2 Mr. W. D. Niven. On certain Definite Integrals. [Ma}* 1, 



along the 40th parallel in North America between the Atlantic Ocean 

 and Salt Lake City. Magnetic observations have been made, with 

 more or less assiduity, at different places in the eastern States for 

 many years past ; but of the immense tract of country lying between 

 the Mississippi and the Pacific Ocean there is only a single determina- 

 tion of one of the three elements indicated on Sir Edward Sabine's 

 maps, viz., a determination of declination at Salt Lake City. A series 

 of observations was made some years since by United States' officers 

 along the Mexican frontier, and a similar series was carried out by 

 the English and American officers employed on the North American 

 Boundary Commission. The present set of observations was made, 

 therefore, along the district which lies midway between the line of 

 observations already run along the northern and southern boundaries 

 of the United States' territory. 



The instruments employed formed part of the admirable magnetic 

 equipment belonging to Owens College, Manchester, and were kindly 

 placed at the disposal of the author by the Council of the College. At 

 the commencement and end of the survey a complete series of observa- 

 tions was made at Kew, which was adopted as a base station ; and a 

 similar set was made at Washington in the magnetic observatory be- 

 longing to Mr. Charles A. Schott, of the United States' Coast Survey 

 Department, with a view of obtaining an independent check on the 

 indications of the instruments. The author learned from Mr. Schott 

 that the Coast Survey Department is gradually accumulating data for a 

 discussion of the magnetic history of the American continent. It has 

 already published a map of declination for 1870, from which it is 

 evident that much has been done in the determination of this par- 

 ticular element since the date of Sir E. Sabine's memoirs, more 

 especially along the Pacific slope, and in the regions to the west and 

 south-west of the Great Salt Lake. As yet, however, no observations 

 have been published relating to any of the districts west of the Missis- 

 sippi visited by the author. 



II. " On certain Definite Integrals occurring in Spherical Har- 

 monic Analysis, and on the Expansion in Series of the 

 Potentials of the Ellipsoid and of the Ellipse." By W. D. 

 NiVEN, M.A., Fellow of Trinity College, Cambridge. Com- 

 municated by J. W. L. Glaisher, M.A., F.R.S. Received 

 April 3, 1879.* 



(Abstract.) 



1. The object of this paper is to explain a general method of cal- 

 culating a class 0£ integrals connected with the expansion of functions 

 in spherical harmonic series. 



