98 



Prof. A. Schuster. 



[May 30, 



modification must he introduced, the summation theorem in that case 

 being 



p = n— 1 



h + J cos P~ cos tf 71 " + 2 cos (ppr/n) cos ' n) = 0, 

 P = i 



the first and second terms representing half the value of the product 

 for p = and p = n respectively. 



There is no corresponding summation theorem in the case of the 

 functions Q% , and the application of the method of least squares leads 

 to a series of normal equations, each of which contains all the other 

 coefficients. This has been one of the great practical difficulties in 

 obtaining an expression for the series of spherical harmonics for the 

 earth's magnetic potential. 



F. E. Neumann has tried to overcome the difficulty by calculating 

 coefficients a ly <u . . . a q in such a way that 



. P = i 



Here pi, /x 2 . . . p, q are the quantities for which the values of the 

 function to be represented are known. Neumann's process is equiva- 

 lent to attaching weights proportional to a p to the different obser- 

 vations, a proceeding against which theoretical objections might be 

 urged. 



2. The expansion in terms of a series of cosines and sines being so 

 much easier than the direct expansion in terms of a series of the 

 functions Qjf, I have endeavoured to obtain the latter series by means 

 of the former. 



It is well known that a function of an angle 6, which is confined 

 to the values lying between and ir 3 may be put either into the form 



« + «i cos 6 + o-2 cos 26+ . . . a p cos2>d +. . ., 

 or into the form 



hi sin d + l>2 sin 20+ . . . b p sin pd+ ... 



The reduction to the series of spherical harmonics is accomplished 

 by calculating and tabulating the coefficients in the series 



cos^e = xiQz +K hl q,Ui+ ■ ■ ■ KQ.' +,...., 



■in** = b^q: +k + iQUi+ ■ ■ ■ b:q: + . . . 



The choice between the cosine and sine series is open to us, but it 

 appears that great simplicity is gained by taking the former series 

 when o- is odd and the latter when or is even. For in that case the 

 coefficients A£ and B£ will all vanish, as long as n is smaller than p. 



