1914-15.] Spherical Harmonic Computation. 
63 
VI. — Formulae and Scheme of Calculation for the Development 
of a Function of Two Variables in Spherical Harmonics. 
By Professor T. Bauschinger, Strassburg. Translated and com- 
municated by The General Secretary, Dr C. G. Knott. 
(MS. received December 7, 1914. Read January 18, 1915.) 
(This paper was read at the Napier Tercentenary Celebration on July 27, 1914. The 
Council, on the suggestion of the Napier Committee, have much pleasure in making it 
accessible in the Society’s Proceedings.) 
When a function has been expressed as a series of spherical harmonics 
with constant coefficients, the determination of these coefficients from given 
values of the function is in the general case one of the most complicated 
operations which can be set before the calculator. 
Since Gauss first carried out these operations in a calculation of this 
kind,* * * § ' efibrts have not been wanting to simplify them and make their 
frequent application possible. The most successful of all in this respect 
was Franz Neumann,]- who showed that by a suitable choice of the 
argument the calculation could be materially shortened. 
For the application of Neumann’s method H. Seeligerj arranged 
the constant coefficients in tables, and thereby made the calculations 
so easy and so obvious that even a non-scientific calculator can carry 
it out. I would now show that some further steps may be taken 
in this direction, with the advantage that in addition to a further 
shortening of the calculation the whole process can be carried out by one 
operation on the calculating machine, since only sums of products have 
to be formed. 
[The given values § of the function /(/x , 0), where cos~V( = ^) is l^he 
polar distance and 0 the longitude, are supposed distributed over a spherical 
surface, such as the earth’s, and the function itself is expressed as 
/(/*,<#>)= 2 "'Y” 
o 
* Burckbardt, Oszillierende Funktionen, pp. 384 ff. 
t Astronomische Nachrichten, Bd. xvi, p. 313 (1838). 
I Silzungsberichte der Konig. layer. Ahademie der Wissenschaften Miinchen : Math.-phys. 
Classe, Band xx, p. 499 (1891). 
§ The part in square brackets has been added by the translator, so as to make the 
notations immediately intelligible to the reader. 
