4(3 Prof. A. H. Leahy. On certain Functions [May 24, 



-j w Hir .P(n,r,i>) cos rq j- 



cos ?w-7 . 2^ - ' v m r P (n,r,v) sin rq, 

 n + r I 



n 



' 



where P(n,m,u) is the "associated function" , , (l 



ct/t '" 



v are put for cos ft', cos ', 7 is the longitude of the second pole referred 

 to the original pole and plane, and u mr v mr are the functions of |3, the 

 angular distance between the poles whose properties are discussed in 

 the paper. When m is zero, i.e., when P(n,m,fi) is a zonal harmonic, 

 the function u mr reduces to P(n,r,fi), if ft is put for cos ft, and v mr is zero. 

 The general equations connecting the functions, and the values of 

 the functions for general and for particular values of TO and r are 

 investigated. 



If f>' and e' are the colatitude and longitude of a point referred to 

 the second pole and any plane through the pole, the integral of the 

 product of any two tesseral harmonics both of the nih order over the 

 surface of a sphere can be expressed concisely in terms of the functions 

 Umr, v mr . The result is 



(J P (n,m,/u!) cos (iwy'+a) . P (n,m,v) cos (re' + p) dS = 

 {cos (7717 + ) cos (rc + p) u mr (ft) sin (m/y + ) sin ( 



if 7 is the longitude of the second pole referred to the plane through 

 the first, e the longitude of the first pole referred to the plane 

 through the second, ft is the angular distance between the poles, and 

 , p are constants. 



The functions u mr , v mr are connected by several equations, bearing a 

 great resemblance to equations connecting tesseral harmonics of the 

 same order. They are of course functions of. n, and should be 

 written, when n may have different values, in the form u^m ,r,v,i >m ,r, but 

 the n is omitted for brevity in most of the results. Some of the most 

 important results are the following, the dashes denoting differential 

 coefficients 



!*",,. sin ft + u' mr cosft + {n(n 



= 2mrv mr cotft ...... (21); 



t,H-i + 2 u' mr (n + r) (nr + 1) mr _j = ...... (24) ; 



(nr + 1) u m>r -\ = 2mcosec(3v mr 



...... (27). 



