental theorems of Green and Stokes and to the extensions 



